Artin's Axioms

Introduction

In this chapter we discuss Artin’s axioms for the representability of functors by algebraic spaces. As references we suggest the papers [ArtinI], [ArtinII], [ArtinVersal].

Some of the notation, conventions, and terminology in this chapter are awkward and may seem backwards to the more experienced reader. This is intentional. Please see Quot, Section 05X6 for an explanation.

Let \(S\) be a locally Noetherian base scheme. Let \[p : \mathcal{X} \longrightarrow (\Sch/S)_{fppf}\] be a category fibred in groupoids. Let \(x_0\) be an object of \(\mathcal{X}\) over a field \(k\) of finite type over \(S\). Throughout this chapter an important role is played by the predeformation category (see Formal Deformation Theory, Definition 06GS) \[\mathcal{F}_{\mathcal{X}, k, x_0} \longrightarrow \{\text{Artinian local }S\text{-algebras with residue field }k\}\] associated to \(x_0\) over \(k\). We introduce the Rim-Schlessinger condition (RS) for \(\mathcal{X}\) and show it guarantees that \(\mathcal{F}_{\mathcal{X}, k, x_0}\) is a deformation category, i.e., \(\mathcal{F}_{\mathcal{X}, k, x_0}\) satisfies (RS) itself. We discuss how \(\mathcal{F}_{\mathcal{X}, k, x_0}\) changes if one replaces \(k\) by a finite extension and we discuss tangent spaces.

Next, we discuss formal objects \(\xi = (\xi_n)\) of \(\mathcal{X}\) which are inverse systems of objects lying over the quotients \(R/\mathfrak m^n\) where \(R\) is a Noetherian complete local \(S\)-algebra whose residue field is of finite type over \(S\). This is the same thing as having a formal object in \(\mathcal{F}_{\mathcal{X}, k, x_0}\) for some \(x_0\) and \(k\). A formal object is called effective when there is an object of \(\mathcal{X}\) over \(R\) which gives rise to the inverse system. A formal object of \(\mathcal{X}\) is called versal if it gives rise to a versal formal object of \(\mathcal{F}_{\mathcal{X}, k, x_0}\). Finally, given a finite type \(S\)-scheme \(U\), an object \(x\) of \(\mathcal{X}\) over \(U\), and a closed point \(u_0 \in U\) we say \(x\) is versal at \(u_0\) if the induced formal object over the complete local ring \(\mathcal{O}_{U, u_0}^\wedge\) is versal.

Having worked through this material we can state Artin’s celebrated theorem: our \(\mathcal{X}\) is an algebraic stack if the following are true

1. \(\mathcal{O}_{S, s}\) is a G-ring for all \(s \in S\)¹,

2. \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) is representable by algebraic spaces,

3. \(\mathcal{X}\) is a stack for the étale topology,

4. \(\mathcal{X}\) is limit preserving,

5. \(\mathcal{X}\) satisfies (RS),

6. tangent spaces and spaces of infinitesimal automorphisms of the deformation categories \(\mathcal{F}_{\mathcal{X}, k, x_0}\) are finite dimensional,

7. formal objects are effective,

8. \(\mathcal{X}\) satisfies openness of versality.

This is Lemma 07Y4; see also Proposition 07Y5 for a slight improvement. There is an analogous proposition characterizing which functors \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) are algebraic spaces, see Section 07Y0.

Here is a rough outline of the proof of Artin’s theorem. First we show that there are plenty of versal formal objects using (RS) and the finite dimensionality of tangent and aut spaces, see for example Formal Deformation Theory, Lemma 06L2. These formal objects are effective by assumption. Effective formal objects can be “approximated” by objects \(x\) over finite type \(S\)-schemes \(U\), see Lemma 07XB. This approximation uses that the local rings of \(S\) are G-rings and that \(\mathcal{X}\) is limit preserving; it is perhaps the most difficult part of the proof relying as it does on general Néron desingularization to approximate formal solutions of algebraic equations over a Noetherian local G-ring by solutions in the henselization. Next openness of versality implies we may (after shrinking \(U\)) assume \(x\) is versal at every closed point of \(U\). Having done all of this we show that \(U \to \mathcal{X}\) is a smooth morphism. Taking sufficiently many \(U \to \mathcal{X}\) we show that we obtain a “smooth atlas” for \(\mathcal{X}\) which shows that \(\mathcal{X}\) is an algebraic stack.

In checking Artin’s axioms for a given category \(\mathcal{X}\) fibred in groupoids, the most difficult step is often to verify openness of versality. For the discussion that follows, assume that \(\mathcal{X}/S\) already satisfies the other conditions listed above. In this chapter we offer two methods that will allow the reader to prove \(\mathcal{X}\) satisfies openness of versality:

1. The first is to assume a stronger Rim-Schlessinger condition, called (RS*) and to assume a stronger version of formal effectiveness, essentially requiring objects over inverse systems of thickenings to be effective. It turns out that under these assumptions, openness of versality comes for free, see Lemma 0CXU. Please observe that here we are using in an essential manner that \(\mathcal{X}\) is defined on that category of all schemes over \(S\), not just the category of Noetherian schemes!

2. The second, following Artin, is to require \(\mathcal{X}\) to come equipped with an obstruction theory. If said obstruction theory “commutes with products” in a suitable sense, then \(\mathcal{X}\) satisfies openness of versality, see Lemma 0CYF.

Obstruction theories can be axiomatized in many different ways and indeed many variants (often adapted to specific moduli stacks) can be found in the literature. We explain a variant using the derived category (which often arises naturally from deformation theory computations done in the literature) in Lemma 07YZ.

In Section 0GE6 we discuss what needs to be modified to make things work for functors defined on the category \((\textit{Noetherian}/S)_\etale\) of locally Noetherian schemes over \(S\).

In the final section of this chapter as an application of Artin’s axioms we prove Artin’s theorem on the existence of contractions, see Section 0GH7. The theorem says roughly that given an algebraic space \(X'\) separated and of finite type over \(S\), a closed subset \(T' \subset |X'|\), and a formal modification \[\mathfrak{f} : X'_{/T'} \longrightarrow \mathfrak{X}\] where \(\mathfrak{X}\) is a Noetherian formal algebraic space over \(S\), there exists a proper morphism \(f : X' \to X\) which “realizes the contraction”. By this we mean that there exists an identification \(\mathfrak{X} = X_{/T}\) such that \(\mathfrak{f} = f_{/T'} : X'_{/T'} \to X_{/T}\) where \(T = f(T')\) and moreover \(f\) is an isomorphism over \(X \setminus T\). The proof proceeds by defining a functor \(F\) on the category of locally Noetherian schemes over \(S\) and proving Artin’s axioms for \(F\). Amusingly, in this application of Artin’s axioms, openness of versality is not the hardest thing to prove; instead the proof that \(F\) is limit preserving requires a lot of work and preliminary results.

Conventions

The conventions we use in this chapter are the same as those in the chapter on algebraic stacks, see Algebraic Stacks, Section 026M. In this chapter the base scheme \(S\) will often be locally Noetherian (although we will always reiterate this condition when stating results).

Predeformation categories

Let \(S\) be a locally Noetherian base scheme. Let \[p : \mathcal{X} \longrightarrow (\Sch/S)_{fppf}\] be a category fibred in groupoids. Let \(k\) be a field and let \(\Spec(k) \to S\) be a morphism of finite type (see Morphisms, Lemma 01TA). We will sometimes simply say that \(k\) is a field of finite type over \(S\). Let \(x_0\) be an object of \(\mathcal{X}\) lying over \(\Spec(k)\). Given \(S\), \(\mathcal{X}\), \(k\), and \(x_0\) we will construct a predeformation category, as defined in Formal Deformation Theory, Definition 06GS. The construction will resemble the construction of Formal Deformation Theory, Remark 06GU.

First, by Morphisms, Lemma 01TA we may pick an affine open \(\Spec(\Lambda) \subset S\) such that \(\Spec(k) \to S\) factors through \(\Spec(\Lambda)\) and the associated ring map \(\Lambda \to k\) is finite. This provides us with the category \(\mathcal{C}_\Lambda\), see Formal Deformation Theory, Definition 06GC. The category \(\mathcal{C}_\Lambda\), up to canonical equivalence, does not depend on the choice of the affine open \(\Spec(\Lambda)\) of \(S\). Namely, \(\mathcal{C}_\Lambda\) is equivalent to the opposite of the category of factorizations [07T3]\[\begin{equation} \Spec(k) \to \Spec(A) \to S \end{equation}\] of the structure morphism such that \(A\) is an Artinian local ring and such that \(\Spec(k) \to \Spec(A)\) corresponds to a ring map \(A \to k\) which identifies \(k\) with the residue field of \(A\).

We let \(\mathcal{F} = \mathcal{F}_{\mathcal{X}, k, x_0}\) be the category whose

1. objects are morphisms \(x_0 \to x\) of \(\mathcal{X}\) where \(p(x) = \Spec(A)\) with \(A\) an Artinian local ring and \(p(x_0) \to p(x) \to S\) a factorization as in (07T3), and

2. morphisms \((x_0 \to x) \to (x_0 \to x')\) are commutative diagrams \[\xymatrix{ x & & x' \ar[ll] \\ & x_0 \ar[lu] \ar[ru] }\] in \(\mathcal{X}\). (Note the reversal of arrows.)

If \(x_0 \to x\) is an object of \(\mathcal{F}\) then writing \(p(x) = \Spec(A)\) we obtain an object \(A\) of \(\mathcal{C}_\Lambda\). We often say that \(x_0 \to x\) or \(x\) lies over \(A\). A morphism of \(\mathcal{F}\) between objects \(x_0 \to x\) lying over \(A\) and \(x_0 \to x'\) lying over \(A'\) corresponds to a morphism \(x' \to x\) of \(\mathcal{X}\), hence a morphism \(p(x' \to x) : \Spec(A') \to \Spec(A)\) which in turn corresponds to a ring map \(A \to A'\). As \(\mathcal{X}\) is a category over the category of schemes over \(S\) we see that \(A \to A'\) is a \(\Lambda\)-algebra homomorphism. Thus we obtain a functor [07T4]\[\begin{equation} p : \mathcal{F} = \mathcal{F}_{\mathcal{X}, k, x_0} \longrightarrow \mathcal{C}_\Lambda. \end{equation}\] We will use the notation \(\mathcal{F}(A)\) to denote the fibre category over an object \(A\) of \(\mathcal{C}_\Lambda\). An object of \(\mathcal{F}(A)\) is simply a morphism \(x_0 \to x\) of \(\mathcal{X}\) such that \(x\) lies over \(\Spec(A)\) and \(x_0 \to x\) lies over \(\Spec(k) \to \Spec(A)\).

Lemma

The functor \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) defined above is a predeformation category.

Proof

We have to show that \(\mathcal{F}\) is (a) cofibred in groupoids over \(\mathcal{C}_\Lambda\) and (b) that \(\mathcal{F}(k)\) is a category equivalent to a category with a single object and a single morphism.

Proof of (a). The fibre categories of \(\mathcal{F}\) over \(\mathcal{C}_\Lambda\) are groupoids as the fibre categories of \(\mathcal{X}\) are groupoids. Let \(A \to A'\) be a morphism of \(\mathcal{C}_\Lambda\) and let \(x_0 \to x\) be an object of \(\mathcal{F}(A)\). Because \(\mathcal{X}\) is fibred in groupoids, we can find a morphism \(x' \to x\) lying over \(\Spec(A') \to \Spec(A)\). Since the composition \(A \to A' \to k\) is equal to the given map \(A \to k\) we see (by uniqueness of pullbacks up to isomorphism) that the pullback via \(\Spec(k) \to \Spec(A')\) of \(x'\) is \(x_0\), i.e., that there exists a morphism \(x_0 \to x'\) lying over \(\Spec(k) \to \Spec(A')\) compatible with \(x_0 \to x\) and \(x' \to x\). This proves that \(\mathcal{F}\) has pushforwards. We conclude by (the dual of) Categories, Lemma 003V.

Proof of (b). If \(A = k\), then \(\Spec(k) = \Spec(A)\) and since \(\mathcal{X}\) is fibred in groupoids over \((\Sch/S)_{fppf}\) we see that given any object \(x_0 \to x\) in \(\mathcal{F}(k)\) the morphism \(x_0 \to x\) is an isomorphism. Hence every object of \(\mathcal{F}(k)\) is isomorphic to \(x_0 \to x_0\). Clearly the only self morphism of \(x_0 \to x_0\) in \(\mathcal{F}\) is the identity.

Let \(S\) be a locally Noetherian base scheme. Let \(F : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism between categories fibred in groupoids over \((\Sch/S)_{fppf}\). Let \(k\) be a field of finite type over \(S\). Let \(x_0\) be an object of \(\mathcal{X}\) lying over \(\Spec(k)\). Set \(y_0 = F(x_0)\) which is an object of \(\mathcal{Y}\) lying over \(\Spec(k)\). Then \(F\) induces a functor [07WJ]\[\begin{equation} F : \mathcal{F}_{\mathcal{X}, k, x_0} \longrightarrow \mathcal{F}_{\mathcal{Y}, k, y_0} \end{equation}\] of categories cofibred over \(\mathcal{C}_\Lambda\). Namely, to the object \(x_0 \to x\) of \(\mathcal{F}_{\mathcal{X}, k, x_0}(A)\) we associate the object \(F(x_0) \to F(x)\) of \(\mathcal{F}_{\mathcal{Y}, k, y_0}(A)\).

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(F : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). Assume either

1. \(F\) is formally smooth on objects (Criteria for Representability, Section 06CZ),

2. \(F\) is representable by algebraic spaces and formally smooth, or

3. \(F\) is representable by algebraic spaces and smooth.

Then for every finite type field \(k\) over \(S\) and object \(x_0\) of \(\mathcal{X}\) over \(k\) the functor (07WJ) is smooth in the sense of Formal Deformation Theory, Definition 06HG.

Proof

Case (1) is a matter of unwinding the definitions. Assumption (2) implies (1) by Criteria for Representability, Lemma 06D3. Assumption (3) implies (2) by More on Morphisms of Spaces, Lemma 04AM and the principle of Algebraic Stacks, Lemma 045F.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \[\xymatrix{ \mathcal{W} \ar[d] \ar[r] & \mathcal{Z} \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{Y} }\] be a \(2\)-fibre product of categories fibred in groupoids over \((\Sch/S)_{fppf}\). Let \(k\) be a finite type field over \(S\) and \(w_0\) an object of \(\mathcal{W}\) over \(k\). Let \(x_0, z_0, y_0\) be the images of \(w_0\) under the morphisms in the diagram. Then \[\xymatrix{ \mathcal{F}_{\mathcal{W}, k, w_0} \ar[d] \ar[r] & \mathcal{F}_{\mathcal{Z}, k, z_0} \ar[d] \\ \mathcal{F}_{\mathcal{X}, k, x_0} \ar[r] & \mathcal{F}_{\mathcal{Y}, k, y_0} }\] is a fibre product of predeformation categories.

Proof

This is a matter of unwinding the definitions. Details omitted.

Pushouts and stacks

In this section we show that algebraic stacks behave well with respect to certain pushouts. The results in this section hold over any base scheme.

The following lemma is also correct when \(Y\), \(X'\), \(X\), \(Y'\) are algebraic spaces.

Lemma

Let \(S\) be a scheme. Let \[\xymatrix{ X \ar[r] \ar[d] & X' \ar[d] \\ Y \ar[r] & Y' }\] be a pushout in the category of schemes over \(S\) where \(X \to X'\) is a thickening and \(X \to Y\) is affine, see More on Morphisms, Lemma 07RT. Let \(\mathcal{Z}\) be an algebraic stack over \(S\). Then the functor of fibre categories \[\mathcal{Z}_{Y'} \longrightarrow \mathcal{Z}_Y \times_{\mathcal{Z}_X} \mathcal{Z}_{X'}\] is an equivalence of categories.

Proof

Let \(y'_1, y'_2\) be objects of the left hand side. The sheaf \(\mathit{Isom}(y'_1, y'_2)\) on the category of schemes over \(Y'\) is representable by an algebraic space \(I\) over \(Y'\), see Algebraic Stacks, Lemma 045G. We conclude that the functor of the lemma is fully faithful as \(Y'\) is the pushout in the category of algebraic spaces as well as the category of schemes, see Pushouts of Spaces, Lemma 07SY.

Let \((y, x', f)\) be an object of the right hand side. Here \(f : y|_X \to x'|_X\) is an isomorphism. To finish the proof we have to construct an object \(y'\) of \(\mathcal{Z}_{Y'}\) whose restrictions to \(Y\) and \(X'\) agree with \(y\) and \(x'\) in a manner compatible with \(f\). In fact, it suffices to construct \(y'\) fppf locally on \(Y'\), see Stacks, Lemma 046N. Choose a representable algebraic stack \(\mathcal{W}\) and a surjective smooth morphism \(\mathcal{W} \to \mathcal{Z}\). Then \[(\Sch/Y)_{fppf} \times_{y, \mathcal{Z}} \mathcal{W} \quad\text{and}\quad (\Sch/X')_{fppf} \times_{x', \mathcal{Z}} \mathcal{W}\] are algebraic stacks representable by algebraic spaces \(V\) and \(U'\) smooth over \(Y\) and \(X'\). The isomorphism \(f\) induces an isomorphism \(\varphi : V \times_Y X \to U' \times_{X'} X\) over \(X\). By Pushouts of Spaces, Lemmas 07VX and 07W3 we see that the pushout \(V' = V \amalg_{V \times_Y X} U'\) is an algebraic space smooth over \(Y'\) whose base change to \(Y\) and \(X'\) recovers \(V\) and \(U'\) in a manner compatible with \(\varphi\).

Let \(W\) be the algebraic space representing \(\mathcal{W}\). The projections \(V \to W\) and \(U' \to W\) agree as morphisms over \(V \times_Y X \cong U' \times_{X'} X\) hence the universal property of the pushout determines a morphism of algebraic spaces \(V' \to W\). Choose a scheme \(Y_1'\) and a surjective étale morphism \(Y_1' \to V'\). Set \(Y_1 = Y \times_{Y'} Y_1'\), \(X_1' = X' \times_{Y'} Y_1'\), \(X_1 = X \times_{Y'} Y_1'\). The composition \[(\Sch/Y_1') \to (\Sch/V') \to (\Sch/W) = \mathcal{W} \to \mathcal{Z}\] corresponds by the \(2\)-Yoneda lemma to an object \(y_1'\) of \(\mathcal{Z}\) over \(Y_1'\) whose restriction to \(Y_1\) and \(X_1'\) agrees with \(y|_{Y_1}\) and \(x'|_{X_1'}\) in a manner compatible with \(f|_{X_1}\). Thus we have constructed our desired object smooth locally over \(Y'\) and we win.

The Rim-Schlessinger condition

The motivation for the following definition comes from Lemma 07WN and Formal Deformation Theory, Definition 06J2 and Lemma 06J5.

Definition

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{Z}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). We say \(\mathcal{Z}\) satisfies condition (RS) if for every pushout \[\xymatrix{ X \ar[r] \ar[d] & X' \ar[d] \\ Y \ar[r] & Y' = Y \amalg_X X' }\] in the category of schemes over \(S\) where

1. \(X\), \(X'\), \(Y\), \(Y'\) are spectra of local Artinian rings,

2. \(X\), \(X'\), \(Y\), \(Y'\) are of finite type over \(S\), and

3. \(X \to X'\) (and hence \(Y \to Y'\)) is a closed immersion

the functor of fibre categories \[\mathcal{Z}_{Y'} \longrightarrow \mathcal{Z}_Y \times_{\mathcal{Z}_X} \mathcal{Z}_{X'}\] is an equivalence of categories.

If \(A\) is an Artinian local ring with residue field \(k\), then any morphism \(\Spec(A) \to S\) is affine and of finite type if and only if the induced morphism \(\Spec(k) \to S\) is of finite type, see Morphisms, Lemmas 01SI and 02HV.

Lemma

Let \(\mathcal{X}\) be an algebraic stack over a locally Noetherian base \(S\). Then \(\mathcal{X}\) satisfies (RS).

Proof

Immediate from the definitions and Lemma 07WN.

Lemma

Let \(S\) be a scheme. Let \(p : \mathcal{X} \to \mathcal{Y}\) and \(q : \mathcal{Z} \to \mathcal{Y}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(\mathcal{X}\), \(\mathcal{Y}\), and \(\mathcal{Z}\) satisfy (RS), then so does \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z}\).

Proof

This is formal. Let \[\xymatrix{ X \ar[r] \ar[d] & X' \ar[d] \\ Y \ar[r] & Y' = Y \amalg_X X' }\] be a diagram as in Definition 07WP. We have to show that \[(\mathcal{X} \times_{\mathcal{Y}} \mathcal{Z})_{Y'} \longrightarrow (\mathcal{X} \times_{\mathcal{Y}} \mathcal{Z})_Y \times_{(\mathcal{X} \times_{\mathcal{Y}} \mathcal{Z})_X} (\mathcal{X} \times_{\mathcal{Y}} \mathcal{Z})_{X'}\] is an equivalence. Using the definition of the \(2\)-fibre product this becomes [07WS]\[\begin{equation} \mathcal{X}_{Y'} \times_{\mathcal{Y}_{Y'}} \mathcal{Z}_{Y'} \longrightarrow (\mathcal{X}_Y \times_{\mathcal{Y}_Y} \mathcal{Z}_Y) \times_{(\mathcal{X}_X \times_{\mathcal{Y}_X} \mathcal{Z}_X)} (\mathcal{X}_{X'} \times_{\mathcal{Y}_{X'}} \mathcal{Z}_{X'}). \end{equation}\] We are given that each of the functors \[\mathcal{X}_{Y'} \to \mathcal{X}_Y \times_{\mathcal{X}_X} \mathcal{X}_{X'}, \quad \mathcal{Y}_{Y'} \to \mathcal{Y}_Y \times_{\mathcal{Y}_X} \mathcal{Y}_{X'}, \quad \mathcal{Z}_{Y'} \to \mathcal{Z}_Y \times_{\mathcal{Z}_X} \mathcal{Z}_{X'}\] are equivalences. An object of the right hand side of (07WS) is a system \[((x_Y, z_Y, \phi_Y), (x_{X'}, z_{X'}, \phi_{X'}), (\alpha, \beta)).\] Then \((x_Y, x_{X'}, \alpha)\) is isomorphic to the image of an object \(x_{Y'}\) in \(\mathcal{X}_{Y'}\) and \((z_Y, z_{X'}, \beta)\) is isomorphic to the image of an object \(z_{Y'}\) of \(\mathcal{Z}_{Y'}\). The pair of morphisms \((\phi_Y, \phi_{X'})\) corresponds to a morphism \(\psi\) between the images of \(x_{Y'}\) and \(z_{Y'}\) in \(\mathcal{Y}_{Y'}\). Then \((x_{Y'}, z_{Y'}, \psi)\) is an object of the left hand side of (07WS) mapping to the given object of the right hand side. This proves that (07WS) is essentially surjective. We omit the proof that it is fully faithful.

Deformation categories

We match the notation introduced above with the notation from the chapter “Formal Deformation Theory”.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\) satisfying (RS). For any field \(k\) of finite type over \(S\) and any object \(x_0\) of \(\mathcal{X}\) lying over \(k\) the predeformation category \(p : \mathcal{F}_{\mathcal{X}, k, x_0} \to \mathcal{C}_\Lambda\) (07T4) is a deformation category, see Formal Deformation Theory, Definition 06J9.

Proof

Set \(\mathcal{F} = \mathcal{F}_{\mathcal{X}, k, x_0}\). Let \(f_1 : A_1 \to A\) and \(f_2 : A_2 \to A\) be ring maps in \(\mathcal{C}_\Lambda\) with \(f_2\) surjective. We have to show that the functor \[\mathcal{F}(A_1 \times_A A_2) \longrightarrow \mathcal{F}(A_1) \times_{\mathcal{F}(A)} \mathcal{F}(A_2)\] is an equivalence, see Formal Deformation Theory, Lemma 06J5. Set \(X = \Spec(A)\), \(X' = \Spec(A_2)\), \(Y = \Spec(A_1)\) and \(Y' = \Spec(A_1 \times_A A_2)\). Note that \(Y' = Y \amalg_X X'\) in the category of schemes, see More on Morphisms, Lemma 07RT. We know that in the diagram of functors of fibre categories \[\xymatrix{ \mathcal{X}_{Y'} \ar[r] \ar[d] & \mathcal{X}_Y \times_{\mathcal{X}_X} \mathcal{X}_{X'} \ar[d] \\ \mathcal{X}_{\Spec(k)} \ar@{=}[r] & \mathcal{X}_{\Spec(k)} }\] the top horizontal arrow is an equivalence by Definition 07WP. Since \(\mathcal{F}(B)\) is the category of objects of \(\mathcal{X}_{\Spec(B)}\) with an identification with \(x_0\) over \(k\) we win.

Remark

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be fibred in groupoids over \((\Sch/S)_{fppf}\). Let \(k\) be a field of finite type over \(S\) and \(x_0\) an object of \(\mathcal{X}\) over \(k\). Let \(p : \mathcal{F} \to \mathcal{C}_\Lambda\) be as in (07T4). If \(\mathcal{F}\) is a deformation category, i.e., if \(\mathcal{F}\) satisfies the Rim-Schlessinger condition (RS), then we see that \(\mathcal{F}\) satisfies Schlessinger’s conditions (S1) and (S2) by Formal Deformation Theory, Lemma 06J7. Let \(\overline{\mathcal{F}}\) be the functor of isomorphism classes, see Formal Deformation Theory, Remarks 06GK (07W5). Then \(\overline{\mathcal{F}}\) satisfies (S1) and (S2) as well, see Formal Deformation Theory, Lemma 06I0. This holds in particular in the situation of Lemma 07WU.

Change of field

This section is the analogue of Formal Deformation Theory, Section 07W7. As pointed out there, to discuss what happens under change of field we need to write \(\mathcal{C}_{\Lambda, k}\) instead of \(\mathcal{C}_\Lambda\). In the following lemma we use the notation \(\mathcal{F}_{l/k}\) introduced in Formal Deformation Theory, Situation 07W8.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Let \(k\) be a field of finite type over \(S\) and let \(l/k\) be a finite extension. Let \(x_0\) be an object of \(\mathcal{X}\) lying over \(\Spec(k)\). Denote by \(x_{l, 0}\) the restriction of \(x_0\) to \(\Spec(l)\). Then there is a canonical functor \[(\mathcal{F}_{\mathcal{X}, k , x_0})_{l/k} \longrightarrow \mathcal{F}_{\mathcal{X}, l, x_{l, 0}}\] of categories cofibred in groupoids over \(\mathcal{C}_{\Lambda, l}\). If \(\mathcal{X}\) satisfies (RS), then this functor is an equivalence.

Proof

Consider a factorization \[\Spec(l) \to \Spec(B) \to S\] as in (07T3). By definition we have \[(\mathcal{F}_{\mathcal{X}, k , x_0})_{l/k}(B) = \mathcal{F}_{\mathcal{X}, k, x_0}(B \times_l k)\] see Formal Deformation Theory, Situation 07W8. Thus an object of this is a morphism \(x_0 \to x\) of \(\mathcal{X}\) lying over the morphism \(\Spec(k) \to \Spec(B \times_l k)\). Choosing a pullback functor for \(\mathcal{X}\) we can associate to \(x_0 \to x\) the morphism \(x_{l, 0} \to x_B\) where \(x_B\) is the restriction of \(x\) to \(\Spec(B)\) (via the morphism \(\Spec(B) \to \Spec(B \times_l k)\) coming from \(B \times_l k \subset B\)). This construction is functorial in \(B\) and compatible with morphisms.

Next, assume \(\mathcal{X}\) satisfies (RS). Consider the diagrams \[\vcenter{ \xymatrix{ l & B \ar[l] \\ k \ar[u] & B \times_l k \ar[l] \ar[u] } } \quad\text{and}\quad \vcenter{ \xymatrix{ \Spec(l) \ar[d] \ar[r] & \Spec(B) \ar[d] \\ \Spec(k) \ar[r] & \Spec(B \times_l k) } }\] The diagram on the left is a fibre product of rings. The diagram on the right is a pushout in the category of schemes, see More on Morphisms, Lemma 07RT. These schemes are all of finite type over \(S\) (see remarks following Definition 07WP). Hence (RS) kicks in to give an equivalence of fibre categories \[\mathcal{X}_{\Spec(B \times_l k)} \longrightarrow \mathcal{X}_{\Spec(k)} \times_{\mathcal{X}_{\Spec(l)}} \mathcal{X}_{\Spec(B)}\] This implies that the functor defined above gives an equivalence of fibre categories. Hence the functor is an equivalence on categories cofibred in groupoids by (the dual of) Categories, Lemma 003Z.

Tangent spaces

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Let \(k\) be a field of finite type over \(S\) and let \(x_0\) be an object of \(\mathcal{X}\) over \(k\). In Formal Deformation Theory, Section 06I1 we have defined the tangent space [07WZ]\[\begin{equation} T\mathcal{F}_{\mathcal{X}, k, x_0} = \left\{ \begin{matrix} \text{isomorphism classes of morphisms}\\ x_0 \to x\text{ over }\Spec(k) \to \Spec(k[\epsilon]) \end{matrix} \right\} \end{equation}\] of the predeformation category \(\mathcal{F}_{\mathcal{X}, k, x_0}\). In Formal Deformation Theory, Section 06JN we have defined [07X0]\[\begin{equation} \text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) = \Ker\left( \text{Aut}_{\Spec(k[\epsilon])}(x'_0) \to \text{Aut}_{\Spec(k)}(x_0) \right) \end{equation}\] where \(x_0'\) is the pullback of \(x_0\) to \(\Spec(k[\epsilon])\). If \(\mathcal{X}\) satisfies the Rim-Schlessinger condition (RS), then \(T\mathcal{F}_{\mathcal{X}, k, x_0}\) comes equipped with a natural \(k\)-vector space structure by Formal Deformation Theory, Lemma 06IH (assumptions hold by Lemma 07WU and Remark 07WV). Moreover, Formal Deformation Theory, Lemma 06JX shows that \(\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})\) has a natural \(k\)-vector space structure such that addition agrees with composition of automorphisms. A natural condition is to ask these vector spaces to have finite dimension.

The following lemma tells us this is true if \(\mathcal{X}\) is locally of finite type over \(S\) (see Morphisms of Stacks, Section 06FR).

Lemma

Let \(S\) be a locally Noetherian scheme. Assume

1. \(\mathcal{X}\) is an algebraic stack,

2. \(U\) is a scheme locally of finite type over \(S\), and

3. \((\Sch/U)_{fppf} \to \mathcal{X}\) is a smooth surjective morphism.

Then, for any \(\mathcal{F} = \mathcal{F}_{\mathcal{X}, k, x_0}\) as in Section 07T2 the tangent space \(T\mathcal{F}\) and infinitesimal automorphism space \(\text{Inf}(\mathcal{F})\) have finite dimension over \(k\).

Proof

Let us write \(\mathcal{U} = (\Sch/U)_{fppf}\). By our definition of algebraic stacks the \(1\)-morphism \(\mathcal{U} \to \mathcal{X}\) is representable by algebraic spaces. Hence in particular the 2-fibre product \[\mathcal{U}_{x_0} = (\Sch/\Spec(k))_{fppf} \times_\mathcal{X} \mathcal{U}\] is representable by an algebraic space \(U_{x_0}\) over \(\Spec(k)\). Then \(U_{x_0} \to \Spec(k)\) is smooth and surjective (in particular \(U_{x_0}\) is nonempty). By Spaces over Fields, Lemma 07W4 we can find a finite extension \(l/k\) and a point \(\Spec(l) \to U_{x_0}\) over \(k\). We have \[(\mathcal{F}_{\mathcal{X}, k , x_0})_{l/k} = \mathcal{F}_{\mathcal{X}, l, x_{l, 0}}\] by Lemma 07WX and the fact that \(\mathcal{X}\) satisfies (RS). Thus we see that \[T\mathcal{F} \otimes_k l \cong T\mathcal{F}_{\mathcal{X}, l, x_{l, 0}} \quad\text{and}\quad \text{Inf}(\mathcal{F}) \otimes_k l \cong \text{Inf}(\mathcal{F}_{\mathcal{X}, l, x_{l, 0}})\] by Formal Deformation Theory, Lemmas 07WB and 07WC (these are applicable by Lemmas 07WQ and 07WU and Remark 07WV). Hence it suffices to prove that \(T\mathcal{F}_{\mathcal{X}, l, x_{l, 0}}\) and \(\text{Inf}(\mathcal{F}_{\mathcal{X}, l, x_{l, 0}})\) have finite dimension over \(l\). Note that \(x_{l, 0}\) comes from a point \(u_0\) of \(\mathcal{U}\) over \(l\).

We interrupt the flow of the argument to show that the lemma for infinitesimal automorphisms follows from the lemma for tangent spaces. Namely, let \(\mathcal{R} = \mathcal{U} \times_\mathcal{X} \mathcal{U}\). Let \(r_0\) be the \(l\)-valued point \((u_0, u_0, \text{id}_{x_0})\) of \(\mathcal{R}\). Combining Lemma 07WL and Formal Deformation Theory, Lemma 06L6 we see that \[\text{Inf}(\mathcal{F}_{\mathcal{X}, l, x_{l, 0}}) \subset T\mathcal{F}_{\mathcal{R}, l, r_0}\] Note that \(\mathcal{R}\) is an algebraic stack, see Algebraic Stacks, Lemma 04TF. Also, \(\mathcal{R}\) is representable by an algebraic space \(R\) smooth over \(U\) (via either projection, see Algebraic Stacks, Lemma 04T5). Choose a scheme \(U'\) and a surjective étale morphism \(U' \to R\). Then \(U'\) is smooth over \(U\), hence locally of finite type over \(S\). As \((\Sch/U')_{fppf} \to \mathcal{R}\) is surjective and smooth, we have reduced the question to the case of tangent spaces.

The functor (07WJ) \[\mathcal{F}_{\mathcal{U}, l, u_0} \longrightarrow \mathcal{F}_{\mathcal{X}, l, x_{l, 0}}\] is smooth by Lemma 07WK. The induced map on tangent spaces \[T\mathcal{F}_{\mathcal{U}, l, u_0} \longrightarrow T\mathcal{F}_{\mathcal{X}, l, x_{l, 0}}\] is \(l\)-linear (by Formal Deformation Theory, Lemma 06IJ) and surjective (as smooth maps of predeformation categories induce surjective maps on tangent spaces by Formal Deformation Theory, Lemma 06HN). Hence it suffices to prove that the tangent space of the deformation space associated to the representable algebraic stack \(\mathcal{U}\) at the point \(u_0\) is finite dimensional. Let \(\Spec(R) \subset U\) be an affine open such that \(u_0 : \Spec(l) \to U\) factors through \(\Spec(R)\) and such that \(\Spec(R) \to S\) factors through \(\Spec(\Lambda) \subset S\). Let \(\mathfrak m_R \subset R\) be the kernel of the \(\Lambda\)-algebra map \(\varphi_0 : R \to l\) corresponding to \(u_0\). Note that \(R\), being of finite type over the Noetherian ring \(\Lambda\), is a Noetherian ring. Hence \(\mathfrak m_R = (f_1, \ldots, f_n)\) is a finitely generated ideal. We have \[T\mathcal{F}_{\mathcal{U}, l, u_0} = \{\varphi : R \to l[\epsilon] \mid \varphi \text{ is a } \Lambda\text{-algebra map and } \varphi \bmod \epsilon = \varphi_0\}\] An element of the right hand side is determined by its values on \(f_1, \ldots, f_n\); hence the dimension is at most \(n\) and we win. Some details omitted.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(p : \mathcal{X} \to \mathcal{Y}\) and \(q : \mathcal{Z} \to \mathcal{Y}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). Assume \(\mathcal{X}\), \(\mathcal{Y}\), \(\mathcal{Z}\) satisfy (RS). Let \(k\) be a field of finite type over \(S\) and let \(w_0\) be an object of \(\mathcal{W} = \mathcal{X} \times_\mathcal{Y} \mathcal{Z}\) over \(k\). Denote by \(x_0, y_0, z_0\) the objects of \(\mathcal{X}, \mathcal{Y}, \mathcal{Z}\) you get from \(w_0\). Then there is a \(6\)-term exact sequence \[\xymatrix{ 0 \ar[r] & \text{Inf}(\mathcal{F}_{\mathcal{W}, k, w_0}) \ar[r] & \text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) \oplus \text{Inf}(\mathcal{F}_{\mathcal{Z}, k, z_0}) \ar[r] & \text{Inf}(\mathcal{F}_{\mathcal{Y}, k, y_0}) \ar[lld] \\ & T\mathcal{F}_{\mathcal{W}, k, w_0} \ar[r] & T\mathcal{F}_{\mathcal{X}, k, x_0} \oplus T\mathcal{F}_{\mathcal{Z}, k, z_0} \ar[r] & T\mathcal{F}_{\mathcal{Y}, k, y_0} }\] of \(k\)-vector spaces.

Proof

By Lemma 07WR we see that \(\mathcal{W}\) satisfies (RS) and hence the lemma makes sense. To see the lemma is true, apply Lemmas 07WL and 07WU and Formal Deformation Theory, Lemma 06L5.

Formal objects

In this section we transfer some of the notions already defined in the chapter “Formal Deformation Theory” to the current setting. In the following we will say “\(R\) is an \(S\)-algebra” to indicate that \(R\) is a ring endowed with a morphism of schemes \(\Spec(R) \to S\).

Definition

Let \(S\) be a locally Noetherian scheme. Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids.

1. A formal object \(\xi = (R, \xi_n, f_n)\) of \(\mathcal{X}\) consists of a Noetherian complete local \(S\)-algebra \(R\), objects \(\xi_n\) of \(\mathcal{X}\) lying over \(\Spec(R/\mathfrak m_R^n)\), and morphisms \(f_n : \xi_n \to \xi_{n + 1}\) of \(\mathcal{X}\) lying over \(\Spec(R/\mathfrak m_R^n) \to \Spec(R/\mathfrak m_R^{n + 1})\) such that \(R/\mathfrak m_R\) is a field of finite type over \(S\).

2. A morphism of formal objects \(a : \xi = (R, \xi_n, f_n) \to \eta = (T, \eta_n, g_n)\) is given by morphisms \(a_n : \xi_n \to \eta_n\) such that for every \(n\) the diagram \[\xymatrix{ \xi_n \ar[r]_{f_n} \ar[d]_{a_n} & \xi_{n + 1} \ar[d]^{a_{n + 1}} \\ \eta_n \ar[r]^{g_n} & \eta_{n + 1} }\] is commutative. Applying the functor \(p\) we obtain a compatible collection of morphisms \(\Spec(R/\mathfrak m_R^n) \to \Spec(T/\mathfrak m_T^n)\) and hence a morphism \(a_0 : \Spec(R) \to \Spec(T)\) over \(S\). We say that \(a\) lies over \(a_0\).

Thus we obtain a category of formal objects of \(\mathcal{X}\).

Remark

Let \(S\) be a locally Noetherian scheme. Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Let \(\xi = (R, \xi_n, f_n)\) be a formal object. Set \(k = R/\mathfrak m\) and \(x_0 = \xi_1\). The formal object \(\xi\) defines a formal object \(\xi\) of the predeformation category \(\mathcal{F}_{\mathcal{X}, k, x_0}\). This follows immediately from Definition 07X4 above, Formal Deformation Theory, Definition 06H3, and our construction of the predeformation category \(\mathcal{F}_{\mathcal{X}, k, x_0}\) in Section 07T2.

If \(F : \mathcal{X} \to \mathcal{Y}\) is a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\), then \(F\) induces a functor between categories of formal objects as well.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(F : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). Let \(\eta = (R, \eta_n, g_n)\) be a formal object of \(\mathcal{Y}\) and let \(\xi_1\) be an object of \(\mathcal{X}\) with \(F(\xi_1) \cong \eta_1\). If \(F\) is formally smooth on objects (see Criteria for Representability, Section 06CZ), then there exists a formal object \(\xi = (R, \xi_n, f_n)\) of \(\mathcal{X}\) such that \(F(\xi) \cong \eta\).

Proof

Note that each of the morphisms \(\Spec(R/\mathfrak m^n) \to \Spec(R/\mathfrak m^{n + 1})\) is a first order thickening of affine schemes over \(S\). Hence the assumption on \(F\) means that we can successively lift \(\xi_1\) to objects \(\xi_2, \xi_3, \ldots\) of \(\mathcal{X}\) endowed with compatible isomorphisms \(\xi_n|_{\Spec(R/\mathfrak m^{n - 1})} \cong \xi_{n - 1}\) and \(F(\xi_n) \cong \eta_n\).

Let \(S\) be a locally Noetherian scheme. Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Suppose that \(x\) is an object of \(\mathcal{X}\) over \(R\), where \(R\) is a Noetherian complete local \(S\)-algebra with residue field of finite type over \(S\). Then we can consider the system of restrictions \(\xi_n = x|_{\Spec(R/\mathfrak m^n)}\) endowed with the natural morphisms \(\xi_1 \to \xi_2 \to \ldots\) coming from transitivity of restriction. Thus \(\xi = (R, \xi_n, \xi_n \to \xi_{n + 1})\) is a formal object of \(\mathcal{X}\). This construction is functorial in the object \(x\). Thus we obtain a functor [07X6]\[\begin{equation} \left\{ \begin{matrix} \text{objects }x\text{ of }\mathcal{X} \text{ such that }p(x) = \Spec(R) \\ \text{where }R\text{ is Noetherian complete local}\\ \text{with }R/\mathfrak m\text{ of finite type over }S \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{formal objects of }\mathcal{X} \end{matrix} \right\} \end{equation}\] To be precise the left hand side is the full subcategory of \(\mathcal{X}\) consisting of objects as indicated and the right hand side is the category of formal objects of \(\mathcal{X}\) as in Definition 07X4.

Definition

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). A formal object \(\xi = (R, \xi_n, f_n)\) of \(\mathcal{X}\) is called effective if it is in the essential image of the functor (07X6).

If the category fibred in groupoids is an algebraic stack, then every formal object is effective as follows from the next lemma.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be an algebraic stack over \(S\). The functor (07X6) is an equivalence.

Proof

Case I: \(\mathcal{X}\) is representable (by a scheme). Say \(\mathcal{X} = (\Sch/X)_{fppf}\) for some scheme \(X\) over \(S\). Unwinding the definitions we have to prove the following: Given a Noetherian complete local \(S\)-algebra \(R\) with \(R/\mathfrak m\) of finite type over \(S\) we have \[\Mor_S(\Spec(R), X) \longrightarrow \lim \Mor_S(\Spec(R/\mathfrak m^n), X)\] is bijective. This follows from Formal Spaces, Lemma 0AQG.

Case II. \(\mathcal{X}\) is representable by an algebraic space. Say \(\mathcal{X}\) is representable by \(X\). Again we have to show that \[\Mor_S(\Spec(R), X) \longrightarrow \lim \Mor_S(\Spec(R/\mathfrak m^n), X)\] is bijective for \(R\) as above. This is Formal Spaces, Lemma 0AQH.

Case III: General case of an algebraic stack. A general remark is that the left and right hand side of (07X6) are categories fibred in groupoids over the category of affine schemes over \(S\) which are spectra of Noetherian complete local rings with residue field of finite type over \(S\). We will also see in the proof below that they form stacks for a certain topology on this category.

We first prove full faithfulness. Let \(R\) be a Noetherian complete local \(S\)-algebra with \(k = R/\mathfrak m\) of finite type over \(S\). Let \(x, x'\) be objects of \(\mathcal{X}\) over \(R\). As \(\mathcal{X}\) is an algebraic stack \(\mathit{Isom}(x, x')\) is representable by an algebraic space \(I\) over \(\Spec(R)\), see Algebraic Stacks, Lemma 045G. Applying Case II to \(I\) over \(\Spec(R)\) implies immediately that (07X6) is fully faithful on fibre categories over \(\Spec(R)\). Hence the functor is fully faithful by Categories, Lemma 003Z.

Essential surjectivity. Let \(\xi = (R, \xi_n, f_n)\) be a formal object of \(\mathcal{X}\). Choose a scheme \(U\) over \(S\) and a surjective smooth morphism \(f : (\Sch/U)_{fppf} \to \mathcal{X}\). For every \(n\) consider the fibre product \[(\Sch/\Spec(R/\mathfrak m^n))_{fppf} \times_{\xi_n, \mathcal{X}, f} (\Sch/U)_{fppf}\] By assumption this is representable by an algebraic space \(V_n\) surjective and smooth over \(\Spec(R/\mathfrak m^n)\). The morphisms \(f_n : \xi_n \to \xi_{n + 1}\) induce cartesian squares \[\xymatrix{ V_{n + 1} \ar[d] & V_n \ar[d] \ar[l] \\ \Spec(R/\mathfrak m^{n + 1}) & \Spec(R/\mathfrak m^n) \ar[l] }\] of algebraic spaces. By Spaces over Fields, Lemma 07W4 we can find a finite separable extension \(k'/k\) and a point \(v'_1 : \Spec(k') \to V_1\) over \(k\). Let \(R \subset R'\) be the finite étale extension whose residue field extension is \(k'/k\) (exists and is unique by Algebra, Lemmas 04GK and 04GM). By the infinitesimal lifting criterion of smoothness (see More on Morphisms of Spaces, Lemma 04AM) applied to \(V_n \to \Spec(R/\mathfrak m^n)\) for \(n = 2, 3, 4, \ldots\) we can successively find morphisms \(v'_n : \Spec(R'/(\mathfrak m')^n) \to V_n\) over \(\Spec(R/\mathfrak m^n)\) fitting into commutative diagrams \[\xymatrix{ \Spec(R'/(\mathfrak m')^{n + 1}) \ar[d]_{v'_{n + 1}} & \Spec(R'/(\mathfrak m')^n) \ar[d]^{v'_n} \ar[l] \\ V_{n + 1} & V_n \ar[l] }\] Composing with the projection morphisms \(V_n \to U\) we obtain a compatible system of morphisms \(u'_n : \Spec(R'/(\mathfrak m')^n) \to U\). By Case I the family \((u'_n)\) comes from a unique morphism \(u' : \Spec(R') \to U\). Denote by \(x'\) the object of \(\mathcal{X}\) over \(\Spec(R')\) we get by applying the \(1\)-morphism \(f\) to \(u'\). By construction, there exists a morphism of formal objects \[(\href{artin.html#artin-equation-approximation}{07X6})(x') = (R', x'|_{\Spec(R'/(\mathfrak m')^n)}, \ldots) \longrightarrow (R, \xi_n, f_n)\] lying over \(\Spec(R') \to \Spec(R)\). Note that \(R' \otimes_R R'\) is a finite product of Noetherian complete local rings to which our current discussion applies. Denote by \(p_0, p_1 : \Spec(R' \otimes_R R') \to \Spec(R')\) the two projections. By the full faithfulness shown above there exists a canonical isomorphism \(\varphi : p_0^*x' \to p_1^*x'\) because we have such isomorphisms over \(\Spec((R' \otimes_R R')/\mathfrak m^n(R' \otimes_R R'))\). We omit the proof that the isomorphism \(\varphi\) satisfies the cocycle condition (see Stacks, Definition 026B). Since \(\{\Spec(R') \to \Spec(R)\}\) is an fppf covering we conclude that \(x'\) descends to an object \(x\) of \(\mathcal{X}\) over \(\Spec(R)\). We omit the proof that \(\xi_n\) is the restriction of \(x\) to \(\Spec(R/\mathfrak m^n)\).

Lemma

Let \(S\) be a scheme. Let \(p : \mathcal{X} \to \mathcal{Y}\) and \(q : \mathcal{Z} \to \mathcal{Y}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If the functor (07X6) is an equivalence for \(\mathcal{X}\), \(\mathcal{Y}\), and \(\mathcal{Z}\), then it is an equivalence for \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z}\).

Proof

The left and the right hand side of (07X6) for \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z}\) are simply the \(2\)-fibre products of the left and the right hand side of (07X6) for \(\mathcal{X}\), \(\mathcal{Z}\) over \(\mathcal{Y}\). Hence the result follows as taking \(2\)-fibre products is compatible with equivalences of categories, see Categories, Lemma 02XB.

Approximation

A fundamental insight of Michael Artin is that you can approximate objects of a limit preserving stack. Namely, given an object \(x\) of the stack over a Noetherian complete local ring, you can find an object \(x_A\) over an algebraic ring which is “close to” \(x\). Here an algebraic ring means a finite type \(S\)-algebra and close means adically close. In this section we present this in a simple, yet general form.

To formulate the result we need to pull together some definitions from different places in the Stacks project. First, in Criteria for Representability, Section 06CT we introduced limit preserving on objects for \(1\)-morphisms of categories fibred in groupoids over the category of schemes. In More on Algebra, Definition 07GH we defined the notion of a G-ring. Let \(S\) be a locally Noetherian scheme. Let \(A\) be an \(S\)-algebra. We say that \(A\) is of finite type over \(S\) or is a finite type \(S\)-algebra if \(\Spec(A) \to S\) is of finite type. In this case \(A\) is a Noetherian ring. Finally, given a ring \(A\) and ideal \(I\) we denote \(\text{Gr}_I(A) = \bigoplus I^n/I^{n + 1}\).

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Let \(x\) be an object of \(\mathcal{X}\) lying over \(\Spec(R)\) where \(R\) is a Noetherian complete local ring with residue field \(k\) of finite type over \(S\). Let \(s \in S\) be the image of \(\Spec(k) \to S\). Assume that (a) \(\mathcal{O}_{S, s}\) is a G-ring and (b) \(p\) is limit preserving on objects. Then for every integer \(N \geq 1\) there exist

1. a finite type \(S\)-algebra \(A\),

2. a maximal ideal \(\mathfrak m_A \subset A\),

3. an object \(x_A\) of \(\mathcal{X}\) over \(\Spec(A)\),

4. an \(S\)-isomorphism \(R/\mathfrak m_R^N \cong A/\mathfrak m_A^N\),

5. an isomorphism \(x|_{\Spec(R/\mathfrak m_R^N)} \cong x_A|_{\Spec(A/\mathfrak m_A^N)}\) compatible with (4), and

6. an isomorphism \(\text{Gr}_{\mathfrak m_R}(R) \cong \text{Gr}_{\mathfrak m_A}(A)\) of graded \(k\)-algebras.

Proof

Choose an affine open \(\Spec(\Lambda) \subset S\) such that \(k\) is a finite \(\Lambda\)-algebra, see Morphisms, Lemma 01TA. We may and do replace \(S\) by \(\Spec(\Lambda)\).

We may write \(R\) as a directed colimit \(R = \colim C_j\) where each \(C_j\) is a finite type \(\Lambda\)-algebra (see Algebra, Lemma 00QN). By assumption (b) the object \(x\) is isomorphic to the restriction of an object over one of the \(C_j\). Hence we may choose a finite type \(\Lambda\)-algebra \(C\), a \(\Lambda\)-algebra map \(C \to R\), and an object \(x_C\) of \(\mathcal{X}\) over \(\Spec(C)\) such that \(x = x_C|_{\Spec(R)}\). The choice of \(C\) is a bookkeeping device and could be avoided. For later use, let us write \(C = \Lambda[y_1, \ldots, y_u]/(f_1, \ldots, f_v)\) and we denote \(\overline{a}_i \in R\) the image of \(y_i\) under the map \(C \to R\). Set \(\mathfrak m_C = C \cap \mathfrak m_R\).

Choose a \(\Lambda\)-algebra surjection \(\Lambda[x_1, \ldots, x_s] \to k\) and denote by \(\mathfrak m'\) the kernel. By the universal property of polynomial rings we may lift this to a \(\Lambda\)-algebra map \(\Lambda[x_1, \ldots, x_s] \to R\). We add some variables (i.e., we increase \(s\) a bit) mapping to generators of \(\mathfrak m_R\). Having done this we see that \(\Lambda[x_1, \ldots, x_s] \to R/\mathfrak m_R^2\) is surjective. Then we see that [07XC]\[\begin{equation} P = \Lambda[x_1, \ldots, x_s]_{\mathfrak m'}^\wedge \longrightarrow R \end{equation}\] is a surjective map of Noetherian complete local rings, see for example Formal Deformation Theory, Lemma 06GZ.

Choose lifts \(a_i \in P\) of \(\overline{a}_i\) we found above. Choose generators \(b_1, \ldots, b_r \in P\) for the kernel of (07XC). Choose \(c_{ji} \in P\) such that \[f_j(a_1, \ldots, a_u) = \sum c_{ji} b_i\] in \(P\) which is possible by the choices made so far. Choose generators \[k_1, \ldots, k_t \in \Ker(P^{\oplus r} \xrightarrow{(b_1, \ldots, b_r)} P)\] and write \(k_i = (k_{i1}, \ldots, k_{ir})\) and \(K = (k_{ij})\) so that \[P^{\oplus t} \xrightarrow{K} P^{\oplus r} \xrightarrow{(b_1, \ldots, b_r)} P \to R \to 0\] is an exact sequence of \(P\)-modules. In particular we have \(\sum k_{ij} b_j = 0\). After possibly increasing \(N\) we may assume \(N - 1\) works in the Artin-Rees lemma for the first two maps of this exact sequence (see More on Algebra, Section 07VD for terminology).

By assumption \(\mathcal{O}_{S, s} = \Lambda_{\Lambda \cap \mathfrak m'}\) is a G-ring. Hence by More on Algebra, Proposition 07PV the ring \(\Lambda[x_1, \ldots, x_s]_{\mathfrak m'}\) is a \(G\)-ring. Hence by Smoothing Ring Maps, Theorem 07QZ there exist an étale ring map \[\Lambda[x_1, \ldots, x_s]_{\mathfrak m'} \to B,\] a maximal ideal \(\mathfrak m_B\) of \(B\) lying over \(\mathfrak m'\), and elements \(a'_i, b'_i, c'_{ij}, k'_{ij} \in B\) such that

1. \(\kappa(\mathfrak m') = \kappa(\mathfrak m_B)\) which implies that \(\Lambda[x_1, \ldots, x_s]_{\mathfrak m'} \subset B_{\mathfrak m_B} \subset P\) and \(P\) is identified with the completion of \(B\) at \(\mathfrak m_B\), see remark preceding Smoothing Ring Maps, Theorem 07QZ,

2. \(a_i - a'_i, b_i - b'_i, c_{ij} - c'_{ij}, k_{ij} - k'_{ij} \in (\mathfrak m')^N P\), and

3. \(f_j(a'_1, \ldots, a'_u) = \sum c'_{ji} b'_i\) and \(\sum k'_{ij}b'_j = 0\).

Set \(A = B/(b'_1, \ldots, b'_r)\) and denote by \(\mathfrak m_A\) the image of \(\mathfrak m_B\) in \(A\). (Note that \(A\) is essentially of finite type over \(\Lambda\); at the end of the proof we will show how to obtain an \(A\) which is of finite type over \(\Lambda\).) There is a ring map \(C \to A\) sending \(y_i \mapsto a'_i\) because the \(a'_i\) satisfy the desired equations modulo \((b'_1, \ldots, b'_r)\). Note that \(A/\mathfrak m_A^N = R/\mathfrak m_R^N\) as quotients of \(P = B^\wedge\) by property (2) above. Set \(x_A = x_C|_{\Spec(A)}\). Since the maps \[C \to A \to A/\mathfrak m_A^N \cong R/\mathfrak m_R^N \quad\text{and}\quad C \to R \to R/\mathfrak m_R^N\] are equal we see that \(x_A\) and \(x\) agree modulo \(\mathfrak m_R^N\) via the isomorphism \(A/\mathfrak m_A^N = R/\mathfrak m_R^N\). At this point we have shown properties (1) – (5) of the statement of the lemma. To see (6) note that \[P^{\oplus t} \xrightarrow{K} P^{\oplus r} \xrightarrow{(b_1, \ldots, b_r)} P \quad\text{and}\quad P^{\oplus t} \xrightarrow{K'} P^{\oplus r} \xrightarrow{(b'_1, \ldots, b'_r)} P\] are two complexes of \(P\)-modules which are congruent modulo \((\mathfrak m')^N\) with the first one being exact. By our choice of \(N\) above we see from More on Algebra, Lemma 07VF that \(R = P/(b_1, \ldots, b_r)\) and \(P/(b'_1, \ldots, b'_r) = B^\wedge/(b'_1, \ldots, b'_r) = A^\wedge\) have isomorphic associated graded algebras, which is what we wanted to show.

This last paragraph of the proof serves to clean up the issue that \(A\) is essentially of finite type over \(S\) and not yet of finite type. The construction above gives \(A = B/(b'_1, \ldots, b'_r)\) and \(\mathfrak m_A \subset A\) with \(B\) étale over \(\Lambda[x_1, \ldots, x_s]_{\mathfrak m'}\). Hence \(A\) is of finite type over the Noetherian ring \(\Lambda[x_1, \ldots, x_s]_{\mathfrak m'}\). Thus we can write \(A = (A_0)_{\mathfrak m'}\) for some finite type \(\Lambda[x_1, \ldots, x_s]\) algebra \(A_0\). Then \(A = \colim (A_0)_f\) where \(f \in \Lambda[x_1, \ldots, x_s] \setminus \mathfrak m'\), see Algebra, Lemma 00CR. Because \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) is limit preserving on objects, we see that \(x_A\) comes from some object \(x_{(A_0)_f}\) over \(\Spec((A_0)_f)\) for an \(f\) as above. After replacing \(A\) by \((A_0)_f\) and \(x_A\) by \(x_{(A_0)_f}\) and \(\mathfrak m_A\) by \((A_0)_f \cap \mathfrak m_A\) the proof is finished.

Limit preserving

The morphism \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) is limit preserving on objects, as defined in Criteria for Representability, Section 06CT, if the functor of the definition below is essentially surjective. However, the example in Examples, Section 07Z0 shows that this isn’t equivalent to being limit preserving.

Definition

Let \(S\) be a scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). We say \(\mathcal{X}\) is limit preserving if for every affine scheme \(T\) over \(S\) which is a limit \(T = \lim T_i\) of a directed inverse system of affine schemes \(T_i\) over \(S\), we have an equivalence \[\colim \mathcal{X}_{T_i} \longrightarrow \mathcal{X}_T\] of fibre categories.

We spell out what this means. First, given objects \(x, y\) of \(\mathcal{X}\) over \(T_i\) we should have \[\Mor_{\mathcal{X}_T}(x|_T, y|_T) = \colim_{i' \geq i} \Mor_{\mathcal{X}_{T_{i'}}}(x|_{T_{i'}}, y|_{T_{i'}})\] and second every object of \(\mathcal{X}_T\) is isomorphic to the restriction of an object over \(T_i\) for some \(i\). Note that the first condition means that the presheaves \(\mathit{Isom}_\mathcal{X}(x, y)\) (see Stacks, Definition 02ZB) are limit preserving.

Lemma

Let \(S\) be a scheme. Let \(p : \mathcal{X} \to \mathcal{Y}\) and \(q : \mathcal{Z} \to \mathcal{Y}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\).

1. If \(\mathcal{X} \to (\Sch/S)_{fppf}\) and \(\mathcal{Z} \to (\Sch/S)_{fppf}\) are limit preserving on objects and \(\mathcal{Y}\) is limit preserving, then \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to (\Sch/S)_{fppf}\) is limit preserving on objects.

2. If \(\mathcal{X}\), \(\mathcal{Y}\), and \(\mathcal{Z}\) are limit preserving, then so is \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z}\).

Proof

This is formal. Proof of (1). Let \(T = \lim_{i \in I} T_i\) be the directed limit of affine schemes \(T_i\) over \(S\). We will prove that the functor \(\colim (\mathcal{X} \times_\mathcal{Y} \mathcal{Z})_{T_i} \to (\mathcal{X} \times_\mathcal{Y} \mathcal{Z})_T\) is essentially surjective. Recall that an object of the fibre product over \(T\) is a quadruple \((T, x, z, \alpha)\) where \(x\) is an object of \(\mathcal{X}\) lying over \(T\), \(z\) is an object of \(\mathcal{Z}\) lying over \(T\), and \(\alpha : p(x) \to q(z)\) is a morphism in the fibre category of \(\mathcal{Y}\) over \(T\). By assumption on \(\mathcal{X}\) and \(\mathcal{Z}\) we can find an \(i\) and objects \(x_i\) and \(z_i\) over \(T_i\) such that \(x_i|_T \cong x\) and \(z_i|_T \cong z\). Then \(\alpha\) corresponds to an isomorphism \(p(x_i)|_T \to q(z_i)|_T\) which comes from an isomorphism \(\alpha_{i'} : p(x_i)|_{T_{i'}} \to q(z_i)|_{T_{i'}}\) by our assumption on \(\mathcal{Y}\). After replacing \(i\) by \(i'\), \(x_i\) by \(x_i|_{T_{i'}}\), and \(z_i\) by \(z_i|_{T_{i'}}\) we see that \((T_i, x_i, z_i, \alpha_i)\) is an object of the fibre product over \(T_i\) which restricts to an object isomorphic to \((T, x, z, \alpha)\) over \(T\) as desired.

We omit the arguments showing that \(\colim (\mathcal{X} \times_\mathcal{Y} \mathcal{Z})_{T_i} \to (\mathcal{X} \times_\mathcal{Y} \mathcal{Z})_T\) is fully faithful in (2).

Lemma

Let \(S\) be a scheme. Let \(\mathcal{X}\) be an algebraic stack over \(S\). Then the following are equivalent

1. \(\mathcal{X}\) is a stack in setoids and \(\mathcal{X} \to (\Sch/S)_{fppf}\) is limit preserving on objects,

2. \(\mathcal{X}\) is a stack in setoids and limit preserving,

3. \(\mathcal{X}\) is representable by an algebraic space locally of finite presentation.

Proof

Under each of the three assumptions \(\mathcal{X}\) is representable by an algebraic space \(X\) over \(S\), see Algebraic Stacks, Proposition 04SZ. It is clear that (1) and (2) are equivalent as a functor between setoids is an equivalence if and only if it is surjective on isomorphism classes. Finally, (1) and (3) are equivalent by Limits of Spaces, Proposition 04AK.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Assume \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) is representable by algebraic spaces and \(\mathcal{X}\) is limit preserving. Then \(\Delta\) is locally of finite type.

Proof

We apply Criteria for Representability, Lemma 07WI. Let \(V\) be an affine scheme locally of finite presentation over \(S\) and let \(\theta\) be an object of \(\mathcal{X} \times \mathcal{X}\) over \(V\). Let \(F_\theta\) be an algebraic space representing \(\mathcal{X} \times_{\Delta, \mathcal{X} \times \mathcal{X}, \theta} (\Sch/V)_{fppf}\) and let \(f_\theta : F_\theta \to V\) be the canonical morphism (see Algebraic Stacks, Section 04SX). It suffices to show that \(F_\theta \to V\) has the corresponding properties. By Lemmas 07XM and 07XN we see that \(F_\theta \to S\) is locally of finite presentation. It follows that \(F_\theta \to V\) is locally of finite type by Morphisms of Spaces, Lemma 0462.

Versality

In the previous section we explained how to approximate objects over complete local rings by algebraic objects. But in order to show that a stack \(\mathcal{X}\) is an algebraic stack, we need to find smooth \(1\)-morphisms from schemes towards \(\mathcal{X}\). Since we are not going to assume a priori that \(\mathcal{X}\) has a representable diagonal, we cannot even speak about smooth morphisms towards \(\mathcal{X}\). Instead, borrowing terminology from deformation theory, we will introduce versal objects.

Definition

Let \(S\) be a locally Noetherian scheme. Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Let \(\xi = (R, \xi_n, f_n)\) be a formal object. Set \(k = R/\mathfrak m\) and \(x_0 = \xi_1\). We will say that \(\xi\) is versal if \(\xi\) as a formal object of \(\mathcal{F}_{\mathcal{X}, k, x_0}\) (Remark 0CXH) is versal in the sense of Formal Deformation Theory, Definition 06HR.

We briefly spell out what this means. With notation as in the definition, suppose given morphisms \(\xi_1 = x_0 \to y \to z\) of \(\mathcal{X}\) lying over closed immersions \(\Spec(k) \to \Spec(A) \to \Spec(B)\) where \(A, B\) are Artinian local rings with residue field \(k\). Suppose given an \(n \geq 1\) and a commutative diagram \[\vcenter{ \xymatrix{ & y \ar[ld] \\ \xi_n & \xi_1 \ar[u] \ar[l] } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & \Spec(A) \ar[ld] \\ \Spec(R/\mathfrak m^n) & \Spec(k) \ar[u] \ar[l] } }\] Versality means that for any data as above there exists an \(m \geq n\) and a commutative diagram \[\vcenter{ \xymatrix{ & & z \ar[lldd] \\ & & y \ar[ld] \ar[u] \\ \xi_m & \xi_n \ar[l] & \xi_1 \ar[u] \ar[l] } } \quad\text{lying over} \vcenter{ \xymatrix{ & & \Spec(B) \ar[lldd] \\ & & \Spec(A) \ar[ld] \ar[u] \\ \Spec(R/\mathfrak m^m) & \Spec(R/\mathfrak m^n) \ar[l] & \Spec(k) \ar[u] \ar[l] } }\] Please compare with Formal Deformation Theory, Remark 06HS.

Let \(S\) be a locally Noetherian scheme. Let \(U\) be a scheme over \(S\) with structure morphism \(U \to S\) locally of finite type. Let \(u_0 \in U\) be a finite type point of \(U\), see Morphisms, Definition 02J1. Set \(k = \kappa(u_0)\). Note that the composition \(\Spec(k) \to S\) is also of finite type, see Morphisms, Lemma 01T3. Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Let \(x\) be an object of \(\mathcal{X}\) which lies over \(U\). Denote by \(x_0\) the pullback of \(x\) by \(u_0\). By the \(2\)-Yoneda lemma \(x\) corresponds to a \(1\)-morphism \[x : (\Sch/U)_{fppf} \longrightarrow \mathcal{X},\] see Algebraic Stacks, Section 04SS. We obtain a morphism of predeformation categories [07XE]\[\begin{equation} \hat x : \mathcal{F}_{(\Sch/U)_{fppf}, k, u_0} \longrightarrow \mathcal{F}_{\mathcal{X}, k, x_0}, \end{equation}\] over \(\mathcal{C}_\Lambda\) see (07WJ).

Definition

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be fibred in groupoids over \((\Sch/S)_{fppf}\). Let \(U\) be a scheme locally of finite type over \(S\). Let \(x\) be an object of \(\mathcal{X}\) lying over \(U\). Let \(u_0\) be a finite type point of \(U\). We say \(x\) is versal at \(u_0\) if the morphism \(\hat x\) (07XE) is smooth, see Formal Deformation Theory, Definition 06HG.

This definition matches our notion of versality for formal objects of \(\mathcal{X}\).

Lemma

With notation as in Definition 07XF, let \(R = \mathcal{O}_{U, u_0}^\wedge\). Let \(\xi\) be the formal object of \(\mathcal{X}\) over \(R\) associated to \(x|_{\Spec(R)}\), see (07X6). Then \[x\text{ is versal at }u_0 \Leftrightarrow \xi\text{ is versal}\]

Proof

Observe that \(\mathcal{O}_{U, u_0}\) is a Noetherian local \(S\)-algebra with residue field \(k\). Hence \(R = \mathcal{O}_{U, u_0}^\wedge\) is an object of \(\mathcal{C}_\Lambda^\wedge\), see Formal Deformation Theory, Definition 06GW. Recall that \(\xi\) is versal if \(\underline{\xi} : \underline{R}|_{\mathcal{C}_\Lambda} \to \mathcal{F}_{\mathcal{X}, k, x_0}\) is smooth and \(x\) is versal at \(u_0\) if \(\hat x : \mathcal{F}_{(\Sch/U)_{fppf}, k, u_0} \to \mathcal{F}_{\mathcal{X}, k, x_0}\) is smooth. There is an identification of predeformation categories \[\underline{R}|_{\mathcal{C}_\Lambda} = \mathcal{F}_{(\Sch/U)_{fppf}, k, u_0},\] see Formal Deformation Theory, Remark 06HC for notation. Namely, given an Artinian local \(S\)-algebra \(A\) with residue field identified with \(k\) we have \[\Mor_{\mathcal{C}_\Lambda^\wedge}(R, A) = \{\varphi \in \Mor_S(\Spec(A), U) \mid \varphi|_{\Spec(k)} = u_0\}\] Unwinding the definitions the reader verifies that the resulting map \[\underline{R}|_{\mathcal{C}_\Lambda} = \mathcal{F}_{(\Sch/U)_{fppf}, k, u_0} \xrightarrow{\hat x} \mathcal{F}_{\mathcal{X}, k, x_0},\] is equal to \(\underline{\xi}\) and we see that the lemma is true.

Here is a sanity check.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(f : U \to V\) be a morphism of schemes locally of finite type over \(S\). Let \(u_0 \in U\) be a finite type point. The following are equivalent

1. \(f\) is smooth at \(u_0\),

2. \(f\) viewed as an object of \((\Sch/V)_{fppf}\) over \(U\) is versal at \(u_0\).

Proof

This is a restatement of More on Morphisms, Lemma 02HX.

It turns out that this notion is well behaved with respect to field extensions.

Lemma

Let \(S\), \(\mathcal{X}\), \(U\), \(x\), \(u_0\) be as in Definition 07XF. Let \(l\) be a field and let \(u_{l, 0} : \Spec(l) \to U\) be a morphism with image \(u_0\), where \(k = \kappa(u_0)\) and \(l/k\) is finite. Set \(x_{l, 0} = x_0|_{\Spec(l)}\). If \(\mathcal{X}\) satisfies (RS) and \(x\) is versal at \(u_0\), then \[\mathcal{F}_{(\Sch/U)_{fppf}, l, u_{l, 0}} \longrightarrow \mathcal{F}_{\mathcal{X}, l, x_{l, 0}}\] is smooth.

Proof

Note that \((\Sch/U)_{fppf}\) satisfies (RS) by Lemma 07WQ. Hence the functor of the lemma is the functor \[(\mathcal{F}_{(\Sch/U)_{fppf}, k , u_0})_{l/k} \longrightarrow (\mathcal{F}_{\mathcal{X}, k , x_0})_{l/k}\] associated to \(\hat x\), see Lemma 07WX. Hence the lemma follows from Formal Deformation Theory, Lemma 07WD.

The following lemma is another sanity check. It more or less signifies that if \(x\) is versal at \(u_0\) as in Definition 07XF, then \(x\) viewed as a morphism from \(U\) to \(\mathcal{X}\) is smooth whenever we make a base change by a scheme.

Lemma

Let \(S\), \(\mathcal{X}\), \(U\), \(x\), \(u_0\) be as in Definition 07XF. Assume

1. \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) is representable by algebraic spaces,

2. \(\Delta\) is locally of finite type (for example if \(\mathcal{X}\) is limit preserving), and

3. \(\mathcal{X}\) has (RS).

Let \(V\) be a scheme locally of finite type over \(S\) and let \(y\) be an object of \(\mathcal{X}\) over \(V\). Form the \(2\)-fibre product \[\xymatrix{ \mathcal{Z} \ar[r] \ar[d] & (\Sch/U)_{fppf} \ar[d]^x \\ (\Sch/V)_{fppf} \ar[r]^y & \mathcal{X} }\] Let \(Z\) be the algebraic space representing \(\mathcal{Z}\) and let \(z_0 \in |Z|\) be a finite type point lying over \(u_0\). If \(x\) is versal at \(u_0\), then the morphism \(Z \to V\) is smooth at \(z_0\).

Proof

(The parenthetical remark in the statement holds by Lemma 0CXI.) Observe that \(Z\) exists by assumption (1) and Algebraic Stacks, Lemma 045G. By assumption (2) we see that \(Z \to V \times_S U\) is locally of finite type. Choose a scheme \(W\), a closed point \(w_0 \in W\), and an étale morphism \(W \to Z\) mapping \(w_0\) to \(z_0\), see Morphisms of Spaces, Definition 06EG. Then \(W\) is locally of finite type over \(S\) and \(w_0\) is a finite type point of \(W\). Let \(l = \kappa(z_0)\). Denote by \(z_{l, 0}\), \(v_{l, 0}\), \(u_{l, 0}\), and \(x_{l, 0}\) the objects of \(\mathcal{Z}\), \((\Sch/V)_{fppf}\), \((\Sch/U)_{fppf}\), and \(\mathcal{X}\) over \(\Spec(l)\) obtained by pullback to \(\Spec(l) = w_0\). Consider \[\xymatrix{ \mathcal{F}_{(\Sch/W)_{fppf}, l, w_0} \ar[r] & \mathcal{F}_{\mathcal{Z}, l, z_{l, 0}} \ar[d] \ar[r] & \mathcal{F}_{(\Sch/U)_{fppf}, l, u_{l, 0}} \ar[d] \\ & \mathcal{F}_{(\Sch/V)_{fppf}, l, v_{l, 0}} \ar[r] & \mathcal{F}_{\mathcal{X}, l, x_{l, 0}} }\] By Lemma 07WL the square is a fibre product of predeformation categories. By Lemma 07XG we see that the right vertical arrow is smooth. By Formal Deformation Theory, Lemma 06HM the left vertical arrow is smooth. By Lemma 07WK we see that the left horizontal arrow is smooth. We conclude that the map \[\mathcal{F}_{(\Sch/W)_{fppf}, l, w_0} \to \mathcal{F}_{(\Sch/V)_{fppf}, l, v_{l, 0}}\] is smooth by Formal Deformation Theory, Lemma 06HM. Thus we conclude that \(W \to V\) is smooth at \(w_0\) by More on Morphisms, Lemma 02HX. This exactly means that \(Z \to V\) is smooth at \(z_0\) and the proof is complete.

We restate the approximation result in terms of versal objects.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Let \(\xi = (R, \xi_n, f_n)\) be a formal object of \(\mathcal{X}\) with \(\xi_1\) lying over \(\Spec(k) \to S\) with image \(s \in S\). Assume

1. \(\xi\) is versal,

2. \(\xi\) is effective,

3. \(\mathcal{O}_{S, s}\) is a G-ring, and

4. \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) is limit preserving on objects.

Then there exist a morphism of finite type \(U \to S\), a finite type point \(u_0 \in U\) with residue field \(k\), and an object \(x\) of \(\mathcal{X}\) over \(U\) such that \(x\) is versal at \(u_0\) and such that \(x|_{\Spec(\mathcal{O}_{U, u_0}/\mathfrak m_{u_0}^n)} \cong \xi_n\).

Proof

Choose an object \(x_R\) of \(\mathcal{X}\) lying over \(\Spec(R)\) whose associated formal object is \(\xi\). Let \(N = 2\) and apply Lemma 07XB. We obtain \(A, \mathfrak m_A, x_A, \ldots\). Let \(\eta = (A^\wedge, \eta_n, g_n)\) be the formal object associated to \(x_A|_{\Spec(A^\wedge)}\). We have a diagram \[\vcenter{ \xymatrix{ & \eta \ar[d] \\ \xi \ar[r] \ar@{..>}[ru] & \xi_2 = \eta_2 } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & A^\wedge \ar[d] \\ R \ar[r] \ar@{..>}[ru] & R/\mathfrak m_R^2 = A/\mathfrak m_A^2 } }\] The versality of \(\xi\) means exactly that we can find the dotted arrows in the diagrams, because we can successively find morphisms \(\xi \to \eta_3\), \(\xi \to \eta_4\), and so on by Formal Deformation Theory, Remark 06HS. The corresponding ring map \(R \to A^\wedge\) is surjective by Formal Deformation Theory, Lemma 06GZ. On the other hand, we have \(\dim_k \mathfrak m_R^n/\mathfrak m_R^{n + 1} = \dim_k \mathfrak m_A^n/\mathfrak m_A^{n + 1}\) for all \(n\) by construction. Hence \(R/\mathfrak m_R^n\) and \(A/\mathfrak m_A^n\) have the same (finite) length as \(\Lambda\)-modules by additivity of length and Formal Deformation Theory, Lemma 06GG. It follows that \(R/\mathfrak m_R^n \to A/\mathfrak m_A^n\) is an isomorphism for all \(n\), hence \(R \to A^\wedge\) is an isomorphism. Thus \(\eta\) is isomorphic to a versal object, hence versal itself. By Lemma 0CXK we conclude that \(x_A\) is versal at the point \(u_0\) of \(U = \Spec(A)\) corresponding to \(\mathfrak m_A\).

Example

In this example we show that the local ring \(\mathcal{O}_{S, s}\) has to be a G-ring in order for the result of Lemma 07XH to be true. Namely, let \(\Lambda\) be a Noetherian ring and let \(\mathfrak m\) be a maximal ideal of \(\Lambda\). Set \(R = \Lambda_\mathfrak m^\wedge\). Let \(\Lambda \to C \to R\) be a factorization with \(C\) of finite type over \(\Lambda\). Set \(S = \Spec(\Lambda)\), \(U = S \setminus \{\mathfrak m\}\), and \(S' = U \amalg \Spec(C)\). Consider the functor \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) defined by the rule \[F(T) = \left\{ \begin{matrix} * & \text{if }T \to S\text{ factors through }S' \\ \emptyset & \text{else} \end{matrix} \right.\] Let \(\mathcal{X} = \mathcal{S}_F\) be the category fibred in sets associated to \(F\), see Algebraic Stacks, Section 04SU. Then \(\mathcal{X} \to (\Sch/S)_{fppf}\) is limit preserving on objects and there exists an effective, versal formal object \(\xi\) over \(R\). Hence if the conclusion of Lemma 07XH holds for \(\mathcal{X}\), then there exists a finite type ring map \(\Lambda \to A\) and a maximal ideal \(\mathfrak m_A\) lying over \(\mathfrak m\) such that

1. \(\kappa(\mathfrak m) = \kappa(\mathfrak m_A)\),

2. \(\Lambda \to A\) and \(\mathfrak m_A\) satisfy condition (4) of Algebra, Lemma 02HT, and

3. there exists a \(\Lambda\)-algebra map \(C \to A\).

Thus \(\Lambda \to A\) is smooth at \(\mathfrak m_A\) by the lemma cited. Slicing \(A\) we may assume that \(\Lambda \to A\) is étale at \(\mathfrak m_A\), see for example More on Morphisms, Lemma 057G or argue directly. Write \(C = \Lambda[y_1, \ldots, y_n]/(f_1, \ldots, f_m)\). Then \(C \to R\) corresponds to a solution in \(R\) of the system of equations \(f_1 = \ldots = f_m = 0\), see Smoothing Ring Maps, Section 07QX. Thus if the conclusion of Lemma 07XH holds for every \(\mathcal{X}\) as above, then a system of equations which has a solution in \(R\) has a solution in the henselization of \(\Lambda_{\mathfrak m}\). In other words, the approximation property holds for \(\Lambda_{\mathfrak m}^h\). This implies that \(\Lambda_{\mathfrak m}^h\) is a G-ring (see also discussion in Smoothing Ring Maps, Section 07BX) which in turn implies that \(\Lambda_{\mathfrak m}\) is a G-ring.

Openness of versality

Next, we come to openness of versality.

Definition

Let \(S\) be a locally Noetherian scheme.

1. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). We say \(\mathcal{X}\) satisfies openness of versality if given a scheme \(U\) locally of finite type over \(S\), an object \(x\) of \(\mathcal{X}\) over \(U\), and a finite type point \(u_0 \in U\) such that \(x\) is versal at \(u_0\), then there exists an open neighbourhood \(u_0 \in U' \subset U\) such that \(x\) is versal at every finite type point of \(U'\).

2. Let \(f : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). We say \(f\) satisfies openness of versality if given a scheme \(U\) locally of finite type over \(S\), an object \(y\) of \(\mathcal{Y}\) over \(U\), openness of versality holds for \((\Sch/U)_{fppf} \times_\mathcal{Y} \mathcal{X}\).

Openness of versality is often the hardest to check. The following example shows that requiring this is necessary however.

Example

Let \(k\) be a field and set \(\Lambda = k[s, t]\). Consider the functor \(F : \Lambda\text{-algebras} \longrightarrow \textit{Sets}\) defined by the rule \[F(A) = \left\{ \begin{matrix} * & \text{if there exist }f_1, \ldots, f_n \in A\text{ such that } \\ & A = (s, t, f_1, \ldots, f_n)\text{ and } f_i s = 0\ \forall i \\ \emptyset & \text{else} \end{matrix} \right.\] Geometrically \(F(A) = *\) means there exists a quasi-compact open neighbourhood \(W\) of \(V(s, t) \subset \Spec(A)\) such that \(s|_W = 0\). Let \(\mathcal{X} \subset (\Sch/\Spec(\Lambda))_{fppf}\) be the full subcategory consisting of schemes \(T\) which have an affine open covering \(T = \bigcup \Spec(A_j)\) with \(F(A_j) = *\) for all \(j\). Then \(\mathcal{X}\) satisfies [0], [1], [2], [3], and [4] but not [5]. Namely, over \(U = \Spec(k[s, t]/(s))\) there exists an object \(x\) which is versal at \(u_0 = (s, t)\) but not at any other point. Details omitted.

Let \(S\) be a locally Noetherian scheme. Let \(f : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). Consider the following property [07XS]\[\begin{equation} \begin{matrix} \text{for all fields }k\text{ of finite type over }S \text{ and all }x_0 \in \Ob(\mathcal{X}_{\Spec(k)})\text{ the}\\ \text{map } \mathcal{F}_{\mathcal{X}, k, x_0} \to \mathcal{F}_{\mathcal{Y}, k, f(x_0)} \text{ of predeformation categories is smooth} \end{matrix} \end{equation}\] We formulate some lemmas around this concept. First we link it with (openness of) versality.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Let \(U\) be a scheme locally of finite type over \(S\). Let \(x\) be an object of \(\mathcal{X}\) over \(U\). Assume that \(x\) is versal at every finite type point of \(U\) and that \(\mathcal{X}\) satisfies (RS). Then \(x : (\Sch/U)_{fppf} \to \mathcal{X}\) satisfies (07XS).

Proof

Let \(\Spec(l) \to U\) be a morphism with \(l\) of finite type over \(S\). Then the image \(u_0 \in U\) is a finite type point of \(U\) and \(l/\kappa(u_0)\) is a finite extension, see discussion in Morphisms, Section 01T9. Hence we see that \(\mathcal{F}_{(\Sch/U)_{fppf}, l, u_{l, 0}} \to \mathcal{F}_{\mathcal{X}, l, x_{l, 0}}\) is smooth by Lemma 07XG.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(f : \mathcal{X} \to \mathcal{Y}\) and \(g : \mathcal{Y} \to \mathcal{Z}\) be composable \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(f\) and \(g\) satisfy (07XS) so does \(g \circ f\).

Proof

This follows formally from Formal Deformation Theory, Lemma 06HM.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(f : \mathcal{X} \to \mathcal{Y}\) and \(\mathcal{Z} \to \mathcal{Y}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(f\) satisfies (07XS) so does the projection \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z} \to \mathcal{Z}\).

Proof

This follows immediately from Lemma 07WL and Formal Deformation Theory, Lemma 06HM.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(f : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(f\) is formally smooth on objects, then \(f\) satisfies (07XS). If \(f\) is representable by algebraic spaces and smooth, then \(f\) satisfies (07XS).

Proof

This is a reformulation of Lemma 07WK.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(f : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). Assume

1. \(f\) is representable by algebraic spaces,

2. \(f\) satisfies (07XS),

3. \(\mathcal{X} \to (\Sch/S)_{fppf}\) is limit preserving on objects, and

4. \(\mathcal{Y}\) is limit preserving.

Then \(f\) is smooth.

Proof

The key ingredient of the proof is More on Morphisms, Lemma 02HX which (almost) says that a morphism of schemes of finite type over \(S\) satisfying (07XS) is a smooth morphism. The other arguments of the proof are essentially bookkeeping.

Let \(V\) be a scheme over \(S\) and let \(y\) be an object of \(\mathcal{Y}\) over \(V\). Let \(Z\) be an algebraic space representing the \(2\)-fibre product \(\mathcal{Z} = \mathcal{X} \times_{f, \mathcal{Y}, y} (\Sch/V)_{fppf}\). We have to show that the projection morphism \(Z \to V\) is smooth, see Algebraic Stacks, Definition 03YK. In fact, it suffices to do this when \(V\) is an affine scheme locally of finite presentation over \(S\), see Criteria for Representability, Lemma 07WI. Then \((\Sch/V)_{fppf}\) is limit preserving by Lemma 07XN. Hence \(Z \to S\) is locally of finite presentation by Lemmas 07XM and 07XN. Choose a scheme \(W\) and a surjective étale morphism \(W \to Z\). Then \(W\) is locally of finite presentation over \(S\).

Since \(f\) satisfies (07XS) we see that so does \(\mathcal{Z} \to (\Sch/V)_{fppf}\), see Lemma 07XV. Next, we see that \((\Sch/W)_{fppf} \to \mathcal{Z}\) satisfies (07XS) by Lemma 07XW. Thus the composition \[(\Sch/W)_{fppf} \to \mathcal{Z} \to (\Sch/V)_{fppf}\] satisfies (07XS) by Lemma 07XU. More on Morphisms, Lemma 02HX shows that the composition \(W \to Z \to V\) is smooth at every finite type point \(w_0\) of \(W\). Since the smooth locus is open we conclude that \(W \to V\) is a smooth morphism of schemes by Morphisms, Lemma 02J4. Thus we conclude that \(Z \to V\) is a smooth morphism of algebraic spaces by definition.

The lemma below is how we will use openness of versality.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Let \(k\) be a finite type field over \(S\) and let \(x_0\) be an object of \(\mathcal{X}\) over \(\Spec(k)\) with image \(s \in S\). Assume

1. \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) is representable by algebraic spaces,

2. \(\mathcal{X}\) satisfies axioms [1], [2], [3] (see Section 07XJ),

3. every formal object of \(\mathcal{X}\) is effective,

4. openness of versality holds for \(\mathcal{X}\), and

5. \(\mathcal{O}_{S, s}\) is a G-ring.

Then there exist a morphism of finite type \(U \to S\) and an object \(x\) of \(\mathcal{X}\) over \(U\) such that \[x : (\Sch/U)_{fppf} \longrightarrow \mathcal{X}\] is smooth and such that there exists a finite type point \(u_0 \in U\) whose residue field is \(k\) and such that \(x|_{u_0} \cong x_0\).

Proof

By axiom [2], Lemma 07WU, and Remark 07WV we see that \(\mathcal{F}_{\mathcal{X}, k, x_0}\) satisfies (S1) and (S2). Since also the tangent space has finite dimension by axiom [3] we deduce from Formal Deformation Theory, Lemma 06IW that \(\mathcal{F}_{\mathcal{X}, k, x_0}\) has a versal formal object \(\xi\). Assumption (3) says \(\xi\) is effective. By axiom [1] and Lemma 07XH there exists a morphism of finite type \(U \to S\), an object \(x\) of \(\mathcal{X}\) over \(U\), and a finite type point \(u_0\) of \(U\) with residue field \(k\) such that \(x\) is versal at \(u_0\) and such that \(x|_{\Spec(k)} \cong x_0\). By openness of versality we may shrink \(U\) and assume that \(x\) is versal at every finite type point of \(U\). We claim that \[x : (\Sch/U)_{fppf} \longrightarrow \mathcal{X}\] is smooth which proves the lemma. Namely, by Lemma 07XT \(x\) satisfies (07XS) whereupon Lemma 07XX finishes the proof.

Axioms

Let \(S\) be a locally Noetherian scheme. Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Here are the axioms we will consider on \(\mathcal{X}\).

1. a set theoretic condition² to be ignored by readers who are not interested in set theoretical issues,

2. \(\mathcal{X}\) is a stack in groupoids for the étale topology,

3. \(\mathcal{X}\) is limit preserving,

4. \(\mathcal{X}\) satisfies the Rim-Schlessinger condition (RS),

5. the spaces \(T\mathcal{F}_{\mathcal{X}, k, x_0}\) and \(\text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0})\) are finite dimensional for every \(k\) and \(x_0\), see (07WZ) and (07X0),

6. the functor (07X6) is an equivalence,

7. \(\mathcal{X}\) and \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) satisfy openness of versality.

Axioms for functors

Let \(S\) be a scheme. Let \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be a functor. Denote by \(\mathcal{X} = \mathcal{S}_F\) the category fibred in sets associated to \(F\), see Algebraic Stacks, Section 04SU. In this section we provide a translation between the material above as it applies to \(\mathcal{X}\), to statements about \(F\).

Let \(S\) be a locally Noetherian scheme. Let \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be a functor. Let \(k\) be a field of finite type over \(S\). Let \(x_0 \in F(\Spec(k))\). The associated predeformation category (07T4) corresponds to the functor \[F_{k, x_0} : \mathcal{C}_\Lambda \longrightarrow \textit{Sets}, \quad A \longmapsto \{x \in F(\Spec(A)) \mid x|_{\Spec(k)} = x_0 \}.\] Recall that we do not distinguish between categories cofibred in sets over \(\mathcal{C}_\Lambda\) and functors \(\mathcal{C}_\Lambda \to \textit{Sets}\), see Formal Deformation Theory, Remarks 06GK (06GP). Given a transformation of functors \(a : F \to G\), setting \(y_0 = a(x_0)\) we obtain a morphism \[F_{k, x_0} \longrightarrow G_{k, y_0}\] see (07WJ). Lemma 07WK tells us that if \(a : F \to G\) is formally smooth (in the sense of More on Morphisms of Spaces, Definition 049S), then \(F_{k, x_0} \longrightarrow G_{k, y_0}\) is smooth as in Formal Deformation Theory, Remark 06HJ.

Lemma 07WN says that if \(Y' = Y \amalg_X X'\) in the category of schemes over \(S\) where \(X \to X'\) is a thickening and \(X \to Y\) is affine, then the map \[F(Y \amalg_X X') \to F(Y) \times_{F(X)} F(X')\] is a bijection, provided that \(F\) is an algebraic space. We say a general functor \(F\) satisfies the Rim-Schlessinger condition or we say \(F\) satisfies (RS) if given any pushout \(Y' = Y \amalg_X X'\) where \(Y, X, X'\) are spectra of Artinian local rings of finite type over \(S\), then \[F(Y \amalg_X X') \to F(Y) \times_{F(X)} F(X')\] is a bijection. Thus every algebraic space satisfies (RS).

Lemma 07WU says that given a functor \(F\) which satisfies (RS), then all \(F_{k, x_0}\) are deformation functors as in Formal Deformation Theory, Definition 06J9, i.e., they satisfy (RS) as in Formal Deformation Theory, Remark 06J6. In particular the tangent space \[TF_{k, x_0} = \{x \in F(\Spec(k[\epsilon])) \mid x|_{\Spec(k)} = x_0\}\] has the structure of a \(k\)-vector space by Formal Deformation Theory, Lemma 06IH.

Lemma 07X1 says that an algebraic space \(F\) locally of finite type over \(S\) gives rise to deformation functors \(F_{k, x_0}\) with finite dimensional tangent spaces \(TF_{k, x_0}\).

A formal object³ \(\xi = (R, \xi_n)\) of \(F\) consists of a Noetherian complete local \(S\)-algebra \(R\) whose residue field is of finite type over \(S\), together with elements \(\xi_n \in F(\Spec(R/\mathfrak m^n))\) such that \(\xi_{n + 1}|_{\Spec(R/\mathfrak m^n)} = \xi_n\). A formal object \(\xi\) defines a formal object \(\xi\) of \(F_{R/\mathfrak m, \xi_1}\). We say \(\xi\) is versal if and only if it is versal in the sense of Formal Deformation Theory, Definition 06HR. A formal object \(\xi = (R, \xi_n)\) is called effective if there exists an \(x \in F(\Spec(R))\) such that \(\xi_n = x|_{\Spec(R/\mathfrak m^n)}\) for all \(n \geq 1\). Lemma 07X8 says that if \(F\) is an algebraic space, then every formal object is effective.

Let \(U\) be a scheme locally of finite type over \(S\) and let \(x \in F(U)\). Let \(u_0 \in U\) be a finite type point. We say that \(x\) is versal at \(u_0\) if and only if \(\xi = (\mathcal{O}_{U, u_0}^\wedge, x|_{\Spec(\mathcal{O}_{U, u_0}/\mathfrak m_{u_0}^n)})\) is a versal formal object in the sense described above.

Let \(S\) be a locally Noetherian scheme. Let \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be a functor. Here are the axioms we will consider on \(F\).

1. a set theoretic condition⁴ to be ignored by readers who are not interested in set theoretical issues,

2. \(F\) is a sheaf for the étale topology,

3. \(F\) is limit preserving,

4. \(F\) satisfies the Rim-Schlessinger condition (RS),

5. every tangent space \(TF_{k, x_0}\) is finite dimensional,

6. every formal object is effective,

7. \(F\) satisfies openness of versality.

Here limit preserving is the notion defined in Limits of Spaces, Definition 049J and openness of versality means the following: Given a scheme \(U\) locally of finite type over \(S\), given \(x \in F(U)\), and given a finite type point \(u_0 \in U\) such that \(x\) is versal at \(u_0\), then there exists an open neighbourhood \(u_0 \in U' \subset U\) such that \(x\) is versal at every finite type point of \(U'\).

Algebraic spaces

The following is our first main result on algebraic spaces.

Proposition

Let \(S\) be a locally Noetherian scheme. Let \(F : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) be a functor. Assume that

1. \(\Delta : F \to F \times F\) is representable by algebraic spaces,

2. \(F\) satisfies axioms [-1], [0], [1], [2], [3], [4], [5] (see Section 07XZ), and

3. \(\mathcal{O}_{S, s}\) is a G-ring for all finite type points \(s\) of \(S\).

Then \(F\) is an algebraic space.

Proof

Lemma 07XY applies to \(F\). Using this we choose, for every finite type field \(k\) over \(S\) and \(x_0 \in F(\Spec(k))\), an affine scheme \(U_{k, x_0}\) of finite type over \(S\) and a smooth morphism \(U_{k, x_0} \to F\) such that there exists a finite type point \(u_{k, x_0} \in U_{k, x_0}\) with residue field \(k\) such that \(x_0\) is the image of \(u_{k, x_0}\). Then \[U = \coprod\nolimits_{k, x_0} U_{k, x_0} \longrightarrow F\] is smooth⁵. To finish the proof it suffices to show this map is surjective, see Bootstrap, Lemma 07WF (this is where we use axiom [0]). By Criteria for Representability, Lemma 07WI it suffices to show that \(U \times_F V \to V\) is surjective for those \(V \to F\) where \(V\) is an affine scheme locally of finite presentation over \(S\). Since \(U \times_F V \to V\) is smooth the image is open. Hence it suffices to show that the image of \(U \times_F V \to V\) contains all finite type points of \(V\), see Morphisms, Lemma 02J4. Let \(v_0 \in V\) be a finite type point. Then \(k = \kappa(v_0)\) is a finite type field over \(S\). Denote by \(x_0\) the composition \(\Spec(k) \xrightarrow{v_0} V \to F\). Then \((u_{k, x_0}, v_0) : \Spec(k) \to U \times_F V\) is a point mapping to \(v_0\) and we win.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(a : F \to G\) be a transformation of functors \((\Sch/S)_{fppf}^{opp} \to \textit{Sets}\). Assume that

1. \(a\) is injective,

2. \(F\) satisfies axioms [0], [1], [2], [4], and [5],

3. \(\mathcal{O}_{S, s}\) is a G-ring for all finite type points \(s\) of \(S\),

4. \(G\) is an algebraic space locally of finite type over \(S\),

Then \(F\) is an algebraic space.

Proof

By Lemma 07X1 the functor \(G\) satisfies [3]. As \(F \to G\) is injective, we conclude that \(F\) also satisfies [3]. Moreover, as \(F \to G\) is injective, we see that given schemes \(U\), \(V\) and morphisms \(U \to F\) and \(V \to F\), then \(U \times_F V = U \times_G V\). Hence \(\Delta : F \to F \times F\) is representable (by schemes) as this holds for \(G\) by assumption. Thus Proposition 07Y1 applies⁶.

Algebraic stacks

Proposition 07Y5 is our first main result on algebraic stacks.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Assume that

1. \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) is representable by algebraic spaces,

2. \(\mathcal{X}\) satisfies axioms [-1], [0], [1], [2], [3] (see Section 07XJ),

3. every formal object of \(\mathcal{X}\) is effective,

4. \(\mathcal{X}\) satisfies openness of versality, and

5. \(\mathcal{O}_{S, s}\) is a G-ring for all finite type points \(s\) of \(S\).

Then \(\mathcal{X}\) is an algebraic stack.

Proof

Lemma 07XY applies to \(\mathcal{X}\). Using this we choose, for every finite type field \(k\) over \(S\) and every isomorphism class of object \(x_0 \in \Ob(\mathcal{X}_{\Spec(k)})\), an affine scheme \(U_{k, x_0}\) of finite type over \(S\) and a smooth morphism \((\Sch/U_{k, x_0})_{fppf} \to \mathcal{X}\) such that there exists a finite type point \(u_{k, x_0} \in U_{k, x_0}\) with residue field \(k\) such that \(x_0\) is the image of \(u_{k, x_0}\). Then \[(\Sch/U)_{fppf} \to \mathcal{X}, \quad\text{with}\quad U = \coprod\nolimits_{k, x_0} U_{k, x_0}\] is smooth⁷. To finish the proof it suffices to show this map is surjective, see Criteria for Representability, Lemma 076V (this is where we use axiom [0]). By Criteria for Representability, Lemma 07WI it suffices to show that \((\Sch/U)_{fppf} \times_\mathcal{X} (\Sch/V)_{fppf} \to (\Sch/V)_{fppf}\) is surjective for those \(y : (\Sch/V)_{fppf} \to \mathcal{X}\) where \(V\) is an affine scheme locally of finite presentation over \(S\). By assumption (1) the fibre product \((\Sch/U)_{fppf} \times_\mathcal{X} (\Sch/V)_{fppf}\) is representable by an algebraic space \(W\). Then \(W \to V\) is smooth, hence the image is open. Hence it suffices to show that the image of \(W \to V\) contains all finite type points of \(V\), see Morphisms, Lemma 02J4. Let \(v_0 \in V\) be a finite type point. Then \(k = \kappa(v_0)\) is a finite type field over \(S\). Set \(x_0 = y|_{\Spec(k)}\), the pullback of \(y\) by \(v_0\). Then \((u_{k, x_0}, v_0)\) will give a morphism \(\Spec(k) \to W\) whose composition with \(W \to V\) is \(v_0\) and we win.

Proposition

Let \(S\) be a locally Noetherian scheme. Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Assume that

1. \(\Delta_\Delta : \mathcal{X} \to \mathcal{X} \times_{\mathcal{X} \times \mathcal{X}} \mathcal{X}\) is representable by algebraic spaces,

2. \(\mathcal{X}\) satisfies axioms [-1], [0], [1], [2], [3], [4], and [5] (see Section 07XJ),

3. \(\mathcal{O}_{S, s}\) is a G-ring for all finite type points \(s\) of \(S\).

Then \(\mathcal{X}\) is an algebraic stack.

Proof

We first prove that \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) is representable by algebraic spaces. To do this it suffices to show that \[\mathcal{Y} = \mathcal{X} \times_{\Delta, \mathcal{X} \times \mathcal{X}, y} (\Sch/V)_{fppf}\] is representable by an algebraic space for any affine scheme \(V\) locally of finite presentation over \(S\) and object \(y\) of \(\mathcal{X} \times \mathcal{X}\) over \(V\), see Criteria for Representability, Lemma 07WH⁸. Observe that \(\mathcal{Y}\) is fibred in setoids (Stacks, Lemma 04SI) and let \(Y : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\), \(T \mapsto \Ob(\mathcal{Y}_T)/\cong\) be the functor of isomorphism classes. We will apply Proposition 07Y1 to see that \(Y\) is an algebraic space.

Note that \(\Delta_\mathcal{Y} : \mathcal{Y} \to \mathcal{Y} \times \mathcal{Y}\) (and hence also \(Y \to Y \times Y\)) is representable by algebraic spaces by condition (1) and Criteria for Representability, Lemma 07WG. Observe that \(Y\) is a sheaf for the étale topology by Stacks, Lemmas 0432 and 05UI, i.e., axiom [0] holds. Also \(Y\) is limit preserving by Lemma 07XM, i.e., we have [1]. Note that \(Y\) has (RS), i.e., axiom [2] holds, by Lemmas 07WQ and 07WR. Axiom [3] for \(Y\) follows from Lemmas 07X1 and 07X2. Axiom [4] follows from Lemmas 07X8 and 07X9. Axiom [5] for \(Y\) follows directly from openness of versality for \(\Delta_\mathcal{X}\) which is part of axiom [5] for \(\mathcal{X}\). Thus all the assumptions of Proposition 07Y1 are satisfied and \(Y\) is an algebraic space.

At this point it follows from Lemma 07Y4 that \(\mathcal{X}\) is an algebraic stack.

Strong Rim-Schlessinger

In the rest of this chapter the following strictly stronger version of the Rim-Schlessinger conditions will play an important role.

Definition

Let \(S\) be a scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). We say \(\mathcal{X}\) satisfies condition (RS*) if given a fibre product diagram \[\xymatrix{ B' \ar[r] & B \\ A' = A \times_B B' \ar[u] \ar[r] & A \ar[u] }\] of \(S\)-algebras, with \(B' \to B\) surjective with square zero kernel, the functor of fibre categories \[\mathcal{X}_{\Spec(A')} \longrightarrow \mathcal{X}_{\Spec(A)} \times_{\mathcal{X}_{\Spec(B)}} \mathcal{X}_{\Spec(B')}\] is an equivalence of categories.

We make some observations: with \(A \to B \leftarrow B'\) as in Definition 07Y8

1. we have \(\Spec(A') = \Spec(A) \amalg_{\Spec(B)} \Spec(B')\) in the category of schemes, see More on Morphisms, Lemma 07RT, and

2. if \(\mathcal{X}\) is an algebraic stack, then \(\mathcal{X}\) satisfies (RS*) by Lemma 0CXP.

If \(S\) is locally Noetherian, then

1. if \(A\), \(B\), \(B'\) are of finite type over \(S\) and \(B\) is finite over \(A\), then \(A'\) is of finite type over \(S\)⁹, and

2. if \(\mathcal{X}\) satisfies (RS*), then \(\mathcal{X}\) satisfies (RS) because (RS) covers exactly those cases of (RS*) where \(A\), \(B\), \(B'\) are Artinian local.

Lemma

Let \(\mathcal{X}\) be an algebraic stack over a base \(S\). Then \(\mathcal{X}\) satisfies (RS*).

Proof

This is implied by Lemma 07WN, see remarks following Definition 07Y8.

Lemma

Let \(S\) be a scheme. Let \(p : \mathcal{X} \to \mathcal{Y}\) and \(q : \mathcal{Z} \to \mathcal{Y}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). If \(\mathcal{X}\), \(\mathcal{Y}\), and \(\mathcal{Z}\) satisfy (RS*), then so does \(\mathcal{X} \times_\mathcal{Y} \mathcal{Z}\).

Proof

The proof is exactly the same as the proof of Lemma 07WR.

Versality and generalizations

We prove that versality is preserved under generalizations for stacks which have (RS*) and are limit preserving. We suggest skipping this section on a first reading.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\) having (RS*). Let \(x\) be an object of \(\mathcal{X}\) over an affine scheme \(U\) of finite type over \(S\). Let \(u \in U\) be a finite type point such that \(x\) is not versal at \(u\). Then there exists a morphism \(x \to y\) of \(\mathcal{X}\) lying over \(U \to T\) satisfying

1. the morphism \(U \to T\) is a first order thickening,

2. we have a short exact sequence \[0 \to \kappa(u) \to \mathcal{O}_T \to \mathcal{O}_U \to 0\]

3. there does not exist a pair \((W, \beta)\) consisting of an open neighbourhood \(W \subset T\) of \(u\) and a morphism \(\beta : y|_W \to x\) such that the composition \[x|_{U \cap W} \xrightarrow{\text{restriction of }x \to y} y|_W \xrightarrow{\beta} x\] is the canonical morphism \(x|_{U \cap W} \to x\).

Proof

Let \(R = \mathcal{O}_{U, u}^\wedge\). Let \(k = \kappa(u)\) be the residue field of \(R\). Let \(\xi\) be the formal object of \(\mathcal{X}\) over \(R\) associated to \(x\). Since \(x\) is not versal at \(u\), we see that \(\xi\) is not versal, see Lemma 0CXK. By the discussion following Definition 0CXJ this means we can find morphisms \(\xi_1 \to x_A \to x_B\) of \(\mathcal{X}\) lying over closed immersions \(\Spec(k) \to \Spec(A) \to \Spec(B)\) where \(A, B\) are Artinian local rings with residue field \(k\), an \(n \geq 1\) and a commutative diagram \[\vcenter{ \xymatrix{ & x_A \ar[ld] \\ \xi_n & \xi_1 \ar[u] \ar[l] } } \quad\text{lying over}\quad \vcenter{ \xymatrix{ & \Spec(A) \ar[ld] \\ \Spec(R/\mathfrak m^n) & \Spec(k) \ar[u] \ar[l] } }\] such that there does not exist an \(m \geq n\) and a commutative diagram \[\vcenter{ \xymatrix{ & & x_B \ar[lldd] \\ & & x_A \ar[ld] \ar[u] \\ \xi_m & \xi_n \ar[l] & \xi_1 \ar[u] \ar[l] } } \quad\text{lying over} \vcenter{ \xymatrix{ & & \Spec(B) \ar[lldd] \\ & & \Spec(A) \ar[ld] \ar[u] \\ \Spec(R/\mathfrak m^m) & \Spec(R/\mathfrak m^n) \ar[l] & \Spec(k) \ar[u] \ar[l] } }\] We may moreover assume that \(B \to A\) is a small extension, i.e., that the kernel \(I\) of the surjection \(B \to A\) is isomorphic to \(k\) as an \(A\)-module. This follows from Formal Deformation Theory, Remark 06HS. Then we simply define \[T = U \amalg_{\Spec(A)} \Spec(B)\] By property (RS*) we find \(y\) over \(T\) whose restriction to \(\Spec(B)\) is \(x_B\) and whose restriction to \(U\) is \(x\) (this gives the arrow \(x \to y\) lying over \(U \to T\)). To finish the proof we verify conditions (1), (2), and (3).

By the construction of the pushout we have a commutative diagram \[\xymatrix{ 0 \ar[r] & I \ar[r] & B \ar[r] & A \ar[r] & 0 \\ 0 \ar[r] & I \ar[r] \ar[u] & \Gamma(T, \mathcal{O}_T) \ar[r] \ar[u] & \Gamma(U, \mathcal{O}_U) \ar[r] \ar[u] & 0 }\] with exact rows. This immediately proves (1) and (2). To finish the proof we will argue by contradiction. Assume we have a pair \((W, \beta)\) as in (3). Since \(\Spec(B) \to T\) factors through \(W\) we get the morphism \[x_B \to y|_W \xrightarrow{\beta} x\] Since \(B\) is Artinian local with residue field \(k = \kappa(u)\) we see that \(x_B \to x\) lies over a morphism \(\Spec(B) \to U\) which factors through \(\Spec(\mathcal{O}_{U, u}/\mathfrak m_u^m)\) for some \(m \geq n\). In other words, \(x_B \to x\) factors through \(\xi_m\) giving a map \(x_B \to \xi_m\). The compatibility condition on the morphism \(\beta\) in condition (3) translates into the condition that \[\xymatrix{ x_B \ar[d] & x_A \ar[d] \ar[l] \\ \xi_m & \xi_n \ar[l] }\] is commutative. This gives the contradiction we were looking for.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Assume

1. \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) is representable by algebraic spaces,

2. \(\mathcal{X}\) has (RS*),

3. \(\mathcal{X}\) is limit preserving.

Let \(x\) be an object of \(\mathcal{X}\) over a scheme \(U\) of finite type over \(S\). Let \(u \leadsto u_0\) be a specialization of finite type points of \(U\) such that \(x\) is versal at \(u_0\). Then \(x\) is versal at \(u\).

Proof

After shrinking \(U\) we may assume \(U\) is affine and \(U\) maps into an affine open \(\Spec(\Lambda)\) of \(S\). If \(x\) is not versal at \(u\) then we may pick \(x \to y\) lying over \(U \to T\) as in Lemma 0G2J. Write \(U = \Spec(R_0)\) and \(T = \Spec(R)\). The morphism \(U \to T\) corresponds to a surjective ring map \(R \to R_0\) whose kernel is an ideal of square zero. By assumption (3) we get that \(y\) comes from an object \(x'\) over \(U' = \Spec(R')\) for some finite type \(\Lambda\)-subalgebra \(R' \subset R\). After increasing \(R'\) we may and do assume that \(R' \to R_0\) is surjective, so that \(U \subset U'\) is a first order thickening. Thus we now have \[x \to y \to x' \text{ lying over } U \to T \to U'\] By assumption (1) there is an algebraic space \(Z\) over \(S\) representing \[(\Sch/U)_{fppf} \times_{x, \mathcal{X}, x'} (\Sch/U')_{fppf}\] see Algebraic Stacks, Lemma 045G. By construction of \(2\)-fibre products, a \(V\)-valued point of \(Z\) corresponds to a triple \((a, a', \alpha)\) consisting of morphisms \(a : V \to U\), \(a' : V \to U'\) and a morphism \(\alpha : a^*x \to (a')^*x'\). We obtain a commutative diagram \[\xymatrix{ U \ar[rd] \ar[rdd] \ar[rrd] \\ & Z \ar[r]_{p'} \ar[d]^p & U' \ar[d] \\ & U \ar[r] & S }\] The morphism \(i : U \to Z\) comes from the isomorphism \(x \to x'|_U\). Let \(z_0 = i(u_0) \in Z\). By Lemma 0CXM we see that \(Z \to U'\) is smooth at \(z_0\). After replacing \(U\) by an affine open neighbourhood of \(u_0\), replacing \(U'\) by the corresponding open, and replacing \(Z\) by the intersection of the inverse images of these opens by \(p\) and \(p'\), we reach the situation where \(Z \to U'\) is smooth along \(i(U)\). Since \(u \leadsto u_0\) the point \(u\) is in this open. Condition (3) of Lemma 0G2J is clearly preserved by shrinking \(U\) (all of the schemes \(U\), \(T\), \(U'\) have the same underlying topological space). Since \(U \to U'\) is a first order thickening of affine schemes, we can choose a morphism \(i' : U' \to Z\) such that \(p' \circ i' = \text{id}_{U'}\) and whose restriction to \(U\) is \(i\) (More on Morphisms of Spaces, Lemma 04AM). Pulling back the universal morphism \(p^*x \to (p')^*x'\) by \(i'\) we obtain a morphism \[x' \to x\] lying over \(p \circ i' : U' \to U\) such that the composition \[x \to x' \to x\] is the identity. Recall that we have \(y \to x'\) lying over the morphism \(T \to U'\). Composing we get a morphism \(y \to x\) whose existence contradicts condition (3) of Lemma 0G2J. This contradiction finishes the proof.

Strong formal effectiveness

In this section we demonstrate how a strong version of effectiveness of formal objects implies openness of versality. The proof of [Bhatt-Algebraize, Theorem 1.1] shows that quasi-compact and quasi-separated algebraic spaces satisfy the strong formal effectiveness discussed in Remark 0CXT. In addition, the theory we develop is nonempty: we use it later to show openness of versality for the stack of coherent sheaves and for moduli of complexes, see Quot, Theorems 09DS and 0DLN.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\) having (RS*). Let \(x\) be an object of \(\mathcal{X}\) over an affine scheme \(U\) of finite type over \(S\). Let \(u_n \in U\), \(n \geq 1\) be finite type points such that (a) there are no specializations \(u_n \leadsto u_m\) for \(n \not = m\), and (b) \(x\) is not versal at \(u_n\) for all \(n\). Then there exist morphisms \[x \to x_1 \to x_2 \to \ldots \quad\text{in }\mathcal{X}\text{ lying over }\quad U \to U_1 \to U_2 \to \ldots\] over \(S\) such that

1. for each \(n\) the morphism \(U \to U_n\) is a first order thickening,

2. for each \(n\) we have a short exact sequence \[0 \to \kappa(u_n) \to \mathcal{O}_{U_n} \to \mathcal{O}_{U_{n - 1}} \to 0\] with \(U_0 = U\) for \(n = 1\),

3. for each \(n\) there does not exist a pair \((W, \alpha)\) consisting of an open neighbourhood \(W \subset U_n\) of \(u_n\) and a morphism \(\alpha : x_n|_W \to x\) such that the composition \[x|_{U \cap W} \xrightarrow{\text{restriction of }x \to x_n} x_n|_W \xrightarrow{\alpha} x\] is the canonical morphism \(x|_{U \cap W} \to x\).

Proof

Since there are no specializations among the points \(u_n\) (and in particular the \(u_n\) are pairwise distinct), for every \(n\) we can find an open \(U' \subset U\) such that \(u_n \in U'\) and \(u_i \not \in U'\) for \(i = 1, \ldots, n - 1\). By Lemma 0G2J for each \(n \geq 1\) we can find \[x \to y_n \quad\text{in }\mathcal{X}\text{ lying over}\quad U \to T_n\] such that

1. the morphism \(U \to T_n\) is a first order thickening,

2. we have a short exact sequence \[0 \to \kappa(u_n) \to \mathcal{O}_{T_n} \to \mathcal{O}_U \to 0\]

3. there does not exist a pair \((W, \beta)\) consisting of an open neighbourhood \(W \subset T_n\) of \(u_n\) and a morphism \(\beta : y_n|_W \to x\) such that the composition \[x|_{U \cap W} \xrightarrow{\text{restriction of }x \to y_n} y_n|_W \xrightarrow{\beta} x\] is the canonical morphism \(x|_{U \cap W} \to x\).

Thus we can define inductively \[U_1 = T_1, \quad U_{n + 1} = U_n \amalg_U T_{n + 1}\] Setting \(x_1 = y_1\) and using (RS*) we find inductively \(x_{n + 1}\) over \(U_{n + 1}\) restricting to \(x_n\) over \(U_n\) and \(y_{n + 1}\) over \(T_{n + 1}\). Property (1) for \(U \to U_n\) follows from the construction of the pushout in More on Morphisms, Lemma 07RT. Property (2) for \(U_n\) similarly follows from property (2) for \(T_n\) by the construction of the pushout. After shrinking to an open neighbourhood \(U'\) of \(u_n\) as discussed above, property (3) for \((U_n, x_n)\) follows from property (3) for \((T_n, y_n)\) simply because the corresponding open subschemes of \(T_n\) and \(U_n\) are isomorphic. Some details omitted.

Remark

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Assume we have

1. an affine open \(\Spec(\Lambda) \subset S\),

2. an inverse system \((R_n)\) of \(\Lambda\)-algebras with surjective transition maps whose kernels are locally nilpotent,

3. a system \((\xi_n)\) of objects of \(\mathcal{X}\) lying over the system \((\Spec(R_n))\).

In this situation, set \(R = \lim R_n\). We say that \((\xi_n)\) is effective if there exists an object \(\xi\) of \(\mathcal{X}\) over \(\Spec(R)\) whose restriction to \(\Spec(R_n)\) gives the system \((\xi_n)\).

It is not the case that every algebraic stack \(\mathcal{X}\) over \(S\) satisfies a strong effectiveness axiom of the form: every system \((\xi_n)\) as in Remark 0CXT is effective. An example is given in Examples, Section 0CXW.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Assume

1. \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) is representable by algebraic spaces,

2. \(\mathcal{X}\) has (RS*),

3. \(\mathcal{X}\) is limit preserving,

4. systems \((\xi_n)\) as in Remark 0CXT where \(\Ker(R_m \to R_n)\) is an ideal of square zero for all \(m \geq n\) are effective.

Then \(\mathcal{X}\) satisfies openness of versality.

Proof

Choose a scheme \(U\) locally of finite type over \(S\), a finite type point \(u_0\) of \(U\), and an object \(x\) of \(\mathcal{X}\) over \(U\) such that \(x\) is versal at \(u_0\). After shrinking \(U\) we may assume \(U\) is affine and \(U\) maps into an affine open \(\Spec(\Lambda)\) of \(S\). Let \(E \subset U\) be the set of finite type points \(u\) such that \(x\) is not versal at \(u\). By Lemma 0G2K if \(u \in E\) then \(u_0\) is not a specialization of \(u\). If openness of versality does not hold, then \(u_0\) is in the closure \(\overline{E}\) of \(E\). By Properties, Lemma 0G2F we may choose a countable subset \(E' \subset E\) with the same closure as \(E\). By Properties, Lemma 0G2R we may assume there are no specializations among the points of \(E'\). Observe that \(E'\) has to be (countably) infinite as \(u_0\) isn’t the specialization of any point of \(E'\) as pointed out above. Thus we can write \(E' = \{u_1, u_2, u_3, \ldots\}\), there are no specializations among the \(u_i\), and \(u_0\) is in the closure of \(E'\).

Choose \(x \to x_1 \to x_2 \to \ldots\) lying over \(U \to U_1 \to U_2 \to \ldots\) as in Lemma 0G2S. Write \(U_n = \Spec(R_n)\) and \(U = \Spec(R_0)\). Set \(R = \lim R_n\). Observe that \(R \to R_0\) is surjective with kernel an ideal of square zero. By assumption (4) we get \(\xi\) over \(\Spec(R)\) whose base change to \(R_n\) is \(x_n\). By assumption (3) we get that \(\xi\) comes from an object \(\xi'\) over \(U' = \Spec(R')\) for some finite type \(\Lambda\)-subalgebra \(R' \subset R\). After increasing \(R'\) we may and do assume that \(R' \to R_0\) is surjective, so that \(U \subset U'\) is a first order thickening. Thus we now have \[x \to x_1 \to x_2 \to \ldots \to \xi' \text{ lying over } U \to U_1 \to U_2 \to \ldots \to U'\] By assumption (1) there is an algebraic space \(Z\) over \(S\) representing \[(\Sch/U)_{fppf} \times_{x, \mathcal{X}, \xi'} (\Sch/U')_{fppf}\] see Algebraic Stacks, Lemma 045G. By construction of \(2\)-fibre products, a \(T\)-valued point of \(Z\) corresponds to a triple \((a, a', \alpha)\) consisting of morphisms \(a : T \to U\), \(a' : T \to U'\) and a morphism \(\alpha : a^*x \to (a')^*\xi'\). We obtain a commutative diagram \[\xymatrix{ U \ar[rd] \ar[rdd] \ar[rrd] \\ & Z \ar[r]_{p'} \ar[d]^p & U' \ar[d] \\ & U \ar[r] & S }\] The morphism \(i : U \to Z\) comes from the isomorphism \(x \to \xi'|_U\). Let \(z_0 = i(u_0) \in Z\). By Lemma 0CXM we see that \(Z \to U'\) is smooth at \(z_0\). After replacing \(U\) by an affine open neighbourhood of \(u_0\), replacing \(U'\) by the corresponding open, and replacing \(Z\) by the intersection of the inverse images of these opens by \(p\) and \(p'\), we reach the situation where \(Z \to U'\) is smooth along \(i(U)\). Note that this also involves replacing \(u_n\) by a subsequence, namely by those indices such that \(u_n\) is in the open. Moreover, condition (3) of Lemma 0G2S is clearly preserved by shrinking \(U\) (all of the schemes \(U\), \(U_n\), \(U'\) have the same underlying topological space). Since \(U \to U'\) is a first order thickening of affine schemes, we can choose a morphism \(i' : U' \to Z\) such that \(p' \circ i' = \text{id}_{U'}\) and whose restriction to \(U\) is \(i\) (More on Morphisms of Spaces, Lemma 04AM). Pulling back the universal morphism \(p^*x \to (p')^*\xi'\) by \(i'\) we obtain a morphism \[\xi' \to x\] lying over \(p \circ i' : U' \to U\) such that the composition \[x \to \xi' \to x\] is the identity. Recall that we have \(x_1 \to \xi'\) lying over the morphism \(U_1 \to U'\). Composing we get a morphism \(x_1 \to x\) whose existence contradicts condition (3) of Lemma 0G2S. This contradiction finishes the proof.

Remark

There is a way to deduce openness of versality of the diagonal of a category fibred in groupoids from a strong formal effectiveness axiom. Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Assume

1. \(\Delta_\Delta : \mathcal{X} \to \mathcal{X} \times_{\mathcal{X} \times \mathcal{X}} \mathcal{X}\) is representable by algebraic spaces,

2. \(\mathcal{X}\) has (RS*),

3. \(\mathcal{X}\) is limit preserving,

4. given an inverse system \((R_n)\) of \(S\)-algebras as in Remark 0CXT where \(\Ker(R_m \to R_n)\) is an ideal of square zero for all \(m \geq n\) the functor \[\mathcal{X}_{\Spec(\lim R_n)} \longrightarrow \lim_n \mathcal{X}_{\Spec(R_n)}\] is fully faithful.

Then \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) satisfies openness of versality. This follows by applying Lemma 0CXU to fibre products of the form \(\mathcal{X} \times_{\Delta, \mathcal{X} \times \mathcal{X}, y} (\Sch/V)_{fppf}\) for any affine scheme \(V\) locally of finite presentation over \(S\) and object \(y\) of \(\mathcal{X} \times \mathcal{X}\) over \(V\). If we ever need this, we will change this remark into a lemma and provide a detailed proof.

Infinitesimal deformations

In this section we discuss a generalization of the notion of the tangent space introduced in Section 07WY. To do this intelligently, we borrow some notation from Formal Deformation Theory, Sections 06I2, 06JD, and 06JN.

Let \(S\) be a scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Given a homomorphism \(A' \to A\) of \(S\)-algebras and an object \(x\) of \(\mathcal{X}\) over \(\Spec(A)\) we write \(\textit{Lift}(x, A')\) for the category of lifts of \(x\) to \(\Spec(A')\). An object of \(\textit{Lift}(x, A')\) is a morphism \(x \to x'\) of \(\mathcal{X}\) lying over \(\Spec(A) \to \Spec(A')\) and morphisms of \(\textit{Lift}(x, A')\) are defined as commutative diagrams. The set of isomorphism classes of \(\textit{Lift}(x, A')\) is denoted \(\text{Lift}(x, A')\). See Formal Deformation Theory, Definition 06JE and Remark 06JF. If \(A' \to A\) is surjective with locally nilpotent kernel we call an element \(x'\) of \(\text{Lift}(x, A')\) a (infinitesimal) deformation of \(x\). In this case the group of infinitesimal automorphisms of \(x'\) over \(x\) is the kernel \[\text{Inf}(x'/x) = \Ker\left( \text{Aut}_{\mathcal{X}_{\Spec(A')}}(x') \to \text{Aut}_{\mathcal{X}_{\Spec(A)}}(x)\right)\] Note that an element of \(\text{Inf}(x'/x)\) is the same thing as a lift of \(\text{id}_x\) over \(\Spec(A')\) for (the category fibred in sets associated to) \(\mathit{Aut}_\mathcal{X}(x')\). Compare with Formal Deformation Theory, Definition 06JP and Formal Deformation Theory, Remark 06JW.

If \(M\) is an \(A\)-module we denote by \(A[M]\) the \(A\)-algebra whose underlying \(A\)-module is \(A \oplus M\) and whose multiplication is given by \((a, m) \cdot (a', m') = (aa', am' + a'm)\). When \(M = A\) this is the ring of dual numbers over \(A\), which we denote \(A[\epsilon]\) as is customary. There is an \(A\)-algebra map \(A[M] \to A\). The pullback of \(x\) to \(\Spec(A[M])\) is called the trivial deformation of \(x\) to \(\Spec(A[M])\).

Lemma

Let \(S\) be a scheme. Let \(f : \mathcal{X} \to \mathcal{Y}\) be a \(1\)-morphism of categories fibred in groupoids over \((\Sch/S)_{fppf}\). Let \[\xymatrix{ B' \ar[r] & B \\ A' \ar[u] \ar[r] & A \ar[u] }\] be a commutative diagram of \(S\)-algebras. Let \(x\) be an object of \(\mathcal{X}\) over \(\Spec(A)\), let \(y\) be an object of \(\mathcal{Y}\) over \(\Spec(B)\), and let \(\phi : f(x)|_{\Spec(B)} \to y\) be a morphism of \(\mathcal{Y}\) over \(\Spec(B)\). Then there is a canonical functor \[\textit{Lift}(x, A') \longrightarrow \textit{Lift}(y, B')\] of categories of lifts induced by \(f\) and \(\phi\). The construction is compatible with compositions of \(1\)-morphisms of categories fibred in groupoids in an obvious manner.

Proof

This lemma proves itself.

Let \(S\) be a base scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). We define a category whose objects are pairs \((x, A' \to A)\) where

1. \(A' \to A\) is a surjection of \(S\)-algebras whose kernel is an ideal of square zero,

2. \(x\) is an object of \(\mathcal{X}\) lying over \(\Spec(A)\).

A morphism \((y, B' \to B) \to (x, A' \to A)\) is given by a commutative diagram \[\xymatrix{ B' \ar[r] & B \\ A' \ar[u] \ar[r] & A \ar[u] }\] of \(S\)-algebras together with a morphism \(x|_{\Spec(B)} \to y\) over \(\Spec(B)\). Let us call this the category of deformation situations.

Lemma

Let \(S\) be a scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Assume \(\mathcal{X}\) satisfies condition (RS*). Let \(A\) be an \(S\)-algebra and let \(x\) be an object of \(\mathcal{X}\) over \(\Spec(A)\).

1. There exists an \(A\)-linear functor \(\text{Inf}_x : \text{Mod}_A \to \text{Mod}_A\) such that given a deformation situation \((x, A' \to A)\) and a lift \(x'\) there is an isomorphism \(\text{Inf}_x(I) \to \text{Inf}(x'/x)\) where \(I = \Ker(A' \to A)\).

2. There exists an \(A\)-linear functor \(T_x : \text{Mod}_A \to \text{Mod}_A\) such that

   1. given \(M\) in \(\text{Mod}_A\) there is a bijection \(T_x(M) \to \text{Lift}(x, A[M])\),

   2. given a deformation situation \((x, A' \to A)\) there is an action \[T_x(I) \times \text{Lift}(x, A') \to \text{Lift}(x, A')\] where \(I = \Ker(A' \to A)\). It is simply transitive if \(\text{Lift}(x, A') \not = \emptyset\).

Proof

We define \(\text{Inf}_x\) as the functor \[\text{Mod}_A \longrightarrow \textit{Sets},\quad M \longrightarrow \text{Inf}(x'_M/x) = \text{Lift}(\text{id}_x, A[M])\] mapping \(M\) to the group of infinitesimal automorphisms of the trivial deformation \(x'_M\) of \(x\) to \(\Spec(A[M])\) or equivalently the group of lifts of \(\text{id}_x\) in \(\mathit{Aut}_\mathcal{X}(x'_M)\). We define \(T_x\) as the functor \[\text{Mod}_A \longrightarrow \textit{Sets},\quad M \longrightarrow \text{Lift}(x, A[M])\] of isomorphism classes of infinitesimal deformations of \(x\) to \(\Spec(A[M])\). We apply Formal Deformation Theory, Lemma 06I6 to \(\text{Inf}_x\) and \(T_x\). This lemma is applicable, since (RS*) tells us that \[\textit{Lift}(x, A[M \times N]) = \textit{Lift}(x, A[M]) \times \textit{Lift}(x, A[N])\] as categories (and trivial deformations match up too).

Let \((x, A' \to A)\) be a deformation situation. Consider the ring map \(g : A' \times_A A' \to A[I]\) defined by the rule \(g(a_1, a_2) = \overline{a_1} \oplus a_2 - a_1\). There is an isomorphism \[A' \times_A A' \longrightarrow A' \times_A A[I]\] given by \((a_1, a_2) \mapsto (a_1, g(a_1, a_2))\). This isomorphism commutes with the projections to \(A'\) on the first factor, and hence with the projections to \(A\). Thus applying (RS*) twice we find equivalences of categories \[\begin{align*} \textit{Lift}(x, A') \times \textit{Lift}(x, A') & = \textit{Lift}(x, A' \times_A A') \\ & = \textit{Lift}(x, A' \times_A A[I]) \\ & = \textit{Lift}(x, A') \times \textit{Lift}(x, A[I]) \end{align*}\] Using these maps and projection onto the last factor of the last product we see that we obtain “difference maps” \[\text{Inf}(x'/x) \times \text{Inf}(x'/x) \longrightarrow \text{Inf}_x(I) \quad\text{and}\quad \text{Lift}(x, A') \times \text{Lift}(x, A') \longrightarrow T_x(I)\] These difference maps satisfy the transitivity rule “\((x'_1 - x'_2) + (x'_2 - x'_3) = x'_1 - x'_3\)” because \[\xymatrix{ A' \times_A A' \times_A A' \ar[rrrrr]_-{(a_1, a_2, a_3) \mapsto (g(a_1, a_2), g(a_2, a_3))} \ar[rrrrrd]_{(a_1, a_2, a_3) \mapsto g(a_1, a_3)} & & & & & A[I] \times_A A[I] = A[I \times I] \ar[d]^{+} \\ & & & & & A[I] }\] is commutative. Inverting the string of equivalences above we obtain an action which is free and transitive provided \(\text{Inf}(x'/x)\), resp. \(\text{Lift}(x, A')\) is nonempty. Note that \(\text{Inf}(x'/x)\) is always nonempty as it is a group.

Remark

Assumptions and notation as in Lemma 07Y9. Suppose \(A \to B\) is a ring map and \(y = x|_{\Spec(B)}\). Let \(M \in \text{Mod}_A\), \(N \in \text{Mod}_B\) and let \(M \to N\) be an \(A\)-linear map. Then there are canonical maps \(\text{Inf}_x(M) \to \text{Inf}_y(N)\) and \(T_x(M) \to T_y(N)\) simply because there is a pullback functor \[\textit{Lift}(x, A[M]) \to \textit{Lift}(y, B[N])\] coming from the ring map \(A[M] \to B[N]\). Similarly, given a morphism of deformation situations \((y, B' \to B) \to (x, A' \to A)\) we obtain a pullback functor \(\textit{Lift}(x, A') \to \textit{Lift}(y, B')\). Since the construction of the action, the addition, and the scalar multiplication on \(\text{Inf}_x\) and \(T_x\) use only morphisms in the categories of lifts (see proof of Formal Deformation Theory, Lemma 06I6) we see that the constructions above are functorial. In other words we obtain \(A\)-linear maps \[\text{Inf}_x(M) \to \text{Inf}_y(N) \quad\text{and}\quad T_x(M) \to T_y(N)\] such that the diagrams \[\vcenter{ \xymatrix{ \text{Inf}_y(J) \ar[r] & \text{Inf}(y'/y) \\ \text{Inf}_x(I) \ar[r] \ar[u] & \text{Inf}(x'/x) \ar[u] } } \quad\text{and}\quad \vcenter{ \xymatrix{ T_y(J) \times \text{Lift}(y, B') \ar[r] & \text{Lift}(y, B') \\ T_x(I) \times \text{Lift}(x, A') \ar[r] \ar[u] & \text{Lift}(x, A') \ar[u] } }\] commute. Here \(I = \Ker(A' \to A)\), \(J = \Ker(B' \to B)\), \(x'\) is a lift of \(x\) to \(A'\) (which may not always exist) and \(y' = x'|_{\Spec(B')}\).

Remark

Assumptions and notation as in Lemma 07Y9. Let \(x', x''\) be lifts of \(x\) to \(A'\). Then we have a composition map \[\text{Inf}(x'/x) \times \Mor_{\textit{Lift}(x, A')}(x', x'') \times \text{Inf}(x''/x) \longrightarrow \Mor_{\textit{Lift}(x, A')}(x', x'').\] Since \(\textit{Lift}(x, A')\) is a groupoid, if \(\Mor_{\textit{Lift}(x, A')}(x', x'')\) is nonempty, then this defines a simply transitive left action of \(\text{Inf}(x'/x)\) on \(\Mor_{\textit{Lift}(x, A')}(x', x'')\) and a simply transitive right action by \(\text{Inf}(x''/x)\). Now the lemma says that \(\text{Inf}(x'/x) = \text{Inf}_x(I) = \text{Inf}(x''/x)\). We claim that the two actions described above agree via these identifications. Namely, either \(x' \not \cong x''\) in which the claim is clear, or \(x' \cong x''\) and in that case we may assume that \(x'' = x'\) in which case the result follows from the fact that \(\text{Inf}(x'/x)\) is commutative. In particular, we obtain a well defined action \[\text{Inf}_x(I) \times \Mor_{\textit{Lift}(x, A')}(x', x'') \longrightarrow \Mor_{\textit{Lift}(x, A')}(x', x'')\] which is simply transitive as soon as \(\Mor_{\textit{Lift}(x, A')}(x', x'')\) is nonempty.

Remark

Let \(S\) be a scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Let \(A\) be an \(S\)-algebra. There is a notion of a short exact sequence \[(x, A_1' \to A) \to (x, A_2' \to A) \to (x, A_3' \to A)\] of deformation situations: we ask that the corresponding maps between the kernels \(I_i = \Ker(A_i' \to A)\) give a short exact sequence \[0 \to I_3 \to I_2 \to I_1 \to 0\] of \(A\)-modules. Note that in this case the map \(A_3' \to A_1'\) factors through \(A\), hence there is a canonical isomorphism \(A_1' = A[I_1]\).

Lemma

Let \(S\) be a scheme. Let \(p : \mathcal{X} \to \mathcal{Y}\) and \(q : \mathcal{Z} \to \mathcal{Y}\) be \(1\)-morphisms of categories fibred in groupoids over \((\Sch/S)_{fppf}\). Assume \(\mathcal{X}\), \(\mathcal{Y}\), \(\mathcal{Z}\) satisfy (RS*). Let \(A\) be an \(S\)-algebra and let \(w\) be an object of \(\mathcal{W} = \mathcal{X} \times_\mathcal{Y} \mathcal{Z}\) over \(A\). Denote by \(x, y, z\) the objects of \(\mathcal{X}, \mathcal{Y}, \mathcal{Z}\) obtained from \(w\). For any \(A\)-module \(M\) there is a \(6\)-term exact sequence \[\xymatrix{ 0 \ar[r] & \text{Inf}_w(M) \ar[r] & \text{Inf}_x(M) \oplus \text{Inf}_z(M) \ar[r] & \text{Inf}_y(M) \ar[lld] \\ & T_w(M) \ar[r] & T_x(M) \oplus T_z(M) \ar[r] & T_y(M) }\] of \(A\)-modules.

Proof

By Lemma 0CXQ we see that \(\mathcal{W}\) satisfies (RS*) and hence \(T_w(M)\) and \(\text{Inf}_w(M)\) are defined. The horizontal arrows are defined using the functoriality of Lemma 07Y7.

Definition of the “boundary” map \(\delta : \text{Inf}_y(M) \to T_w(M)\). Choose isomorphisms \(p(x) \to y\) and \(y \to q(z)\) such that \(w = (x, z, p(x) \to y \to q(z))\) in the description of the \(2\)-fibre product of Categories, Lemma 0041 and more precisely Categories, Lemma 0040. Let \(x', y', z', w'\) denote the trivial deformation of \(x, y, z, w\) over \(A[M]\). By pullback we get isomorphisms \(y' \to p(x')\) and \(q(z') \to y'\). An element \(\alpha \in \text{Inf}_y(M)\) is the same thing as an automorphism \(\alpha : y' \to y'\) over \(A[M]\) which restricts to the identity on \(y\) over \(A\). Thus setting \[\delta(\alpha) = (x', z', p(x') \to y' \xrightarrow{\alpha} y' \to q(z'))\] we obtain an object of \(T_w(M)\). This is a map of \(A\)-modules by Formal Deformation Theory, Lemma 06I7.

The rest of the proof is exactly the same as the proof of Formal Deformation Theory, Lemma 06L5.

Remark

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Assume \(\mathcal{X}\) has (RS*). Let \(k\) be a field of finite type over \(S\) and let \(x_0\) be an object of \(\mathcal{X}\) over \(\Spec(k)\). Then we have equalities of \(k\)-vector spaces \[T\mathcal{F}_{\mathcal{X}, k, x_0} = T_{x_0}(k) \quad\text{and}\quad \text{Inf}(\mathcal{F}_{\mathcal{X}, k, x_0}) = \text{Inf}_{x_0}(k)\] where the spaces on the left hand side of the equality signs are given in (07WZ) and (07X0) and the spaces on the right hand side are given by Lemma 07Y9.

Remark

Assumptions and notation as in Lemma 07Y9. Choose an affine open \(\Spec(\Lambda) \subset S\) such that \(\Spec(A) \to S\) corresponds to a ring map \(\Lambda \to A\). Consider the ring map \[A \longrightarrow A[\Omega_{A/\Lambda}], \quad a \longmapsto (a, \text{d}_{A/\Lambda}(a))\] Pulling back \(x\) along the corresponding morphism \(\Spec(A[\Omega_{A/\Lambda}]) \to \Spec(A)\) we obtain a deformation \(x_{can}\) of \(x\) over \(A[\Omega_{A/\Lambda}]\). We call this the canonical element \[x_{can} \in T_x(\Omega_{A/\Lambda}) = \text{Lift}(x, A[\Omega_{A/\Lambda}]).\] Next, assume that \(\Lambda\) is Noetherian and \(\Lambda \to A\) is of finite type. Let \(k = \kappa(\mathfrak p)\) be a residue field at a finite type point \(u_0\) of \(U = \Spec(A)\). Let \(x_0 = x|_{u_0}\). By (RS*) and the fact that \(A[k] = A \times_k k[k]\) the space \(T_x(k)\) is the tangent space to the deformation functor \(\mathcal{F}_{\mathcal{X}, k, x_0}\). Via \[T\mathcal{F}_{U, k, u_0} = \text{Der}_\Lambda(A, k) = \Hom_A(\Omega_{A/\Lambda}, k)\] (see Formal Deformation Theory, Example 06IC) and functoriality of \(T_x\) the canonical element produces the map on tangent spaces induced by the object \(x\) over \(U\). Namely, \(\theta \in T\mathcal{F}_{U, k, u_0}\) maps to \(T_x(\theta)(x_{can})\) in \(T_x(k) = T\mathcal{F}_{\mathcal{X}, k, x_0}\).

Remark

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Assume \(\mathcal{X}\) satisfies condition (RS*). Let \(A\) be an \(S\)-algebra such that \(\Spec(A) \to S\) maps into an affine open and let \(x, y\) be objects of \(\mathcal{X}\) over \(\Spec(A)\). Further, let \(A \to B\) be a ring map and let \(\alpha : x|_{\Spec(B)} \to y|_{\Spec(B)}\) be a morphism of \(\mathcal{X}\) over \(\Spec(B)\). Consider the ring map \[B \longrightarrow B[\Omega_{B/A}], \quad b \longmapsto (b, \text{d}_{B/A}(b))\] Pulling back \(\alpha\) along the corresponding morphism \(\Spec(B[\Omega_{B/A}]) \to \Spec(B)\) we obtain a morphism \(\alpha_{can}\) between the pullbacks of \(x\) and \(y\) over \(B[\Omega_{B/A}]\). On the other hand, we can pull back \(\alpha\) by the morphism \(\Spec(B[\Omega_{B/A}]) \to \Spec(B)\) corresponding to the injection of \(B\) into the first summand of \(B[\Omega_{B/A}]\). By the discussion of Remark 07YB we can take the difference \[\varphi(x, y, \alpha) = \alpha_{can} - \alpha|_{\Spec(B[\Omega_{B/A}])} \in \text{Inf}_{x|_{\Spec(B)}}(\Omega_{B/A}).\] We will call this the canonical automorphism. It depends on all the ingredients \(A\), \(x\), \(y\), \(A \to B\) and \(\alpha\).

Obstruction theories

In this section we describe what an obstruction theory is. Contrary to the spaces of infinitesimal deformations and infinitesimal automorphisms, an obstruction theory is an additional piece of data. The formulation is motivated by the results of Lemma 07Y9 and Remark 07YA.

Definition

Let \(S\) be a locally Noetherian base. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). An obstruction theory is given by the following data

1. for every \(S\)-algebra \(A\) such that \(\Spec(A) \to S\) maps into an affine open and every object \(x\) of \(\mathcal{X}\) over \(\Spec(A)\) an \(A\)-linear functor \[\mathcal{O}_x : \text{Mod}_A \to \text{Mod}_A\] of obstruction modules,

2. for \((x, A)\) as in (1), a ring map \(A \to B\), \(M \in \text{Mod}_A\), \(N \in \text{Mod}_B\), and an \(A\)-linear map \(M \to N\) there is an induced \(A\)-linear map \(\mathcal{O}_x(M) \to \mathcal{O}_y(N)\) where \(y = x|_{\Spec(B)}\), and

3. for every deformation situation \((x, A' \to A)\) an obstruction element \(o_x(A') \in \mathcal{O}_x(I)\) where \(I = \Ker(A' \to A)\).

These data are subject to the following conditions

1. the functoriality maps turn the obstruction modules into a functor from the category of triples \((x, A, M)\) to sets,

2. for every morphism of deformation situations \((y, B' \to B) \to (x, A' \to A)\) the element \(o_x(A')\) maps to \(o_y(B')\), and

3. we have \[\text{Lift}(x, A') \not = \emptyset \Leftrightarrow o_x(A') = 0\] for every deformation situation \((x, A' \to A)\).

This last condition explains the terminology. The module \(\mathcal{O}_x(I)\) is called the obstruction module. The element \(o_x(A')\) is the obstruction. Most obstruction theories have additional properties, and in order to make them useful additional conditions are needed. Moreover, this is just a sample definition, for example in the definition we could consider only deformation situations of finite type over \(S\).

One of the main reasons for introducing obstruction theories is to check openness of versality. An example of this type of result is Lemma 0CYF below. The initial idea to do this is due to Artin, see the papers of Artin mentioned in the introduction. It has been taken up for example in the work by Flenner [Flenner], Hall [Hall-coherent], Hall and Rydh [rydh_axioms], Olsson [olsson_deformation], Olsson and Starr [olsson-starr], and Lieblich [lieblich-complexes] (random order of references). Moreover, for particular categories fibred in groupoids, often authors develop a little bit of theory adapted to the problem at hand. We will develop this theory later.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Assume

1. \(\Delta : \mathcal{X} \to \mathcal{X} \times \mathcal{X}\) is representable by algebraic spaces,

2. \(\mathcal{X}\) has (RS*),

3. \(\mathcal{X}\) is limit preserving,

4. there exists an obstruction theory¹⁰,

5. for an object \(x\) of \(\mathcal{X}\) over \(\Spec(A)\) and \(A\)-modules \(M_n\), \(n \geq 1\) we have

   1. \(T_x(\prod M_n) = \prod T_x(M_n)\),

   2. \(\mathcal{O}_x(\prod M_n) \to \prod \mathcal{O}_x(M_n)\) is injective.

Then \(\mathcal{X}\) satisfies openness of versality.

Proof

We prove this by verifying condition (4) of Lemma 0CXU. Let \((\xi_n)\) and \((R_n)\) be as in Remark 0CXT such that \(\Ker(R_m \to R_n)\) is an ideal of square zero for all \(m \geq n\). Set \(A = R_1\) and \(x = \xi_1\). Set \(M_n = \Ker(R_n \to R_1)\). Then \(M_n\) is an \(A\)-module. Set \(R = \lim R_n\). Let \[\tilde R = \{(r_1, r_2, r_3, \ldots) \in \prod R_n \text{ such that all have the same image in }A\}\] Then \(\tilde R \to A\) is surjective with kernel \(M = \prod M_n\). There is a map \(R \to \tilde R\) and a map \(\tilde R \to A[M]\), \((r_1, r_2, r_3, \ldots) \mapsto (r_1, r_2 - r_1, r_3 - r_2, \ldots)\). Together these give a short exact sequence \[(x, R \to A) \to (x, \tilde R \to A) \to (x, A[M])\] of deformation situations, see Remark 07YE. The associated sequence of kernels \(0 \to \lim M_n \to M \to M \to 0\) is the canonical sequence computing the limit of the system of modules \((M_n)\).

Let \(o_x(\tilde R) \in \mathcal{O}_x(M)\) be the obstruction element. Since we have the lifts \(\xi_n\) we see that \(o_x(\tilde R)\) maps to zero in \(\mathcal{O}_x(M_n)\). By assumption (5)(b) we see that \(o_x(\tilde R) = 0\). Choose a lift \(\tilde \xi\) of \(x\) to \(\Spec(\tilde R)\). Let \(\tilde \xi_n\) be the restriction of \(\tilde \xi\) to \(\Spec(R_n)\). There exist elements \(t_n \in T_x(M_n)\) such that \(t_n \cdot \tilde \xi_n = \xi_n\) by Lemma 07Y9 part (2)(b). By assumption (5)(a) we can find \(t \in T_x(M)\) mapping to \(t_n\) in \(T_x(M_n)\). After replacing \(\tilde \xi\) by \(t \cdot \tilde \xi\) we find that \(\tilde \xi\) restricts to \(\xi_n\) over \(\Spec(R_n)\) for all \(n\). In particular, since \(\xi_{n + 1}\) restricts to \(\xi_n\) over \(\Spec(R_n)\), the restriction \(\overline{\xi}\) of \(\tilde \xi\) to \(\Spec(A[M])\) has the property that it restricts to the trivial deformation over \(\Spec(A[M_n])\) for all \(n\). Hence by assumption (5)(a) we find that \(\overline{\xi}\) is the trivial deformation of \(x\). By axiom (RS*) applied to \(R = \tilde R \times_{A[M]} A\) this implies that \(\tilde \xi\) is the pullback of a deformation \(\xi\) of \(x\) over \(R\). This finishes the proof.

Example

Let \(S = \Spec(\Lambda)\) for some Noetherian ring \(\Lambda\). Let \(W \to S\) be a morphism of schemes. Let \(\mathcal{F}\) be a quasi-coherent \(\mathcal{O}_W\)-module flat over \(S\). Consider the functor \[F : (\Sch/S)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad T/S \longrightarrow H^0(W_T, \mathcal{F}_T)\] where \(W_T = T \times_S W\) is the base change and \(\mathcal{F}_T\) is the pullback of \(\mathcal{F}\) to \(W_T\). If \(T = \Spec(A)\) we will write \(W_T = W_A\), etc. Let \(\mathcal{X} \to (\Sch/S)_{fppf}\) be the category fibred in groupoids associated to \(F\). Then \(\mathcal{X}\) has an obstruction theory. Namely,

1. given \(A\) over \(\Lambda\) and \(x \in H^0(W_A, \mathcal{F}_A)\) we set \(\mathcal{O}_x(M) = H^1(W_A, \mathcal{F}_A \otimes_A M)\),

2. given a deformation situation \((x, A' \to A)\) we let \(o_x(A') \in \mathcal{O}_x(I)\) be the image of \(x\) under the boundary map \[H^0(W_A, \mathcal{F}_A) \longrightarrow H^1(W_A, \mathcal{F}_A \otimes_A I)\] coming from the short exact sequence of modules \[0 \to \mathcal{F}_A \otimes_A I \to \mathcal{F}_{A'} \to \mathcal{F}_A \to 0.\]

We have omitted some details, in particular the construction of the short exact sequence above (it uses that \(W_A\) and \(W_{A'}\) have the same underlying topological space) and the explanation for why flatness of \(\mathcal{F}\) over \(S\) implies that the sequence above is short exact.

Example

Let \(S = \Spec(\Lambda)\) for some Noetherian ring \(\Lambda\). Say \(\mathcal{X} = (\Sch/X)_{fppf}\) with \(X = \Spec(R)\) and \(R = \Lambda[x_1, \ldots, x_n]/J\). The naive cotangent complex \(\NL_{R/\Lambda}\) is (canonically) homotopy equivalent to \[J/J^2 \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} R\text{d}x_i,\] see Algebra, Lemma 00S1. Consider a deformation situation \((x, A' \to A)\). Denote by \(I\) the kernel of \(A' \to A\). The object \(x\) corresponds to \((a_1, \ldots, a_n)\) with \(a_i \in A\) such that \(f(a_1, \ldots, a_n) = 0\) in \(A\) for all \(f \in J\). Set \[\begin{align*} \mathcal{O}_x(I) & = \Hom_R(J/J^2, I)/\Hom_R(R^{\oplus n}, I) \\ & = \Ext^1_R(\NL_{R/\Lambda}, I) \\ & = \Ext^1_A(\NL_{R/\Lambda} \otimes_R A, I). \end{align*}\] Choose lifts \(a_i' \in A'\) of \(a_i\) in \(A\). Then \(o_x(A')\) is the class of the map \(J/J^2 \to I\) defined by sending \(f \in J\) to \(f(a_1', \ldots, a'_n) \in I\). We omit the verification that \(o_x(A')\) is independent of choices. It is clear that if \(o_x(A') = 0\) then the map lifts. Finally, functoriality is straightforward. Thus we obtain an obstruction theory. We observe that \(o_x(A')\) can be described a bit more canonically as the composition \[\NL_{R/\Lambda} \to \NL_{A/\Lambda} \to \NL_{A/A'} = I[1]\] in \(D(A)\), see Algebra, Lemma 07BP for the last identification.

Naive obstruction theories

The title of this section refers to the fact that we will use the naive cotangent complex in this section. Let \((x, A' \to A)\) be a deformation situation for a given category fibred in groupoids over a locally Noetherian scheme \(S\). The key Example 07YI suggests that any obstruction theory should be closely related to maps in \(D(A)\) with target the naive cotangent complex of \(A\). Working this out we find a criterion for versality in Lemma 07YM which leads to a criterion for openness of versality in Lemma 07YN. We introduce a notion of a naive obstruction theory in Definition 07YP to try to formalize the notion a bit further.

In the following we will use the naive cotangent complex as defined in Algebra, Section 00S0. In particular, if \(A' \to A\) is a surjection of \(\Lambda\)-algebras with square zero kernel \(I\), then there are maps \[\NL_{A'/\Lambda} \to \NL_{A/\Lambda} \to \NL_{A/A'}\] whose composition is homotopy equivalent to zero (see Algebra, Remark 07VC). This doesn’t form a distinguished triangle in general as we are using the naive cotangent complex and not the full one. There is a homotopy equivalence \(\NL_{A/A'} \to I[1]\) (the complex consisting of \(I\) placed in degree \(-1\), see Algebra, Lemma 07BP). Finally, note that there is a canonical map \(\NL_{A/\Lambda} \to \Omega_{A/\Lambda}\).

Lemma

Let \(A \to k\) be a ring map with \(k\) a field. Let \(E \in D^-(A)\). Then \(\Ext^i_A(E, k) = \Hom_k(H^{-i}(E \otimes^\mathbf{L} k), k)\).

Proof

Omitted. Hint: Replace \(E\) by a bounded above complex of free \(A\)-modules and compute both sides.

Lemma

Let \(\Lambda \to A \to k\) be finite type ring maps of Noetherian rings with \(k = \kappa(\mathfrak p)\) for some prime \(\mathfrak p\) of \(A\). Let \(\xi : E \to \NL_{A/\Lambda}\) be a morphism of \(D^{-}(A)\) such that \(H^{-1}(\xi \otimes^{\mathbf{L}} k)\) is not surjective. Then there exists a surjection \(A' \to A\) of \(\Lambda\)-algebras such that

1. \(I = \Ker(A' \to A)\) has square zero and is isomorphic to \(k\) as an \(A\)-module,

2. \(\Omega_{A'/\Lambda} \otimes k = \Omega_{A/\Lambda} \otimes k\), and

3. \(E \to \NL_{A/A'}\) is zero.

Proof

Let \(f \in A\), \(f \not \in \mathfrak p\). Suppose that \(A'' \to A_f\) satisfies (a), (b), (c) for the induced map \(E \otimes_A A_f \to \NL_{A_f/\Lambda}\), see Algebra, Lemma 00S7. Then we can set \(A' = A'' \times_{A_f} A\) and get a solution. Namely, it is clear that \(A' \to A\) satisfies (a) because \(\Ker(A' \to A) = \Ker(A'' \to A_f) = I\). Pick \(f'' \in A''\) lifting \(f\). Then the localization of \(A'\) at \((f'', f)\) is isomorphic to \(A''\) (for example by More on Algebra, Lemma 01Z8). Thus (b) and (c) are clear for \(A'\) too. In this way we see that we may replace \(A\) by the localization \(A_f\) (finitely many times). In particular (after such a replacement) we may assume that \(\mathfrak p\) is a maximal ideal of \(A\), see Morphisms, Lemma 01TA.

Choose a presentation \(A = \Lambda[x_1, \ldots, x_n]/J\). Then \(\NL_{A/\Lambda}\) is (canonically) homotopy equivalent to \[J/J^2 \longrightarrow \bigoplus\nolimits_{i = 1, \ldots, n} A\text{d}x_i,\] see Algebra, Lemma 00S1. After localizing if necessary (using Nakayama’s lemma) we can choose generators \(f_1, \ldots, f_m\) of \(J\) such that \(f_j \otimes 1\) form a basis for \(J/J^2 \otimes_A k\). Moreover, after renumbering, we can assume that the images of \(\text{d}f_1, \ldots, \text{d}f_r\) form a basis for the image of \(J/J^2 \otimes k \to \bigoplus k\text{d}x_i\) and that \(\text{d}f_{r + 1}, \ldots, \text{d}f_m\) map to zero in \(\bigoplus k\text{d}x_i\). With these choices the space \[H^{-1}(\NL_{A/\Lambda} \otimes^{\mathbf{L}}_A k) = H^{-1}(\NL_{A/\Lambda} \otimes_A k)\] has basis \(f_{r + 1} \otimes 1, \ldots, f_m \otimes 1\). Changing basis once again we may assume that the image of \(H^{-1}(\xi \otimes^{\mathbf{L}} k)\) is contained in the \(k\)-span of \(f_{r + 1} \otimes 1, \ldots, f_{m - 1} \otimes 1\). Set \[A' = \Lambda[x_1, \ldots, x_n]/(f_1, \ldots, f_{m - 1}, \mathfrak pf_m)\] By construction \(A' \to A\) satisfies (a). Since \(\text{d}f_m\) maps to zero in \(\bigoplus k\text{d}x_i\) we see that (b) holds. Finally, by construction the induced map \(E \to \NL_{A/A'} = I[1]\) induces the zero map \(H^{-1}(E \otimes_A^\mathbf{L} k) \to I \otimes_A k\). By Lemma 07YK we see that the composition is zero.

The following lemma is our key technical result.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\) satisfying (RS*). Let \(U = \Spec(A)\) be an affine scheme of finite type over \(S\) which maps into an affine open \(\Spec(\Lambda)\). Let \(x\) be an object of \(\mathcal{X}\) over \(U\). Let \(\xi : E \to \NL_{A/\Lambda}\) be a morphism of \(D^{-}(A)\). Assume

1. for every deformation situation \((x, A' \to A)\) we have: \(x\) lifts to \(\Spec(A')\) if and only if \(E \to \NL_{A/\Lambda} \to \NL_{A/A'}\) is zero, and

2. there is an isomorphism of functors \(T_x(-) \to \Ext^0_A(E, -)\) such that \(E \to \NL_{A/\Lambda} \to \Omega^1_{A/\Lambda}\) corresponds to the canonical element (see Remark 07YC).

Let \(u_0 \in U\) be a finite type point with residue field \(k = \kappa(u_0)\). Consider the following statements

1. \(x\) is versal at \(u_0\), and

2. \(\xi : E \to \NL_{A/\Lambda}\) induces a surjection \(H^{-1}(E \otimes_A^{\mathbf{L}} k) \to H^{-1}(\NL_{A/\Lambda} \otimes_A^{\mathbf{L}} k)\) and an injection \(H^0(E \otimes_A^{\mathbf{L}} k) \to H^0(\NL_{A/\Lambda} \otimes_A^{\mathbf{L}} k)\).

Then we always have (2) \(\Rightarrow\) (1) and we have (1) \(\Rightarrow\) (2) if \(u_0\) is a closed point.

Proof

Let \(\mathfrak p = \Ker(A \to k)\) be the prime corresponding to \(u_0\).

Assume that \(x\) is versal at \(u_0\) and that \(u_0\) is a closed point of \(U\). If \(H^{-1}(\xi \otimes_A^{\mathbf{L}} k)\) is not surjective, then let \(A' \to A\) be an extension with kernel \(I\) as in Lemma 07YL. Because \(u_0\) is a closed point, we see that \(I\) is a finite \(A\)-module, hence that \(A'\) is a finite type \(\Lambda\)-algebra (this fails if \(u_0\) is not closed). In particular \(A'\) is Noetherian. By property (c) for \(A'\) and (i) for \(\xi\) we see that \(x\) lifts to an object \(x'\) over \(A'\). Let \(\mathfrak p' \subset A'\) be the kernel of the surjective map to \(k\). By Artin-Rees (Algebra, Lemma 00IN) there exists an \(n > 1\) such that \((\mathfrak p')^n \cap I = 0\). Then we see that \[B' = A'/(\mathfrak p')^n \longrightarrow A/\mathfrak p^n = B\] is a small, essential extension of local Artinian rings, see Formal Deformation Theory, Lemma 06H0. On the other hand, as \(x\) is versal at \(u_0\) and as \(x'|_{\Spec(B')}\) is a lift of \(x|_{\Spec(B)}\), there exists an integer \(m \geq n\) and a map \(q : A/\mathfrak p^m \to B'\) such that the composition \(A/\mathfrak p^m \to B' \to B\) is the quotient map. Since the maximal ideal of \(B'\) has \(n\)th power equal to zero, this \(q\) factors through \(B\) which contradicts the fact that \(B' \to B\) is an essential surjection. This contradiction shows that \(H^{-1}(\xi \otimes_A^{\mathbf{L}} k)\) is surjective.

Assume that \(x\) is versal at \(u_0\). By Lemma 07YK the map \(H^0(\xi \otimes_A^{\mathbf{L}} k)\) is dual to the map \(\Ext^0_A(\NL_{A/\Lambda}, k) \to \Ext^0_A(E, k)\). Note that \[\Ext^0_A(\NL_{A/\Lambda}, k) = \text{Der}_\Lambda(A, k) \quad\text{and}\quad T_x(k) = \Ext^0_A(E, k)\] Condition (ii) assures us the map \(\Ext^0_A(\NL_{A/\Lambda}, k) \to \Ext^0_A(E, k)\) sends a tangent vector \(\theta\) to \(U\) at \(u_0\) to the corresponding infinitesimal deformation of \(x_0\), see Remark 07YC. Hence if \(x\) is versal, then this map is surjective, see Formal Deformation Theory, Lemma 06IU. Hence \(H^0(\xi \otimes_A^{\mathbf{L}} k)\) is injective. This finishes the proof of (1) \(\Rightarrow\) (2) in case \(u_0\) is a closed point.

For the rest of the proof assume \(H^{-1}(E \otimes_A^\mathbf{L} k) \to H^{-1}(\NL_{A/\Lambda} \otimes_A^\mathbf{L} k)\) is surjective and \(H^0(E \otimes_A^\mathbf{L} k) \to H^0(\NL_{A/\Lambda} \otimes_A^\mathbf{L} k)\) is injective. Set \(R = A_\mathfrak p^\wedge\) and let \(\eta\) be the formal object over \(R\) associated to \(x|_{\Spec(R)}\). The map \(d\underline{\eta}\) on tangent spaces is surjective because it is identified with the dual of the injective map \(H^0(E \otimes_A^{\mathbf{L}} k) \to H^0(\NL_{A/\Lambda} \otimes_A^{\mathbf{L}} k)\) (see previous paragraph). According to Formal Deformation Theory, Lemma 06IU it suffices to prove the following: Let \(C' \to C\) be a small extension of finite type Artinian local \(\Lambda\)-algebras with residue field \(k\). Let \(R \to C\) be a \(\Lambda\)-algebra map compatible with identifications of residue fields. Let \(y = x|_{\Spec(C)}\) and let \(y'\) be a lift of \(y\) to \(C'\). To show: we can lift the \(\Lambda\)-algebra map \(R \to C\) to \(R \to C'\).

Observe that it suffices to lift the \(\Lambda\)-algebra map \(A \to C\). Let \(I = \Ker(C' \to C)\). Note that \(I\) is a \(1\)-dimensional \(k\)-vector space. The obstruction \(ob\) to lifting \(A \to C\) is an element of \(\Ext^1_A(\NL_{A/\Lambda}, I)\), see Example 07YI. By Lemma 07YK and our assumption the map \(\xi\) induces an injection \[\Ext^1_A(\NL_{A/\Lambda}, I) \longrightarrow \Ext^1_A(E, I)\] By the construction of \(ob\) and (i) the image of \(ob\) in \(\Ext^1_A(E, I)\) is the obstruction to lifting \(x\) to \(A \times_C C'\). By (RS*) the fact that \(y/C\) lifts to \(y'/C'\) implies that \(x\) lifts to \(A \times_C C'\). Hence \(ob = 0\) and we are done.

The key lemma above allows us to conclude that we have openness of versality in some cases.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\) satisfying (RS*). Let \(U = \Spec(A)\) be an affine scheme of finite type over \(S\) which maps into an affine open \(\Spec(\Lambda)\). Let \(x\) be an object of \(\mathcal{X}\) over \(U\). Let \(\xi : E \to \NL_{A/\Lambda}\) be a morphism of \(D^{-}(A)\). Assume

1. for every deformation situation \((x, A' \to A)\) we have: \(x\) lifts to \(\Spec(A')\) if and only if \(E \to \NL_{A/\Lambda} \to \NL_{A/A'}\) is zero,

2. there is an isomorphism of functors \(T_x(-) \to \Ext^0_A(E, -)\) such that \(E \to \NL_{A/\Lambda} \to \Omega^1_{A/\Lambda}\) corresponds to the canonical element (see Remark 07YC),

3. the cohomology groups of \(E\) are finite \(A\)-modules.

If \(x\) is versal at a closed point \(u_0 \in U\), then there exists an open neighbourhood \(u_0 \in U' \subset U\) such that \(x\) is versal at every finite type point of \(U'\).

Proof

Let \(C\) be the cone of \(\xi\) so that we have a distinguished triangle \[E \to \NL_{A/\Lambda} \to C \to E[1]\] in \(D^{-}(A)\). By Lemma 07YM the assumption that \(x\) is versal at \(u_0\) implies that \(H^{-1}(C \otimes^\mathbf{L} k) = 0\). By More on Algebra, Lemma 068U there exists an \(f \in A\) not contained in the prime corresponding to \(u_0\) such that \(H^{-1}(C \otimes^\mathbf{L}_A M) = 0\) for any \(A_f\)-module \(M\). Using Lemma 07YM again we see that we have versality for all finite type points of the open \(D(f) \subset U\).

The technical lemmas above suggest the following definition.

Definition

Let \(S\) be a locally Noetherian base. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Assume that \(\mathcal{X}\) satisfies (RS*). A naive obstruction theory is given by the following data

1. for every \(S\)-algebra \(A\) such that \(\Spec(A) \to S\) maps into an affine open \(\Spec(\Lambda) \subset S\) and every object \(x\) of \(\mathcal{X}\) over \(\Spec(A)\) we are given an object \(E_x \in D^-(A)\) and a map \(\xi_x : E_x \to \NL_{A/\Lambda}\),

2. given \((x, A)\) as in (07YQ) there are transformations of functors \[\text{Inf}_x( - ) \to \Ext^{-1}_A(E_x, -) \quad\text{and}\quad T_x(-) \to \Ext^0_A(E_x, -)\]

3. for \((x, A)\) as in (07YQ) and a ring map \(A \to B\) setting \(y = x|_{\Spec(B)}\) there is a functoriality map \(E_x \to E_y\) in \(D(A)\).

These data are subject to the following conditions

1. in the situation of (07YS) the diagram \[\xymatrix{ E_y \ar[r]_{\xi_y} & \NL_{B/\Lambda} \\ E_x \ar[u] \ar[r]^{\xi_x} & \NL_{A/\Lambda} \ar[u] }\] is commutative in \(D(A)\),

2. given \((x, A)\) as in (07YQ) and \(A \to B \to C\) setting \(y = x|_{\Spec(B)}\) and \(z = x|_{\Spec(C)}\) the composition of the functoriality maps \(E_x \to E_y\) and \(E_y \to E_z\) is the functoriality map \(E_x \to E_z\),

3. the maps of (07YR) are isomorphisms compatible with the functoriality maps and the maps of Remark 07YA,

4. the composition \(E_x \to \NL_{A/\Lambda} \to \Omega_{A/\Lambda}\) corresponds to the canonical element of \(T_x(\Omega_{A/\Lambda}) = \Ext^0(E_x, \Omega_{A/\Lambda})\), see Remark 07YC,

5. given a deformation situation \((x, A' \to A)\) with \(I = \Ker(A' \to A)\) the composition \(E_x \to \NL_{A/\Lambda} \to \NL_{A/A'}\) is zero in \[\Hom_A(E_x, \NL_{A/A'}) = \Ext^0_A(E_x, \NL_{A/A'}) = \Ext^1_A(E_x, I)\] if and only if \(x\) lifts to \(A'\).

Thus we see in particular that we obtain an obstruction theory as in Section 07YF by setting \(\mathcal{O}_x( - ) = \Ext^1_A(E_x, -)\).

Lemma

Let \(S\) and \(\mathcal{X}\) be as in Definition 07YP and let \(\mathcal{X}\) be endowed with a naive obstruction theory. Let \(A \to B\) and \(y \to x\) be as in (07YS). Let \(k\) be a \(B\)-algebra which is a field. Then the functoriality map \(E_x \to E_y\) induces bijections \[H^i(E_x \otimes_A^{\mathbf{L}} k) \to H^i(E_y \otimes_B^{\mathbf{L}} k)\] for \(i = 0, 1\).

Proof

Let \(z = x|_{\Spec(k)}\). Then (RS*) implies that \[\textit{Lift}(x, A[k]) = \textit{Lift}(z, k[k]) \quad\text{and}\quad \textit{Lift}(y, B[k]) = \textit{Lift}(z, k[k])\] because \(A[k] = A \times_k k[k]\) and \(B[k] = B \times_k k[k]\). Hence the properties of a naive obstruction theory imply that the functoriality map \(E_x \to E_y\) induces bijections \(\Ext^i_A(E_x, k) \to \Ext^i_B(E_y, k)\) for \(i = -1, 0\). By Lemma 07YK our maps \(H^i(E_x \otimes_A^{\mathbf{L}} k) \to H^i(E_y \otimes_B^{\mathbf{L}} k)\), \(i = 0, 1\) induce isomorphisms on dual vector spaces hence are isomorphisms.

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(p : \mathcal{X} \to (\Sch/S)_{fppf}\) be a category fibred in groupoids. Assume that \(\mathcal{X}\) satisfies (RS*) and that \(\mathcal{X}\) has a naive obstruction theory. Then openness of versality holds for \(\mathcal{X}\) provided the complexes \(E_x\) of Definition 07YP have finitely generated cohomology groups for pairs \((A, x)\) where \(A\) is of finite type over \(S\).

Proof

Let \(U\) be a scheme locally of finite type over \(S\), let \(x\) be an object of \(\mathcal{X}\) over \(U\), and let \(u_0\) be a finite type point of \(U\) such that \(x\) is versal at \(u_0\). We may first shrink \(U\) to an affine scheme such that \(u_0\) is a closed point and such that \(U \to S\) maps into an affine open \(\Spec(\Lambda)\). Say \(U = \Spec(A)\). Let \(\xi_x : E_x \to \NL_{A/\Lambda}\) be the obstruction map. At this point we may apply Lemma 07YN to conclude.

A dual notion

Let \((x, A' \to A)\) be a deformation situation for a given category \(\mathcal{X}\) fibred in groupoids over a locally Noetherian scheme \(S\). Assume \(\mathcal{X}\) has an obstruction theory, see Definition 07YG. In practice one often has a complex \(K^\bullet\) of \(A\)-modules and isomorphisms of functors \[\text{Inf}_x(-) \to H^0(K^\bullet \otimes_A^\mathbf{L} -),\quad T_x(-) \to H^1(K^\bullet \otimes_A^\mathbf{L} -),\quad \mathcal{O}_x(-) \to H^2(K^\bullet \otimes_A^\mathbf{L} -)\] In this section we formalize this a little bit and show how this leads to a verification of openness of versality in some cases.

Example

Let \(\Lambda, S, W, \mathcal{F}\) be as in Example 07YH. Assume that \(W \to S\) is proper and \(\mathcal{F}\) coherent. By Cohomology of Schemes, Remark 07VL there exists a finite complex of finite projective \(\Lambda\)-modules \(N^\bullet\) which universally computes the cohomology of \(\mathcal{F}\). In particular the obstruction spaces from Example 07YH are \(\mathcal{O}_x(M) = H^1(N^\bullet \otimes_\Lambda M)\). Hence with \(K^\bullet = N^\bullet \otimes_\Lambda A[-1]\) we see that \(\mathcal{O}_x(M) = H^2(K^\bullet \otimes_A^\mathbf{L} M)\).

Situation

Let \(S\) be a locally Noetherian scheme. Let \(\mathcal{X}\) be a category fibred in groupoids over \((\Sch/S)_{fppf}\). Assume that \(\mathcal{X}\) has (RS*) so that we can speak of the functor \(T_x(-)\), see Lemma 07Y9. Let \(U = \Spec(A)\) be an affine scheme of finite type over \(S\) which maps into an affine open \(\Spec(\Lambda)\). Let \(x\) be an object of \(\mathcal{X}\) over \(U\). Assume we are given

1. a complex of \(A\)-modules \(K^\bullet\),

2. a transformation of functors \(T_x(-) \to H^1(K^\bullet \otimes_A^\mathbf{L} -)\),

3. for every deformation situation \((x, A' \to A)\) with kernel \(I = \Ker(A' \to A)\) an element \(o_x(A') \in H^2(K^\bullet \otimes_A^\mathbf{L} I)\)

satisfying the following (minimal) conditions

1. the transformation \(T_x(-) \to H^1(K^\bullet \otimes_A^\mathbf{L} -)\) is an isomorphism,

2. given a morphism \((x, A'' \to A) \to (x, A' \to A)\) of deformation situations the element \(o_x(A')\) maps to the element \(o_x(A'')\) via the map \(H^2(K^\bullet \otimes_A^\mathbf{L} I) \to H^2(K^\bullet \otimes_A^\mathbf{L} I')\) where \(I' = \Ker(A'' \to A)\), and

3. \(x\) lifts to an object over \(\Spec(A')\) if and only if \(o_x(A') = 0\).

It is possible to incorporate infinitesimal automorphisms as well, but we refrain from doing so in order to get the sharpest possible result.

In Situation 07YX an important role will be played by \(K^\bullet \otimes_A^\mathbf{L} \NL_{A/\Lambda}\). Suppose we are given an element \(\xi \in H^1(K^\bullet \otimes_A^\mathbf{L} \NL_{A/\Lambda})\). Then (1) for any surjection \(A' \to A\) of \(\Lambda\)-algebras with kernel \(I\) of square zero the canonical map \(\NL_{A/\Lambda} \to \NL_{A/A'} = I[1]\) sends \(\xi\) to an element \(\xi_{A'} \in H^2(K^\bullet \otimes_A^\mathbf{L} I)\) and (2) the map \(\NL_{A/\Lambda} \to \Omega_{A/\Lambda}\) sends \(\xi\) to an element \(\xi_{can}\) of \(H^1(K^\bullet \otimes_A^\mathbf{L} \Omega_{A/\Lambda})\).

Lemma

In Situation 07YX, assume furthermore that

1. given a short exact sequence of deformation situations as in Remark 07YE and a lift \(x'_2 \in \text{Lift}(x, A_2')\) then \(o_x(A_3') \in H^2(K^\bullet \otimes_A^\mathbf{L} I_3)\) equals \(\partial\theta\) where \(\theta \in H^1(K^\bullet \otimes_A^\mathbf{L} I_1)\) is the element corresponding to \(x'_2|_{\Spec(A_1')}\) via \(A_1' = A[I_1]\) and the given map \(T_x(-) \to H^1(K^\bullet \otimes_A^\mathbf{L} -)\).

In this case there exists an element \(\xi \in H^1(K^\bullet \otimes_A^\mathbf{L} \NL_{A/\Lambda})\) such that

1. for every deformation situation \((x, A' \to A)\) we have \(\xi_{A'} = o_x(A')\), and

2. \(\xi_{can}\) matches the canonical element of Remark 07YC via the given transformation \(T_x(-) \to H^1(K^\bullet \otimes_A^\mathbf{L} -)\).

Proof

Choose an \(\alpha : \Lambda[x_1, \ldots, x_n] \to A\) with kernel \(J\). Write \(P = \Lambda[x_1, \ldots, x_n]\). In the rest of this proof we work with \[\NL(\alpha) = (J/J^2 \longrightarrow \bigoplus A \text{d}x_i)\] which is permissible by Algebra, Lemma 00S1 and More on Algebra, Lemma 064I. Consider the element \(o_x(P/J^2) \in H^2(K^\bullet \otimes_A^\mathbf{L} J/J^2)\) and consider the quotient \[C = (P/J^2 \times \bigoplus A \text{d}x_i)/(J/J^2)\] where \(J/J^2\) is embedded diagonally. Note that \(C \to A\) is a surjection with kernel \(\bigoplus A\text{d}x_i\). Moreover there is a section \(A \to C\) to \(C \to A\) given by mapping the class of \(f \in P\) to the class of \((f, \text{d}f)\) in the pushout. For later use, denote by \(x_C\) the pullback of \(x\) along the corresponding morphism \(\Spec(C) \to \Spec(A)\). Thus we see that \(o_x(C) = 0\). We conclude that \(o_x(P/J^2)\) maps to zero in \(H^2(K^\bullet \otimes_A^\mathbf{L} \bigoplus A\text{d}x_i)\). It follows that there exists some element \(\xi \in H^1(K^\bullet \otimes_A^\mathbf{L} \NL(\alpha))\) mapping to \(o_x(P/J^2)\).

Note that for any deformation situation \((x, A' \to A)\) there exists a \(\Lambda\)-algebra map \(P/J^2 \to A'\) compatible with the augmentations to \(A\). Hence the element \(\xi\) satisfies the first property of the lemma by construction and property (ii) of Situation 07YX.

Note that our choice of \(\xi\) was well defined up to the choice of an element of \(H^1(K^\bullet \otimes_A^\mathbf{L} \bigoplus A\text{d}x_i)\). We will show that after modifying \(\xi\) by an element of the aforementioned group we can arrange it so that the second assertion of the lemma is true. Let \(C' \subset C\) be the image of \(P/J^2\) under the \(\Lambda\)-algebra map \(P/J^2 \to C\) (inclusion of first factor). Observe that \(\Ker(C' \to A) = \Im(J/J^2 \to \bigoplus A\text{d}x_i)\). Set \(\overline{C} = A[\Omega_{A/\Lambda}]\). The map \(P/J^2 \times \bigoplus A \text{d}x_i \to \overline{C}\), \((f, \sum f_i \text{d}x_i) \mapsto (f \bmod J, \sum f_i \text{d}x_i)\) factors through a surjective map \(C \to \overline{C}\). Then \[(x, \overline{C} \to A) \to (x, C \to A) \to (x, C' \to A)\] is a short exact sequence of deformation situations. The associated splitting \(\overline{C} = A[\Omega_{A/\Lambda}]\) (from Remark 07YE) equals the given splitting above. Moreover, the section \(A \to C\) composed with the map \(C \to \overline{C}\) is the map \((1, \text{d}) : A \to A[\Omega_{A/\Lambda}]\) of Remark 07YC. Thus \(x_C\) restricts to the canonical element \(x_{can}\) of \(T_x(\Omega_{A/\Lambda}) = \text{Lift}(x, A[\Omega_{A/\Lambda}])\). By condition (iv) we conclude that \(o_x(P/J^2)\) maps to \(\partial x_{can}\) in \[H^1(K^\bullet \otimes_A^\mathbf{L} \Im(J/J^2 \to \bigoplus A\text{d}x_i))\] By construction \(\xi\) maps to \(o_x(P/J^2)\). It follows that \(x_{can}\) and \(\xi_{can}\) map to the same element in the displayed group which means (by the long exact cohomology sequence) that they differ by an element of \(H^1(K^\bullet \otimes_A^\mathbf{L} \bigoplus A\text{d}x_i)\) as desired.

Lemma

In Situation 07YX assume that (iv) of Lemma 07YY holds and that \(K^\bullet\) is a perfect object of \(D(A)\). In this case, if \(x\) is versal at a closed point \(u_0 \in U\) then there exists an open neighbourhood \(u_0 \in U' \subset U\) such that \(x\) is versal at every finite type point of \(U'\).

Proof

We may assume that \(K^\bullet\) is a finite complex of finite projective \(A\)-modules. Thus the derived tensor product with \(K^\bullet\) is the same as simply tensoring with \(K^\bullet\). Let \(E^\bullet\) be the dual perfect complex to \(K^\bullet\), see More on Algebra, Lemma 07VI. (So \(E^n = \Hom_A(K^{-n}, A)\) with differentials the transpose of the differentials of \(K^\bullet\).) Let \(E \in D^{-}(A)\) denote the object represented by the complex \(E^\bullet[-1]\). Let \(\xi \in H^1(\text{Tot}(K^\bullet \otimes_A \NL_{A/\Lambda}))\) be the element constructed in Lemma 07YY and denote by \(\xi : E = E^\bullet[-1] \to \NL_{A/\Lambda}\) the corresponding map (loc.cit.). We claim that the pair \((E, \xi)\) satisfies all the assumptions of Lemma 07YN which finishes the proof.

Namely, assumption (i) of Lemma 07YN follows from conclusion (1) of Lemma 07YY and the fact that \(H^2(K^\bullet \otimes_A^\mathbf{L} -) = \Ext^1(E, -)\) by loc.cit. Assumption (ii) of Lemma 07YN follows from conclusion (2) of Lemma 07YY and the fact that \(H^1(K^\bullet \otimes_A^\mathbf{L} -) = \Ext^0(E, -)\) by loc.cit. Assumption (iii) of Lemma 07YN is clear.

Limit preserving functors on Noetherian schemes

It is sometimes convenient to consider functors or stacks defined only on the full subcategory of (locally) Noetherian schemes. In this section we discuss this in the case of algebraic spaces.

Let \(S\) be a locally Noetherian scheme. Let us be a bit pedantic in order to line up our categories correctly; people who are ignoring set theoretical issues can just replace the sets of schemes we choose by the collection of all schemes in what follows. As in Topologies, Remark 03FF we choose a category \(\Sch_\alpha\) of schemes containing \(S\) such that we obtain big sites \((\Sch/S)_{Zar}\), \((\Sch/S)_\etale\), \((\Sch/S)_{smooth}\), \((\Sch/S)_{syntomic}\), and \((\Sch/S)_{fppf}\) all with the same underlying category \(\Sch_\alpha/S\). Denote by \[\textit{Noetherian}_\alpha \subset \Sch_\alpha\] the full subcategory consisting of locally Noetherian schemes. This determines a full subcategory \[\textit{Noetherian}_\alpha/S \subset \Sch_\alpha/S\] For \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\) we have

1. if \(f : X \to Y\) is a morphism of \(\Sch_\alpha/S\) with \(Y\) in \(\textit{Noetherian}_\alpha/S\) and \(f\) locally of finite type, then \(X\) is in \(\textit{Noetherian}_\alpha/S\),

2. for morphisms \(f : X \to Y\) and \(g : Z \to Y\) of \(\textit{Noetherian}_\alpha/S\) with \(f\) locally of finite type the fibre product \(X \times_Y Z\) in \(\textit{Noetherian}_\alpha/S\) exists and agrees with the fibre product in \(\Sch_\alpha/S\),

3. if \(\{X_i \to X\}_{i \in I}\) is a covering of \((\Sch/S)_\tau\) and \(X\) is in \(\textit{Noetherian}_\alpha/S\), then the objects \(X_i\) are in \(\textit{Noetherian}_\alpha/S\)

4. the category \(\textit{Noetherian}_\alpha/S\) endowed with the set of coverings of \((\Sch/S)_\tau\) whose objects are in \(\textit{Noetherian}_\alpha/S\) is a site we will denote \((\textit{Noetherian}/S)_\tau\),

5. the inclusion functor \((\textit{Noetherian}/S)_\tau \to (\Sch/S)_\tau\) is fully faithful, continuous, and cocontinuous.

By Sites, Lemmas 00XO and 00XR we obtain a morphism of topoi \[g_\tau : \Sh((\textit{Noetherian}/S)_\tau) \longrightarrow \Sh((\Sch/S)_\tau)\] whose pullback functor is the restriction of sheaves along the inclusion functor \((\textit{Noetherian}/S)_\tau \to (\Sch/S)_\tau\).

Remark

The site \((\textit{Noetherian}/S)_\tau\) does not have fibre products. Hence we have to be careful in working with sheaves. For example, the continuous inclusion functor \((\textit{Noetherian}/S)_\tau \to (\Sch/S)_\tau\) does not define a morphism of sites. See Examples, Section 0GE8 for an example in case \(\tau = fppf\).

Let \(F : (\textit{Noetherian}/S)_\tau^{opp} \to \textit{Sets}\) be a functor. We say \(F\) is limit preserving if for any directed limit of affine schemes \(X = \lim X_i\) of \((\textit{Noetherian}/S)_\tau\) we have \(F(X) = \colim F(X_i)\).

Lemma

Let \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). Restricting along the inclusion functor \((\textit{Noetherian}/S)_\tau \to (\Sch/S)_\tau\) defines an equivalence of categories between

1. the category of limit preserving sheaves on \((\Sch/S)_\tau\) and

2. the category of limit preserving sheaves on \((\textit{Noetherian}/S)_\tau\)

Proof

Let \(F : (\textit{Noetherian}/S)_\tau^{opp} \to \textit{Sets}\) be a functor which is both limit preserving and a sheaf. By Topologies, Lemmas 0EUW and 0GDW there exists a unique functor \(F' : (\Sch/S)_\tau^{opp} \to \textit{Sets}\) which is limit preserving, a sheaf, and restricts to \(F\). In fact, the construction of \(F'\) in Topologies, Lemma 0EUW is functorial in \(F\) and this construction is a quasi-inverse to restriction. Some details omitted.

Lemma

Let \(X\) be an object of \((\textit{Noetherian}/S)_\tau\). If the functor of points \(h_X : (\textit{Noetherian}/S)_\tau^{opp} \to \textit{Sets}\) is limit preserving, then \(X\) is locally of finite presentation over \(S\).

Proof

Let \(V \subset X\) be an affine open subscheme which maps into an affine open \(U \subset S\). We may write \(V = \lim V_i\) as a directed limit of affine schemes \(V_i\) of finite presentation over \(U\), see Algebra, Lemma 00QN. By assumption, the arrow \(V \to X\) factors as \(V \to V_i \to X\) for some \(i\). After increasing \(i\) we may assume \(V_i \to X\) factors through \(V\) as the inverse image of \(V \subset X\) in \(V_i\) eventually becomes equal to \(V_i\) by Limits, Lemma 01Z4. Then the identity morphism \(V \to V\) factors through \(V_i\) for some \(i\) in the category of schemes over \(U\). Thus \(V \to U\) is of finite presentation; the corresponding algebra fact is that if \(B\) is an \(A\)-algebra such that \(\text{id} : B \to B\) factors through a finitely presented \(A\)-algebra, then \(B\) is of finite presentation over \(A\) (nice exercise). Hence \(X\) is locally of finite presentation over \(S\).

The following lemma has a variant for transformations representable by algebraic spaces.

Lemma

Let \(\tau \in \{Zariski, \etale, smooth, syntomic, fppf\}\). Let \(F', G' : (\Sch/S)_\tau^{opp} \to \textit{Sets}\) be limit preserving and sheaves. Let \(a' : F' \to G'\) be a transformation of functors. Denote by \(a : F \to G\) the restriction of \(a' : F' \to G'\) to \((\textit{Noetherian}/S)_\tau\). The following are equivalent

1. \(a'\) is representable (as a transformation of functors, see Categories, Definition 001X), and

2. for every object \(V\) of \((\textit{Noetherian}/S)_\tau\) and every map \(V \to G\) the fibre product \(F \times_G V : (\textit{Noetherian}/S)_\tau^{opp} \to \textit{Sets}\) is a representable functor, and

3. same as in (2) but only for \(V\) affine finite type over \(S\) mapping into an affine open of \(S\).

Proof

Assume (1). By Limits of Spaces, Lemma 0GDY the transformation \(a'\) is limit preserving¹¹. Take \(\xi : V \to G\) as in (2). Set \(V' = V\), viewed as an object of \((\Sch/S)_\tau\). Since \(G\) is the restriction of \(G'\) to \((\textit{Noetherian}/S)_\tau\) we see that \(\xi \in G(V)\) corresponds to \(\xi' \in G'(V')\). By assumption \(V' \times_{\xi', G'} F'\) is representable by a scheme \(U'\). The morphism of schemes \(U' \to V'\) corresponding to the projection \(V' \times_{\xi', G'} F' \to V'\) is locally of finite presentation by Limits of Spaces, Lemma 049M and Limits, Proposition 01ZC. Hence \(U'\) is a locally Noetherian scheme and therefore \(U'\) is isomorphic to an object \(U\) of \((\textit{Noetherian}/S)_\tau\). Then \(U\) represents \(F \times_G V\) as desired.

The implication (2) \(\Rightarrow\) (3) is immediate. Assume (3). We will prove (1). Let \(T\) be an object of \((\Sch/S)_\tau\) and let \(T \to G'\) be a morphism. We have to show the functor \(F' \times_{G'} T\) is representable by a scheme \(X\) over \(T\). Let \(\mathcal{B}\) be the set of affine opens of \(T\) which map into an affine open of \(S\). This is a basis for the topology of \(T\). Below we will show that for \(W \in \mathcal{B}\) the fibre product \(F' \times_{G'} W\) is representable by a scheme \(X_W\) over \(W\). If \(W_1 \subset W_2\) in \(\mathcal{B}\), then we obtain an isomorphism \(X_{W_1} \to X_{W_2} \times_{W_2} W_1\) because both \(X_{W_1}\) and \(X_{W_2} \times_{W_2} W_1\) represent the functor \(F' \times_{G'} W_1\). These isomorphisms are canonical and satisfy the cocycle condition mentioned in Constructions, Lemma 01LH. Hence we can glue the schemes \(X_W\) to a scheme \(X\) over \(T\). Compatibility of the glueing maps with the maps \(X_W \to F'\) provides us with a map \(X \to F'\). The resulting map \(X \to F' \times_{G'} T\) is an isomorphism as we may check this locally on \(T\) (as source and target of this arrow are sheaves for the Zariski topology).

Let \(W\) be an affine scheme which maps into an affine open \(U \subset S\). Let \(W \to G'\) be a map. Still assuming (3) we have to show that \(F' \times_{G'} W\) is representable by a scheme. We may write \(W = \lim V'_i\) as a directed limit of affine schemes \(V'_i\) of finite presentation over \(U\), see Algebra, Lemma 00QN. Since \(V'_i\) is of finite type over a Noetherian scheme, we see that \(V'_i\) is a Noetherian scheme. Set \(V_i = V'_i\), viewed as an object of \((\textit{Noetherian}/S)_\tau\). As \(G'\) is limit preserving, we can choose an \(i\) and a map \(V'_i \to G'\) such that \(W \to G'\) is the composition \(W \to V'_i \to G'\). Since \(G\) is the restriction of \(G'\) to \((\textit{Noetherian}/S)_\tau\) the morphism \(V'_i \to G'\) is the same thing as a morphism \(V_i \to G\) (see above). By assumption (3) the functor \(F \times_G V_i\) is representable by an object \(X_i\) of \((\textit{Noetherian}/S)_\tau\). The functor \(F \times_G V_i\) is limit preserving as it is the restriction of \(F' \times_{G'} V'_i\) and this functor is limit preserving by Limits of Spaces, Lemma 0GDZ, the assumption that \(F'\) and \(G'\) are limit preserving, and Limits, Remark 05LX which tells us that the functor of points of \(V'_i\) is limit preserving. By Lemma 0GE4 we conclude that \(X_i\) is locally of finite presentation over \(S\). Set \(X'_i = X_i\), viewed as an object of \((\Sch/S)_\tau\). Then we see that \(F' \times_{G'} V'_i\) and the functor of points \(h_{X'_i}\) are both extensions of \(h_{X_i} : (\textit{Noetherian}/S)_\tau^{opp} \to \textit{Sets}\) to limit preserving sheaves on \((\Sch/S)_\tau\). By the equivalence of categories of Lemma 0GE3 we deduce that \(X'_i\) represents \(F' \times_{G'} V'_i\). Then finally \[F' \times_{G'} W = F' \times_{G'} V'_i \times_{V'_i} W = X'_i \times_{V'_i} W\] is representable as desired.

Algebraic spaces in the Noetherian setting

Let \(S\) be a locally Noetherian scheme. Let \((\textit{Noetherian}/S)_\etale \subset (\Sch/S)_\etale\) denote the site studied in Section 0GE1. Let \(F : (\textit{Noetherian}/S)_\etale^{opp} \to \textit{Sets}\) be a functor, i.e., \(F\) is a presheaf on \((\textit{Noetherian}/S)_\etale\). In this setting all the axioms [-1], [0], [1], [2], [3], [4], [5] of Section 07XZ make sense. We will review them one by one and make sure the reader knows exactly what we mean.

Axiom [-1]. This is a set theoretic condition to be ignored by readers who are not interested in set theoretic questions. It makes sense for \(F\) since it concerns the evaluation of \(F\) on spectra of fields of finite type over \(S\) which are objects of \((\textit{Noetherian}/S)_\etale\).

Axiom [0]. This is the axiom that \(F\) is a sheaf on \((\textit{Noetherian}/S)_\etale\), i.e., satisfies the sheaf condition for étale coverings.

Axiom [1]. This is the axiom that \(F\) is limit preserving as defined in Section 0GE1: for any directed limit of affine schemes \(X = \lim X_i\) of \((\textit{Noetherian}/S)_\etale\) we have \(F(X) = \colim F(X_i)\).

Axiom [2]. This is the axiom that \(F\) satisfies the Rim-Schlessinger condition (RS). Looking at the definition of condition (RS) in Definition 07WP and the discussion in Section 07XZ we see that this means: given any pushout \(Y' = Y \amalg_X X'\) of schemes of finite type over \(S\) where \(Y, X, X'\) are spectra of Artinian local rings, then \[F(Y \amalg_X X') \to F(Y) \times_{F(X)} F(X')\] is a bijection. This condition makes sense as the schemes \(X\), \(X'\), \(Y\), and \(Y'\) are in \((\text{Noetherian}/S)_\etale\) since they are of finite type over \(S\).

Axiom [3]. This is the axiom that every tangent space \(TF_{k, x_0}\) is finite dimensional. This makes sense as the tangent spaces \(TF_{k, x_0}\) are constructed from evaluations of \(F\) at \(\Spec(k)\) and \(\Spec(k[\epsilon])\) with \(k\) a field of finite type over \(S\) and hence are obtained by evaluating at objects of the category \((\textit{Noetherian}/S)_\etale\).

Axiom [4]. This is the axiom that every formal object is effective. Looking at the discussion in Sections 07X3 and 07XZ we see that this involves evaluating our functor at Noetherian schemes only and hence this condition makes sense for \(F\).

Axiom [5]. This is the axiom stating that \(F\) satisfies openness of versality. Recall that this means the following: Given a scheme \(U\) locally of finite type over \(S\), given \(x \in F(U)\), and given a finite type point \(u_0 \in U\) such that \(x\) is versal at \(u_0\), then there exists an open neighbourhood \(u_0 \in U' \subset U\) such that \(x\) is versal at every finite type point of \(U'\). As before, verifying this only involves evaluating our functor at Noetherian schemes.

Proposition

Let \(S\) be a locally Noetherian scheme. Let \(F : (\textit{Noetherian}/S)_\etale^{opp} \to \textit{Sets}\) be a functor. Assume that

1. \(\Delta : F \to F \times F\) is representable (as a transformation of functors, see Categories, Definition 001X),

2. \(F\) satisfies axioms [-1], [0], [1], [2], [3], [4], [5] (see above), and

3. \(\mathcal{O}_{S, s}\) is a G-ring for all finite type points \(s\) of \(S\).

Then there exists a unique algebraic space \(F' : (\Sch/S)_{fppf}^{opp} \to \textit{Sets}\) whose restriction to \((\textit{Noetherian}/S)_\etale\) is \(F\) (see proof for elucidation).

Proof

Recall that the sites \((\Sch/S)_{fppf}\) and \((\Sch/S)_\etale\) have the same underlying category, see discussion in Section 0GE1. Similarly the sites \((\textit{Noetherian}/S)_\etale\) and \((\textit{Noetherian}/S)_{fppf}\) have the same underlying category. By axioms [0] and [1] the functor \(F\) is a sheaf and limit preserving. Let \(F' : (\Sch/S)_\etale^{opp} \to \textit{Sets}\) be the unique extension of \(F\) which is a sheaf (for the étale topology) and which is limit preserving, see Lemma 0GE3. Then \(F'\) satisfies axioms [0] and [1] as given in Section 07XZ. By Lemma 0GE5 we see that \(\Delta' : F' \to F' \times F'\) is representable (by schemes). On the other hand, it is immediately clear that \(F'\) satisfies axioms [-1], [2], [3], [4], [5] of Section 07XZ as each of these involves only evaluating \(F'\) at objects of \((\textit{Noetherian}/S)_\etale\) and we’ve assumed the corresponding conditions for \(F\). Whence \(F'\) is an algebraic space by Proposition 07Y1.

Artin’s theorem on contractions

In this section we will freely use the language of formal algebraic spaces, see Formal Spaces, Section 0AHX. Artin’s theorem on contractions is one of the two main theorems of Artin’s paper [ArtinII]; the first one is his theorem on dilatations which we stated and proved in Algebraization of Formal Spaces, Section 0GDT.

Situation

Let \(S\) be a locally Noetherian scheme. Let \(X'\) be an algebraic space locally of finite type over \(S\). Let \(T' \subset |X'|\) be a closed subset. Let \(U' \subset X'\) be the open subspace with \(|U'| = |X'| \setminus T'\). Let \(W\) be a locally Noetherian formal algebraic space over \(S\) with \(W_{red}\) locally of finite type over \(S\). Finally, we let \[g : X'_{/T'} \longrightarrow W\] be a formal modification, see Algebraization of Formal Spaces, Definition 0GDK. Recall that \(X'_{/T'}\) denotes the formal completion of \(X'\) along \(T'\), see Formal Spaces, Section 0AIX.

In the situation above our goal is to prove that there exists a proper morphism \(f : X' \to X\) of algebraic spaces over \(S\), a closed subset \(T \subset |X|\), and an isomorphism \(a : X_{/T} \to W\) of formal algebraic spaces such that

1. \(T'\) is the inverse image of \(T\) by \(|f| : |X'| \to |X|\),

2. \(f : X' \to X\) maps \(U'\) isomorphically to an open subspace \(U\) of \(X\), and

3. \(g = a \circ f_{/T}\) where \(f_{/T} : X'_{/T'} \to X_{/T}\) is the induced morphism.

Let us say that \((f : X' \to X, T, a)\) is a solution.

We will follow Artin’s strategy by constructing a functor \(F\) on the category of locally Noetherian schemes over \(S\), showing that \(F\) is an algebraic space using Proposition 0GE7, and proving that setting \(X = F\) works.

Remark

In particular, we cannot prove that the desired result is true for every Situation 0GH8 because we will need to assume the local rings of \(S\) are G-rings. If you can prove the result in general or if you have a counter example, please let us know at stacks.project@gmail.com.

In Situation 0GH8 let \(V\) be a locally Noetherian scheme over \(S\). The value of our functor \(F\) on \(V\) will be all triples \[(Z, u' : V \setminus Z \to U', \hat x : V_{/Z} \to W)\] satisfying the following conditions

1. \(Z \subset V\) is a closed subset,

2. \(u' : V \setminus Z \to U'\) is a morphism over \(S\),

3. \(\hat x : V_{/Z} \to W\) is an adic morphism of formal algebraic spaces over \(S\),

4. \(u'\) and \(\hat x\) are compatible (see below).

The compatibility condition is the following: pulling back the formal modification \(g\) we obtain a formal modification \[X'_{/T'} \times_{g, W, \hat x} V_{/Z} \longrightarrow V_{/Z}\] See Algebraization of Formal Spaces, Lemma 0GDN. By the main theorem on dilatations (Algebraization of Formal Spaces, Theorem 0GDU), there is a unique proper morphism \(V' \to V\) of algebraic spaces which is an isomorphism over \(V \setminus Z\) such that \(V'_{/Z} \to V_{/Z}\) is isomorphic to the displayed arrow. In other words, for some morphism \(\hat x' : V'_{/Z} \to X'_{/T'}\) we have a cartesian diagram \[\xymatrix{ V'_{/Z} \ar[r] \ar[d]_{\hat x'} & V_{/Z} \ar[d]^{\hat x} \\ X'_{/T'} \ar[r]^g & W }\] of formal algebraic spaces. We will think of \(V \setminus Z\) as an open subspace of \(V'\) without further mention. The compatibility condition is that there should be a morphism \(x' : V' \to X'\) restricting to \(u'\) and \(\hat x'\) over \(V \setminus Z \subset V'\) and \(V'_{/Z}\). In other words, such that the diagram \[\xymatrix{ V \setminus Z \ar[r] \ar[d]_{u'} & V' \ar[d]^{x'} & V'_{/Z} \ar[l] \ar[d]^{\hat x'} \ar[r] & V_{/Z} \ar[d]^{\hat x} \\ U' \ar[r] & X' & X'_{/T'} \ar[r]^g \ar[l] & W }\] is commutative. Observe that by Algebraization of Formal Spaces, Lemma 0AR7 the morphism \(x'\) is unique if it exists. We will indicate this situation by saying “\(V' \to V\), \(\hat x'\), and \(x'\) witness the compatibility between \(u'\) and \(\hat x\)”.

Remark

In Situation 0GH8 let \(V\) be a locally Noetherian scheme over \(S\). Let \((Z, u', \hat x)\) be a triple satisfying (1), (2), and (3) above. We want to explain a way to think about the compatibility condition (4). It will not be mathematically precise as we are going to use a fictitious category \(\textit{An}_S\) of analytic spaces over \(S\) and a fictitious analytification functor \[\left\{ \begin{matrix} \text{locally Noetherian formal} \\ \text{algebraic spaces over }S \end{matrix} \right\} \longrightarrow \textit{An}_S, \quad\quad Y \longmapsto Y^{an}\] For example if \(Y = \text{Spf}(k[[t]])\) over \(S = \Spec(k)\), then \(Y^{an}\) should be thought of as an open unit disc. If \(Y = \Spec(k)\), then \(Y^{an}\) is a single point. The category \(\textit{An}_S\) should have open and closed immersions and we should be able to take the open complement of a closed. Given \(Y\) the morphism \(Y_{red} \to Y\) should induce a closed immersion \(Y_{red}^{an} \to Y^{an}\). We set \(Y^{rig} = Y^{an} \setminus Y_{red}^{an}\) equal to its open complement. If \(Y\) is an algebraic space and if \(Z \subset Y\) is closed, then the morphism \(Y_{/Z} \to Y\) should induce an open immersion \(Y_{/Z}^{an} \to Y^{an}\) which in turn should induce an open immersion \[can : (Y_{/Z})^{rig} \longrightarrow (Y \setminus Z)^{an}\] Also, given a formal modification \(g : Y' \to Y\) of locally Noetherian formal algebraic spaces, the induced morphism \(g^{rig} : (Y')^{rig} \to Y^{rig}\) should be an isomorphism. Given \(\text{An}_S\) and the analytification functor, we can consider the requirement that \[\xymatrix{ (V_{/Z})^{rig} \ar[rr]_{can} \ar[d]_{(g^{rig})^{-1} \circ \hat x^{an}} & & (V \setminus Z)^{an} \ar[d]^{(u')^{an}} \\ (X'_{/T'})^{rig} \ar[rr]^{can} & & (X' \setminus T')^{an} }\] commutes. This makes sense as \(g^{rig} : (X'_{/T'})^{rig} \to W^{rig}\) is an isomorphism and \(U' = X' \setminus T'\). Finally, under some assumptions of faithfulness of the analytification functor, this requirement will be equivalent to the compatibility condition formulated above. We hope this will motivate the reader to think of the compatibility of \(u'\) and \(\hat x\) as the requirement that some maps be equal, rather than asking for the existence of a certain commutative diagram.

Lemma

In Situation 0GH8 the rule \(F\) that sends a locally Noetherian scheme \(V\) over \(S\) to the set of triples \((Z, u', \hat x)\) satisfying the compatibility condition and which sends a morphism \(\varphi : V_2 \to V_1\) of locally Noetherian schemes over \(S\) to the map \[F(\varphi) : F(V_1) \longrightarrow F(V_2)\] sending an element \((Z_1, u'_1, \hat x_1)\) of \(F(V_1)\) to \((Z_2, u'_2, \hat x_2)\) in \(F(V_2)\) given by

1. \(Z_2 \subset V_2\) is the inverse image of \(Z_1\) by \(\varphi\),

2. \(u'_2\) is the composition of \(u'_1\) and \(\varphi|_{V_2 \setminus Z_2} : V_2 \setminus Z_2 \to V_1 \setminus Z_1\),

3. \(\hat x_2\) is the composition of \(\hat x_1\) and \(\varphi_{/Z_2} : V_{2, /Z_2} \to V_{1, /Z_1}\)

is a contravariant functor.

Proof

To see the compatibility condition between \(u'_2\) and \(\hat x_2\), let \(V'_1 \to V_1\), \(\hat x'_1\), and \(x'_1\) witness the compatibility between \(u'_1\) and \(\hat x_1\). Set \(V'_2 = V_2 \times_{V_1} V'_1\), set \(\hat x'_2\) equal to the composition of \(\hat x'_1\) and \(V'_{2, /Z_2} \to V'_{1, /Z_1}\), and set \(x'_2\) equal to the composition of \(x'_1\) and \(V'_2 \to V'_1\). Then \(V'_2 \to V_2\), \(\hat x'_2\), and \(x'_2\) witness the compatibility between \(u'_2\) and \(\hat x_2\). We omit the detailed verification.

Lemma

In Situation 0GH8 if there exists a solution \((f : X' \to X, T, a)\) then there is a functorial bijection \(F(V) = \Mor_S(V, X)\) on the category of locally Noetherian schemes \(V\) over \(S\).

Proof

Let \(V\) be a locally Noetherian scheme over \(S\). Let \(x : V \to X\) be a morphism over \(S\). Then we get an element \((Z, u', \hat x)\) in \(F(V)\) as follows

1. \(Z \subset V\) is the inverse image of \(T\) by \(x\),

2. \(u' : V \setminus Z \to U' = U\) is the restriction of \(x\) to \(V \setminus Z\),

3. \(\hat x : V_{/Z} \to W\) is the composition of \(x_{/Z} : V_{/Z} \to X_{/T}\) with the isomorphism \(a : X_{/T} \to W\).

This triple satisfies the compatibility condition because we can take \(V' = V \times_{x, X} X'\), we can take \(\hat x'\) to be the completion of the projection \(x' : V' \to X'\).

Conversely, suppose given an element \((Z, u', \hat x)\) of \(F(V)\). We claim there is a unique morphism \(x : V \to X\) compatible with \(u'\) and \(\hat x\). Namely, let \(V' \to V\), \(\hat x'\), and \(x'\) witness the compatibility between \(u'\) and \(\hat x\). Then Algebraization of Formal Spaces, Proposition 0GI3 is exactly the result we need to find a unique morphism \(x : V \to X\) agreeing with \(\hat x\) over \(V_{/Z}\) and with \(x'\) over \(V'\) (and a fortiori agreeing with \(u'\) over \(V \setminus Z\)).

We omit the verification that the two constructions above define inverse bijections between their respective domains.

Lemma

In Situation 0GH8 if there exists an algebraic space \(X\) locally of finite type over \(S\) and a functorial bijection \(F(V) = \Mor_S(V, X)\) on the category of locally Noetherian schemes \(V\) over \(S\), then \(X\) is a solution.

Proof

We have to construct a proper morphism \(f : X' \to X\), a closed subset \(T \subset |X|\), and an isomorphism \(a : X_{/T} \to W\) with properties (1), (2), (3) listed just below Situation 0GH8.

The discussion in this proof is a bit pedantic because we want to carefully match the underlying categories. In this paragraph we explain how the adventurous reader can proceed less timidly. Namely, the reader may extend our definition of the functor \(F\) to all locally Noetherian algebraic spaces over \(S\). Doing so the reader may then conclude that \(F\) and \(X\) agree as functors on the category of these algebraic spaces, i.e., \(X\) represents \(F\). Then one considers the universal object \((T, u', \hat x)\) in \(F(X)\). Then the reader will find that for the triple \(X'' \to X\), \(\hat x'\), \(x'\) witnessing the compatibility between \(u'\) and \(\hat x\) the morphism \(x' : X'' \to X'\) is an isomorphism and this will produce \(f : X' \to X\) by inverting \(x'\). Finally, we already have \(T \subset |X|\) and the reader may show that \(\hat x\) is an isomorphism which can serve as the last ingredient namely \(a\).

Denote by \(h_X(-) = \Mor_S(-, X)\) the functor of points of \(X\) restricted to the category \((\textit{Noetherian}/S)_\etale\) of Section 0GE1. By Limits of Spaces, Remark 05N0 the algebraic spaces \(X\) and \(X'\) are limit preserving. Hence so are the restrictions \(h_X\) and \(h_{X'}\). To construct \(f\) it therefore suffices to construct a transformation \(h_{X'} \to h_X = F\), see Lemma 0GE3. Thus let \(V \to S\) be an object of \((\textit{Noetherian}/S)_\etale\) and let \(\tilde x : V \to X'\) be in \(h_{X'}(V)\). Then we get an element \((Z, u', \hat x)\) in \(F(V)\) as follows

1. \(Z \subset V\) is the inverse image of \(T'\) by \(\tilde x\),

2. \(u' : V \setminus Z \to U'\) is the restriction of \(\tilde x\) to \(V \setminus Z\),

3. \(\hat x : V_{/Z} \to W\) is the composition of \(\tilde x_{/Z} : V_{/Z} \to X'_{/T'}\) with \(g : X'_{/T'} \to W\).

This triple satisfies the compatibility condition: first we always obtain \(V' \to V\) and \(\hat x' : V'_{/Z} \to X'_{/T'}\) for free (see discussion preceding Lemma 0GHA). Then we just define \(x' : V' \to X'\) to be the composition of \(V' \to V\) and the morphism \(\tilde x : V \to X'\). We omit the verification that this works.

If \(\xi : V \to X\) is an étale morphism where \(V\) is a scheme, then we obtain \(\xi = (Z, u', \hat x) \in F(V) = h_X(V) = X(V)\). Of course, if \(\varphi : V' \to V\) is a further étale morphism of schemes, then \((Z, u', \hat x)\) pulled back to \(F(V')\) corresponds to \(\xi \circ \varphi\). The closed subset \(T \subset |X|\) is just defined as the closed subset such that \(\xi : V \to X\) for \(\xi = (Z, u', \hat x)\) pulls \(T\) back to \(Z\).

Consider Noetherian schemes \(V\) over \(S\) and a morphism \(\xi : V \to X\) corresponding to \((Z, u', \hat x)\) as above. Then we see that \(\xi(V)\) is set theoretically contained in \(T\) if and only if \(V = Z\) (as topological spaces). Hence we see that \(X_{/T}\) agrees with \(W\) as a functor. This produces the isomorphism \(a : X_{/T} \to W\). (We’ve omitted a small detail here which is that for the locally Noetherian formal algebraic spaces \(X_{/T}\) and \(W\) it suffices to check one gets an isomorphism after evaluating on locally Noetherian schemes over \(S\).)

We omit the proof of conditions (1), (2), and (3).

Remark

In Situation 0GH8, let \(V\) be a locally Noetherian scheme over \(S\). Let \((Z_i, u'_i, \hat x_i) \in F(V)\) for \(i = 1, 2\). Let \(V'_i \to V\), \(\hat x'_i\) and \(x'_i\) witness the compatibility between \(u'_i\) and \(\hat x_i\) for \(i = 1, 2\).

Set \(V' = V'_1 \times_V V'_2\). Let \(E' \to V'\) denote the equalizer of the morphisms \[V' \to V'_1 \xrightarrow{x'_1} X' \quad\text{and}\quad V' \to V'_2 \xrightarrow{x'_2} X'\] Set \(Z = Z_1 \cap Z_2\). Let \(E_W \to V_{/Z}\) be the equalizer of the morphisms \[V_{/Z} \to V_{/Z_1} \xrightarrow{\hat x_1} W \quad\text{and}\quad V_{/Z} \to V_{/Z_2} \xrightarrow{\hat x_2} W\] Observe that \(E' \to V\) is separated and locally of finite type and that \(E_W\) is a locally Noetherian formal algebraic space separated over \(V\). The compatibilities between the various morphisms involved show that

1. \(\Im(E' \to V) \cap (Z_1 \cup Z_2)\) is contained in \(Z = Z_1 \cap Z_2\),

2. the morphism \(E' \times_V (V \setminus Z) \to V \setminus Z\) is a monomorphism and is equal to the equalizer of the restrictions of \(u'_1\) and \(u'_2\) to \(V \setminus (Z_1 \cup Z_2)\),

3. the morphism \(E'_{/Z} \to V_{/Z}\) factors through \(E_W\) and the diagram \[\xymatrix{ E'_{/Z} \ar[r] \ar[d] & X'_{/T'} \ar[d]^g \\ E_W \ar[r] & W }\] is cartesian. In particular, the morphism \(E'_{/Z} \to E_W\) is a formal modification as the base change of \(g\),

4. \(E'\), \((E' \to V)^{-1}Z\), and \(E'_{/Z} \to E_W\) form a triple as in Situation 0GH8 with base scheme the locally Noetherian scheme \(V\),

5. given a morphism \(\varphi : A \to V\) of locally Noetherian schemes, the following are equivalent

   1. \((Z_1, u'_1, \hat x_1)\) and \((Z_2, u'_2, \hat x_2)\) restrict to the same element of \(F(A)\),

   2. \(A \setminus \varphi^{-1}(Z) \to V \setminus Z\) factors through \(E' \times_V (V \setminus Z)\) and \(A_{/\varphi^{-1}(Z)} \to V_{/Z}\) factors through \(E_W\).

We conclude, using Lemmas 0GHB and 0GHC, that if there is a solution \(E \to V\) for the triple in (4), then \(E\) represents \(F \times_{\Delta, F \times F} V\) on the category of locally Noetherian schemes over \(V\).

Lemma

In Situation 0GH8 assume given a closed subset \(Z \subset S\) such that

1. the inverse image of \(Z\) in \(X'\) is \(T'\),

2. \(U' \to S \setminus Z\) is a closed immersion,

3. \(W \to S_{/Z}\) is a closed immersion.

Then there exists a solution \((f : X' \to X, T, a)\) and moreover \(X \to S\) is a closed immersion.

Proof

Suppose we have a closed subscheme \(X \subset S\) such that \(X \cap (S \setminus Z) = U'\) and \(X_{/Z} = W\). Then \(X\) represents the functor \(F\) (some details omitted) and hence is a solution. To find \(X\) is clearly a local question on \(S\). In this way we reduce to the case discussed in the next paragraph.

Assume \(S = \Spec(A)\) is affine. Let \(I \subset A\) be the radical ideal cutting out \(Z\). Write \(I = (f_1, \ldots, f_r)\). By assumption we are given

1. the closed immersion \(U' \to S \setminus Z\) determines ideals \(J_i \subset A[1/f_i]\) such that \(J_i\) and \(J_j\) generate the same ideal in \(A[1/f_if_j]\),

2. the closed immersion \(W \to S_{/Z}\) is the map \(\text{Spf}(A^\wedge/J') \to \text{Spf}(A^\wedge)\) for some ideal \(J' \subset A^\wedge\) in the \(I\)-adic completion \(A^\wedge\) of \(A\).

To finish the proof we need to find an ideal \(J \subset A\) such that \(J_i = J[1/f_i]\) and \(J' = JA^\wedge\). By More on Algebra, Proposition 05ER it suffices to show that \(J_i\) and \(J'\) generate the same ideal in \(A^\wedge[1/f_i]\) for all \(i\).

Recall that \(A' = H^0(X', \mathcal{O})\) is a finite \(A\)-algebra whose formation commutes with flat base change (Cohomology of Spaces, Lemmas 08AS and 073K). Set \(J'' = \Ker(A \to A')\)¹². We have \(J_i = J''A[1/f_i]\) as follows from base change to the spectrum of \(A[1/f_i]\). Observe that we have a commutative diagram \[\xymatrix{ X' \ar[d] & X'_{/T'} \times_{S_{/Z}} \text{Spf}(A^\wedge) \ar[l] \ar[d] & X'_{/T'} \times_W \text{Spf}(A^\wedge/J') \ar@{=}[l] \ar[d] \\ \Spec(A) & \text{Spf}(A^\wedge) \ar[l] & \text{Spf}(A^\wedge/J') \ar[l] }\] The middle vertical arrow is the completion of the left vertical arrow along the obvious closed subsets. By the theorem on formal functions we have \[(A')^\wedge = \Gamma(X' \times_S \Spec(A^\wedge), \mathcal{O}) = \lim H^0(X' \times_S \Spec(A/I^n), \mathcal{O})\] See Cohomology of Spaces, Theorem 08AZ. From the diagram we conclude that \(J'\) maps to zero in \((A')^\wedge\). Hence \(J' \subset J'' A^\wedge\). Consider the arrows \[X'_{/T'} \to \text{Spf}(A^\wedge/J''A^\wedge) \to \text{Spf}(A^\wedge/J') = W\] We know the composition \(g\) is a formal modification (in particular rig-étale and rig-surjective) and the second arrow is a closed immersion (in particular an adic monomorphism). Hence \(X'_{/T'} \to \text{Spf}(A^\wedge/J''A^\wedge)\) is rig-surjective and rig-étale, see Algebraization of Formal Spaces, Lemmas 0GH5 and 0GH4. Applying Algebraization of Formal Spaces, Lemmas 0GH6 and 0AQT we conclude that \(\text{Spf}(A^\wedge/J''A^\wedge) \to W\) is rig-étale and rig-surjective. By Algebraization of Formal Spaces, Lemma 0GD6 we conclude that \(I^n J'' A^\wedge \subset J'\) for some \(n > 0\). It follows that \(J'' A^\wedge[1/f_i] = J' A^\wedge[1/f_i]\) and we deduce \(J_i A^\wedge[1/f_i] = J' A^\wedge[1/f_i]\) for all \(i\) as desired.

Lemma

In Situation 0GH8 assume \(X' \to S\) and \(W \to S\) are separated. Then the diagonal \(\Delta : F \to F \times F\) is representable by closed immersions.

Proof

Combine Lemma 0GHE with the discussion in Remark 0GHD.

Lemma

In Situation 0GH8 the functor \(F\) satisfies the sheaf property for all étale coverings of locally Noetherian schemes over \(S\).

Proof

Omitted. Hint: morphisms may be defined étale locally.

Lemma

In Situation 0GH8 the functor \(F\) is limit preserving: for any directed limit \(V = \lim V_\lambda\) of Noetherian affine schemes over \(S\) we have \(F(V) = \colim F(V_\lambda)\).

Proof

This is an absurdly long proof. Much of it consists of standard arguments on limits and étale localization. We urge the reader to skip ahead to the last part of the proof where something interesting happens.

Let \(V = \lim_{\lambda \in \Lambda} V_\lambda\) be a directed limit of schemes over \(S\) with \(V\) and \(V_\lambda\) Noetherian and with affine transition morphisms. See Limits, Section 01YV for material on limits of schemes. We will prove that \(\colim F(V_\lambda) \to F(V)\) is bijective.

Proof of injectivity: notation. Let \(\lambda \in \Lambda\) and \(\xi_{\lambda, 1}, \xi_{\lambda, 2} \in F(V_\lambda)\) be elements which restrict to the same element of \(F(V)\). Write \(\xi_{\lambda, 1} = (Z_{\lambda, 1}, u'_{\lambda, 1}, \hat x_{\lambda, 1})\) and \(\xi_{\lambda, 2} = (Z_{\lambda, 2}, u'_{\lambda, 2}, \hat x_{\lambda, 2})\).

Proof of injectivity: agreement of \(Z_{\lambda, i}\). Since \(Z_{\lambda, 1}\) and \(Z_{\lambda, 2}\) restrict to the same closed subset of \(V\), we may after increasing \(\lambda\) assume \(Z_{\lambda, 1} = Z_{\lambda, 2}\), see Limits, Lemma 0CUF and Topology, Lemma 0A2P. Let us denote the common value \(Z_\lambda \subset V_\lambda\), for \(\mu \geq \lambda\) denote \(Z_\mu \subset V_\mu\) the inverse image in \(V_\mu\) and denote \(Z\) the inverse image in \(V\). We will use below that \(Z = \lim_{\mu \geq \lambda} Z_\mu\) as schemes if we view \(Z\) and \(Z_\mu\) as reduced closed subschemes.

Proof of injectivity: agreement of \(u'_{\lambda, i}\). Since \(U'\) is locally of finite type over \(S\) and since the restrictions of \(u'_{\lambda, 1}\) and \(u'_{\lambda, 2}\) to \(V \setminus Z\) are the same, we may after increasing \(\lambda\) assume \(u'_{\lambda, 1} = u'_{\lambda, 2}\), see Limits, Proposition 01ZC. Let us denote the common value \(u'_\lambda\) and denote \(u'\) the restriction to \(V \setminus Z\).

Proof of injectivity: restatement. At this point we have \(\xi_{\lambda, 1} = (Z_\lambda, u'_\lambda, \hat x_{\lambda, 1})\) and \(\xi_{\lambda, 2} = (Z_\lambda, u'_\lambda, \hat x_{\lambda, 2})\). The main problem we face in this part of the proof is to show that the morphisms \(\hat x_{\lambda, 1}\) and \(\hat x_{\lambda, 2}\) become the same after increasing \(\lambda\).

Proof of injectivity: agreement of \(\hat x_{\lambda, i}|_{Z_\lambda}\). Consider the morphisms \(\hat x_{\lambda, 1}|_{Z_\lambda}, \hat x_{\lambda, 2}|_{Z_\lambda} : Z_\lambda \to W_{red}\). These morphisms restrict to the same morphism \(Z \to W_{red}\). Since \(W_{red}\) is a scheme locally of finite type over \(S\) we see using Limits, Proposition 01ZC that after replacing \(\lambda\) by a bigger index we may assume \(\hat x_{\lambda, 1}|_{Z_\lambda} = \hat x_{\lambda, 2}|_{Z_\lambda} : Z_\lambda \to W_{red}\).

Proof of injectivity: end. Next, we are going to apply the discussion in Remark 0GHD to \(V_\lambda\) and the two elements \(\xi_{\lambda, 1}, \xi_{\lambda, 2} \in F(V_\lambda)\). This gives us

1. \(e_\lambda : E_\lambda' \to V_\lambda\) separated and locally of finite type,

2. \(e_\lambda^{-1}(V_\lambda \setminus Z_\lambda) \to V_\lambda \setminus Z_\lambda\) is an isomorphism,

3. a monomorphism \(E_{W, \lambda} \to V_{\lambda, /Z_\lambda}\) which is the equalizer of \(\hat x_{\lambda, 1}\) and \(\hat x_{\lambda, 2}\),

4. a formal modification \(E'_{\lambda, /Z_\lambda} \to E_{W, \lambda}\)

Assertion (2) holds by assertion (2) in Remark 0GHD and the preparatory work we did above getting \(u'_{\lambda, 1} = u'_{\lambda, 2} = u'_\lambda\). Since \(Z_\lambda = (V_{\lambda, /Z_\lambda})_{red}\) factors through \(E_{W, \lambda}\) because \(\hat x_{\lambda, 1}|_{Z_\lambda} = \hat x_{\lambda, 2}|_{Z_\lambda}\) we see from Formal Spaces, Lemma 0GHZ that \(E_{W, \lambda} \to V_{\lambda, /Z_\lambda}\) is a closed immersion. Then we see from assertion (4) in Remark 0GHD and Lemma 0GHE applied to the triple \(E_\lambda'\), \(e_\lambda^{-1}(Z_\lambda)\), \(E'_{\lambda, /Z_\lambda} \to E_{W, \lambda}\) over \(V_\lambda\) that there exists a closed immersion \(E_\lambda \to V_\lambda\) which is a solution for this triple. Next we use assertion (5) in Remark 0GHD which combined with Lemma 0GHB says that \(E_\lambda\) is the “equalizer” of \(\xi_{\lambda, 1}\) and \(\xi_{\lambda, 2}\). In particular, we see that \(V \to V_\lambda\) factors through \(E_\lambda\). Then using Limits, Proposition 01ZC once more we find \(\mu \geq \lambda\) such that \(V_\mu \to V_\lambda\) factors through \(E_\lambda\) and hence the pullbacks of \(\xi_{\lambda, 1}\) and \(\xi_{\lambda, 2}\) to \(V_\mu\) are the same as desired.

Proof of surjectivity: statement. Let \(\xi = (Z, u', \hat x)\) be an element of \(F(V)\). We have to find a \(\lambda \in \Lambda\) and an element \(\xi_\lambda \in F(V_\lambda)\) restricting to \(\xi\).

Proof of surjectivity: the question is étale local. By the unicity proved in the previous part of the proof and by the sheaf property of \(F\) in Lemma 0GHG, the problem is local on \(V\) in the étale topology. More precisely, let \(v \in V\). We claim it suffices to find an étale morphism \((\tilde V, \tilde v) \to (V, v)\) and some \(\lambda\), an étale morphism \(\tilde V_\lambda \to V_\lambda\), and some element \(\tilde \xi_\lambda \in F(\tilde V_\lambda)\) such that \(\tilde V = \tilde V_\lambda \times_{V_\lambda} V\) and \(\xi|_{\tilde V} = \tilde \xi_\lambda|_{\tilde V}\). We omit a detailed proof of this claim¹³.

Proof of surjectivity: rephrasing the problem. Recall that any étale morphism \((\tilde V, \tilde v) \to (V, v)\) with \(\tilde V\) affine is the base change of an étale morphism \(\tilde V_\lambda \to V_\lambda\) with \(\tilde V_\lambda\) affine for some \(\lambda\), see for example Topologies, Lemma 049N. Given \(\tilde V_\lambda\) we have \(\tilde V = \lim_{\mu \geq \lambda} \tilde V_\lambda \times_{V_\lambda} V_\mu\). Hence given \((\tilde V, \tilde v) \to (V, v)\) étale with \(\tilde V\) affine, we may replace \((V, v)\) by \((\tilde V, \tilde v)\) and \(\xi\) by the restriction of \(\xi\) to \(\tilde V\).

Proof of surjectivity: reduce to base being affine. In particular, suppose \(\tilde S \subset S\) is an affine open subscheme such that \(v \in V\) maps to a point of \(\tilde S\). Then we may according to the previous paragraph, replace \(V\) by \(\tilde V = \tilde S \times_S V\). Of course, if we do this, it suffices to solve the problem for the functor \(F\) restricted to the category of locally Noetherian schemes over \(\tilde S\). This functor is of course the functor associated to the whole situation base changed to \(\tilde S\). Thus we may and do assume \(S = \Spec(R)\) is a Noetherian affine scheme for the rest of the proof.

Proof of surjectivity: easy case. If \(v \in V \setminus Z\), then we can take \(\tilde V = V \setminus Z\). This descends to an open subscheme \(\tilde V_\lambda \subset V_\lambda\) for some \(\lambda\) by Limits, Lemma 01Z4. Next, after increasing \(\lambda\) we may assume there is a morphism \(u'_\lambda : \tilde V_\lambda \to U'\) restricting to \(u'\). Taking \(\tilde \xi_\lambda = (\emptyset, u'_\lambda, \emptyset)\) gives the desired element of \(F(\tilde V_\lambda)\).

Proof of surjectivity: hard case and reduction to affine \(W\). The most difficult case comes from considering \(v \in Z \subset V\). We claim that we can reduce this to the case where \(W\) is an affine formal scheme; we urge the reader to skip this argument¹⁴. Namely, we can choose an étale morphism \(\tilde W \to W\) where \(\tilde W\) is an affine formal algebraic space such that the image of \(v\) by \(\hat x : V_{/Z} \to W\) is in the image of \(\tilde W \to W\) (on reductions). Then the morphisms \[p : \tilde W \times_{W, g} X'_{/T'} \longrightarrow X'_{/T'}\] and \[q : \tilde W \times_{W, \hat x} V_{/Z} \to V_{/Z}\] are étale morphisms of locally Noetherian formal algebraic spaces. By (an easy case of) Algebraization of Formal Spaces, Theorem 0ARB there exists a morphism \(\tilde X' \to X'\) of algebraic spaces which is locally of finite type, is an isomorphism over \(U'\), and such that \(\tilde X'_{/T'} \to X'_{/T'}\) is isomorphic to \(p\). By Algebraization of Formal Spaces, Lemma 0GI4 the morphism \(\tilde X' \to X'\) is étale. Let \(\tilde T' \subset |\tilde X'|\) denote the inverse image of \(T'\). Let \(\tilde U' \subset \tilde X'\) denote the complementary open subspace. Let \(\tilde g : \tilde X'_{/\tilde T'} \to \tilde W\) denote the formal modification which is the base change of \(g\) by \(\tilde W \to W\). Then we see that \[\tilde X',\ \tilde T',\ \tilde U',\ \tilde W, \ \tilde g : \tilde X'_{/\tilde T'} \to \tilde W\] is another example of Situation 0GH8. Let \(\tilde F\) denote the functor constructed from this triple. There is a transformation of functors \[\tilde F \longrightarrow F\] constructed using the morphisms \(\tilde X' \to X'\) and \(\tilde W \to W\) in the obvious manner; details omitted.

Proof of surjectivity: hard case and reduction to affine \(W\), part 2. By the same theorem as used above, there exists a morphism \(\tilde V \to V\) of algebraic spaces which is locally of finite type, is an isomorphism over \(V \setminus Z\) and such that \(\tilde V_{/Z} \to V_{/Z}\) is isomorphic to \(q\). Let \(\tilde Z \subset \tilde V\) denote the inverse image of \(Z\). By Algebraization of Formal Spaces, Lemmas 0GI4 and 0ARW the morphism \(\tilde V \to V\) is étale and separated. In particular \(\tilde V\) is a (locally Noetherian) scheme, see for example Morphisms of Spaces, Proposition 03XX. We have the morphism \(u'\) which we may view as a morphism \[\tilde u' : \tilde V \setminus \tilde Z \longrightarrow \tilde U'\] where \(\tilde U' \subset \tilde X'\) is the open mapping isomorphically to \(U'\). We have a morphism \[\tilde {\hat x} : \tilde V_{/\tilde Z} = \tilde W \times_{W, \hat x} V_{/Z} \longrightarrow \tilde W\] Namely, here we just use the projection. Thus we have the triple \[\tilde \xi = (\tilde Z, \tilde u', \tilde {\hat x}) \in \tilde F(\tilde V)\] We omit proving the compatibility condition; hints: if \(V' \to V\), \(\hat x'\), and \(x'\) witness the compatibility between \(u'\) and \(\hat x\), then one sets \(\tilde V' = V' \times_V \tilde V\) which comes with morphisms \(\tilde{\hat x}'\) and \(\tilde x'\) and shows this works. The image of \(\tilde \xi\) under the transformation \(\tilde F \to F\) is the restriction of \(\xi\) to \(\tilde V\).

Proof of surjectivity: hard case and reduction to affine \(W\), part 3. By our choice of \(\tilde W \to W\), there is an affine open \(\tilde V_{open} \subset \tilde V\) (we’re running out of notation) whose image in \(V\) contains our chosen point \(v \in V\). Now by the case studied in the next paragraph and the remarks made earlier, we can descend \(\tilde \xi|_{\tilde V_{open}}\) to some element \(\tilde \xi_\lambda\) of \(\tilde F\) over \(\tilde V_{\lambda, open}\) for some étale morphism \(\tilde V_{\lambda, open} \to V_\lambda\) whose base change to \(V\) is \(\tilde V_{open}\). Applying the transformation of functors \(\tilde F \to F\) we obtain the element of \(F(\tilde V_{\lambda, open})\) we were looking for. This reduces us to the case discussed in the next paragraph.

Proof of surjectivity: the case of an affine \(W\). We have \(v \in Z \subset V\) and \(W\) is an affine formal algebraic space. Recall that \[\xi = (Z, u', \hat x) \in F(V)\] We may still replace \(V\) by an étale neighbourhood of \(v\). In particular we may and do assume \(V\) and \(V_\lambda\) are affine.

Proof of surjectivity: descending \(Z\). We can find a \(\lambda\) and a closed subscheme \(Z_\lambda \subset V_\lambda\) such that \(Z\) is the base change of \(Z_\lambda\) to \(V\). See Limits, Lemma 01ZM. Warning: we don’t know (and in general it won’t be true) that \(Z_\lambda\) is a reduced closed subscheme of \(V_\lambda\). For \(\mu \geq \lambda\) denote \(Z_\mu \subset V_\mu\) the scheme theoretic inverse image in \(V_\mu\). We will use below that \(Z = \lim_{\mu \geq \lambda} Z_\mu\) as schemes.

Proof of surjectivity: descending \(u'\). Since \(U'\) is locally of finite type over \(S\) we may assume after increasing \(\lambda\) that there exists a morphism \(u'_\lambda : V_\lambda \setminus Z_\lambda \to U'\) whose restriction to \(V \setminus Z\) is \(u'\). See Limits, Proposition 01ZC. For \(\mu \geq \lambda\) we will denote \(u'_\mu\) the restriction of \(u'_\lambda\) to \(V_\mu \setminus Z_\mu\).

Proof of surjectivity: descending a witness. Let \(V' \to V\), \(\hat x'\), and \(x'\) witness the compatibility between \(u'\) and \(\hat x\). Using the same references as above we may assume (after increasing \(\lambda\)) that there exists a morphism \(V'_\lambda \to V_\lambda\) of finite type whose base change to \(V\) is \(V' \to V\). After increasing \(\lambda\) we may assume \(V'_\lambda \to V_\lambda\) is proper (Limits, Lemma 081F). Next, we may assume \(V'_\lambda \to V_\lambda\) is an isomorphism over \(V_\lambda \setminus Z_\lambda\) (Limits, Lemma 081E). Next, we may assume there is a morphism \(x'_\lambda : V'_\lambda \to X'\) whose restriction to \(V'\) is \(x'\). Increasing \(\lambda\) again we may assume \(x'_\lambda\) agrees with \(u'_\lambda\) over \(V_\lambda \setminus Z_\lambda\). For \(\mu \geq \lambda\) we denote \(V'_\mu\) and \(x'_\mu\) the base change of \(V'_\lambda\) and the restriction of \(x'_\lambda\).

Proof of surjectivity: algebra. Write \(W = \text{Spf}(B)\), \(V = \Spec(A)\), and for \(\mu \geq \lambda\) write \(V_\mu = \Spec(A_\mu)\). Let \(I_\mu \subset A_\mu\) and \(I \subset A\) denote the ideals cutting out \(Z_\mu\) and \(Z\). Then \(I_\lambda A_\mu = I_\mu\) and \(I_\lambda A = I\). The morphism \(\hat x\) determines and is determined by a continuous ring map \[(\hat x)^\sharp : B \longrightarrow A^\wedge\] where \(A^\wedge\) is the \(I\)-adic completion of \(A\). To finish the proof we have to show that this map descends to a map into \(A_\mu^\wedge\) for some sufficiently large \(\mu\) where \(A_\mu^\wedge\) is the \(I_\mu\)-adic completion of \(A_\mu\). This is a nontrivial fact; Artin writes in his paper [ArtinII]: “Since the data (3.5) involve \(I\)-adic completions, which do not commute with direct limits, the verification is somewhat delicate. It is an algebraic analogue of a convergence proof in analysis.”

Proof of surjectivity: algebra, more rings. Let us denote \[C_\mu = \Gamma(V'_\mu, \mathcal{O}) \quad\text{and}\quad C = \Gamma(V', \mathcal{O})\] Observe that \(A \to C\) and \(A_\mu \to C_\mu\) are finite ring maps as \(V' \to V\) and \(V'_\mu \to V_\mu\) are proper morphisms, see Cohomology of Spaces, Lemma 08AS. Since \(V = \lim V_\mu\) and \(V' = \lim V'_\mu\) we have \[A = \colim A_\mu \quad\text{and}\quad C = \colim C_\mu\] by Limits, Lemma 01Z0¹⁵. For an element \(a \in I\), resp. \(a \in I_\mu\) the maps \(A_a \to C_a\), resp. \((A_\mu)_a \to (C_\mu)_a\) are isomorphisms by flat base change (Cohomology of Spaces, Lemma 073K). Hence the kernel and cokernel of \(A \to C\) are supported on \(V(I)\) and similarly for \(A_\mu \to C_\mu\). We conclude the kernel and cokernel of \(A \to C\) are annihilated by a power of \(I\) and the kernel and cokernel of \(A_\mu \to C_\mu\) are annihilated by a power of \(I_\mu\), see Algebra, Lemma 00L6.

Proof of surjectivity: algebra, more ring maps. Let \(Z_n \subset V\) and \(Z_{\mu, n} \subset V_\mu\) denote the respective \(n\)th infinitesimal neighbourhoods of \(Z\) and \(Z_\mu\). By the theorem on formal functions (Cohomology of Spaces, Theorem 08AZ) we have \[C^\wedge = \lim_n H^0(V' \times_V Z_n, \mathcal{O}) \quad\text{and}\quad C_\mu^\wedge = \lim_n H^0(V'_\mu \times_{V_\mu} Z_{\mu, n}, \mathcal{O})\] where \(C^\wedge\) and \(C_\mu^\wedge\) are the completions with respect to \(I\) and \(I_\mu\). Combining the completion of the morphism \(x'_\mu : V'_\mu \to X'\) with the morphism \(g : X'_{/T'} \to W\) we obtain \[g \circ x'_{\mu, /Z_\mu} : V'_{\mu, /Z_\mu} = \colim V_\mu' \times_{V_\mu} Z_{\mu, n} \longrightarrow W\] and hence by the description of the completion \(C_\mu^\wedge\) above we obtain a continuous ring homomorphism \[(g \circ x'_{\mu, /Z_\mu})^\sharp : B \longrightarrow C_\mu^\wedge\] The fact that \(V' \to V\), \(\hat x'\), \(x'\) witness the compatibility between \(u'\) and \(\hat x\) implies the commutativity of the following diagram \[\xymatrix{ C_\mu^\wedge \ar[r] & C^\wedge \\ B \ar[u]^{(g \circ x'_{\mu, /Z_\mu})^\sharp} \ar[r]^{(\hat x)^\sharp} & A^\wedge \ar[u] }\]

Proof of surjectivity: more algebra arguments. Recall that the finite \(A\)-modules \(\Ker(A \to C)\) and \(\Coker(A \to C)\) are annihilated by a power of \(I\) and similarly the finite \(A_\mu\)-modules \(\Ker(A_\mu \to C_\mu)\) and \(\Coker(A_\mu \to C_\mu)\) are annihilated by a power of \(I_\mu\). This implies that these modules are equal to their completions. Since \(I\)-adic completion on the category of finite \(A\)-modules is exact (see Algebra, Section 0BNH) it follows that we have \[\Coker(A^\wedge \to C^\wedge) = \Coker(A \to C)\] and similarly for kernels and for the maps \(A_\mu \to C_\mu\). Of course we also have \[\Ker(A \to C) = \colim \Ker(A_\mu \to C_\mu) \quad\text{and}\quad \Coker(A \to C) = \colim \Coker(A_\mu \to C_\mu)\] Recall that \(S = \Spec(R)\) is affine. All of the ring maps above are \(R\)-algebra homomorphisms as all of the morphisms are morphisms over \(S\). By Algebraization of Formal Spaces, Lemma 0GC6 we see that \(B\) is topologically of finite type over \(R\). Say \(B\) is topologically generated by \(b_1, \ldots, b_n\). Pick some \(\mu\) (for example \(\lambda\)) and consider the elements \[\text{images of } (g \circ x'_{\mu, /Z_\mu})^\sharp(b_1) , \ldots, (g \circ x'_{\mu, /Z_\mu})^\sharp(b_n) \text{ in }\Coker(A_\mu \to C_\mu)\] The image of these elements in \(\Coker(A \to C)\) are zero by the commutativity of the square above. Since \(\Coker(A \to C) = \colim \Coker(A_\mu \to C_\mu)\) and these cokernels are equal to their completions we see that after increasing \(\mu\) we may assume these images are all zero. This means that the continuous homomorphism \((g \circ x'_{\mu, /Z_\mu})^\sharp\) has image contained in \(\Im(A_\mu \to C_\mu)\). Choose elements \(a_{\mu, j} \in (A_\mu)^\wedge\) mapping to \((g \circ x'_{\mu, /Z_\mu})^\sharp(b_j)\) in \((C_\mu)^\wedge\). Then \(a_{\mu, j} \in A_\mu^\wedge\) and \((\hat x)^\sharp(b_j) \in A^\wedge\) map to the same element of \(C^\wedge\) by the commutativity of the square above. Since \(\Ker(A \to C) = \colim \Ker(A_\mu \to C_\mu)\) and these kernels are equal to their completions, we may after increasing \(\mu\) adjust our choices of \(a_{\mu, j}\) such that the image of \(a_{\mu, j}\) in \(A^\wedge\) is equal to \((\hat x)^\sharp(b_j)\).

Proof of surjectivity: final algebra arguments. Let \(\mathfrak b \subset B\) be the ideal of topologically nilpotent elements. Let \(J \subset R[x_1, \ldots, x_n]\) be the ideal consisting of those \(h(x_1, \ldots, x_n)\) such that \(h(b_1, \ldots, b_n) \in \mathfrak b\). Then we get a continuous surjection of topological \(R\)-algebras \[\Phi : R[x_1, \ldots, x_n]^\wedge \longrightarrow B,\quad x_j \longmapsto b_j\] where the completion on the left hand side is with respect to \(J\). Since \(R[x_1, \ldots, x_n]\) is Noetherian we can choose generators \(h_1, \ldots, h_m\) for \(J\). By the commutativity of the square above we see that \(h_j(a_{\mu, 1}, \ldots, a_{\mu, n})\) is an element of \(A_\mu^\wedge\) whose image in \(A^\wedge\) is contained in \(IA^\wedge\). Namely, the ring map \((\hat x)^\sharp\) is continuous and \(IA^\wedge\) is the ideal of topological nilpotent elements of \(A^\wedge\) because \(A^\wedge/IA^\wedge = A/I\) is reduced. (See Algebra, Section 0BNH for results on completion in Noetherian rings.) Since \(A/I = \colim A_\mu/I_\mu\) we conclude that after increasing \(\mu\) we may assume \(h_j(a_{\mu, 1}, \ldots, a_{\mu, n})\) is in \(I_\mu A_\mu^\wedge\). In particular the elements \(h_j(a_{\mu, 1}, \ldots, a_{\mu, n})\) of \(A_\mu^\wedge\) are topologically nilpotent in \(A_\mu^\wedge\). Thus we obtain a continuous \(R\)-algebra homomorphism \[\Psi : R[x_1, \ldots, x_n]^\wedge \longrightarrow A_\mu^\wedge,\quad x_j \longmapsto a_{\mu, j}\] In order to conclude what we want, we need to see if \(\Ker(\Phi)\) is annihilated by \(\Psi\). This may not be true, but we can achieve this after increasing \(\mu\). Indeed, since \(R[x_1, \ldots, x_n]^\wedge\) is Noetherian, we can choose generators \(g_1, \ldots, g_l\) of the ideal \(\Ker(\Phi)\). Then we see that \[\Psi(g_1), \ldots, \Psi(g_l) \in \Ker(A_\mu^\wedge \to C_\mu^\wedge) = \Ker(A_\mu \to C_\mu)\] map to zero in \(\Ker(A \to C) = \colim \Ker(A_\mu \to C_\mu)\). Hence increasing \(\mu\) as before we get the desired result.

Proof of surjectivity: mopping up. The continuous ring homomorphism \(B \to (A_\mu)^\wedge\) constructed above determines a morphism \(\hat x_\mu : V_{\mu, /Z_\mu} \to W\). The compatibility of \(\hat x_\mu\) and \(u'_\mu\) follows from the fact that the ring map \(B \to (A_\mu)^\wedge\) is by construction compatible with the ring map \(A_\mu \to C_\mu\). In fact, the compatibility will be witnessed by the proper morphism \(V'_\mu \to V_\mu\) and the morphisms \(x'_\mu\) and \(\hat x'_\mu = x'_{\mu, /Z_\mu}\) we used in the construction. This finishes the proof.

Lemma

In Situation 0GH8 the functor \(F\) satisfies the Rim-Schlessinger condition (RS).

Proof

Recall that the condition only involves the evaluation \(F(V)\) of the functor \(F\) on schemes \(V\) over \(S\) which are spectra of Artinian local rings and the restriction maps \(F(V_2) \to F(V_1)\) for morphisms \(V_1 \to V_2\) of schemes over \(S\) which are spectra of Artinian local rings. Thus let \(V/S\) be the spectrum of an Artinian local ring. If \(\xi = (Z, u', \hat x) \in F(V)\) then either \(Z = \emptyset\) or \(Z = V\) (set theoretically). In the first case we see that \(\hat x\) is a morphism from the empty formal algebraic space into \(W\). In the second case we see that \(u'\) is a morphism from the empty scheme into \(X'\) and we see that \(\hat x : V \to W\) is a morphism into \(W\). We conclude that \[F(V) = U'(V) \amalg W(V)\] and moreover for \(V_1 \to V_2\) as above the induced map \(F(V_2) \to F(V_1)\) is compatible with this decomposition. Hence it suffices to prove that both \(U'\) and \(W\) satisfy the Rim-Schlessinger condition. For \(U'\) this follows from Lemma 07WQ. To see that it is true for \(W\), we write \(W = \colim W_n\) as in Formal Spaces, Lemma 0GHM. Say \(V = \Spec(A)\) with \((A, \mathfrak m)\) an Artinian local ring. Pick \(n \geq 1\) such that \(\mathfrak m^n = 0\). Then we have \(W(V) = W_n(V)\). Hence we see that the Rim-Schlessinger condition for \(W\) follows from the Rim-Schlessinger condition for \(W_n\) for all \(n\) (which in turn follows from Lemma 07WQ).

Lemma

In Situation 0GH8 the tangent spaces of the functor \(F\) are finite dimensional.

Proof

In the proof of Lemma 0GI6 we have seen that \(F(V) = U'(V) \amalg W(V)\) if \(V\) is the spectrum of an Artinian local ring. The tangent spaces are computed entirely from evaluations of \(F\) on such schemes over \(S\). Hence it suffices to prove that the tangent spaces of the functors \(U'\) and \(W\) are finite dimensional. For \(U'\) this follows from Lemma 07X1. Write \(W = \colim W_n\) as in the proof of Lemma 0GI6. Then we see that the tangent spaces of \(W\) are equal to the tangent spaces of \(W_2\), as to get at the tangent space we only need to evaluate \(W\) on spectra of Artinian local rings \((A, \mathfrak m)\) with \(\mathfrak m^2 = 0\). Then again we see that the tangent spaces of \(W_2\) have finite dimension by Lemma 07X1.

Lemma

In Situation 0GH8 assume \(X' \to S\) is separated. Then every formal object for \(F\) is effective.

Proof

A formal object \(\xi = (R, \xi_n)\) of \(F\) consists of a Noetherian complete local \(S\)-algebra \(R\) whose residue field is of finite type over \(S\), together with elements \(\xi_n \in F(\Spec(R/\mathfrak m^n))\) for all \(n\) such that \(\xi_{n + 1}|_{\Spec(R/\mathfrak m^n)} = \xi_n\). By the discussion in the proof of Lemma 0GI6 we see that either \(\xi\) is a formal object of \(U'\) or a formal object of \(W\). In the first case we see that \(\xi\) is effective by Lemma 07X8. The second case is the interesting case. Set \(V = \Spec(R)\). We will construct an element \((Z, u', \hat x) \in F(V)\) whose image in \(F(\Spec(R/\mathfrak m^n))\) is \(\xi_n\) for all \(n \geq 1\).

We may view the collection of elements \(\xi_n\) as a morphism \[\xi : \text{Spf}(R) \longrightarrow W\] of locally Noetherian formal algebraic spaces over \(S\). Observe that \(\xi\) is not an adic morphism in general. To fix this, let \(I \subset R\) be the ideal corresponding to the formal closed subspace \[\text{Spf}(R) \times_{\xi, W} W_{red} \subset \text{Spf}(R)\] Note that \(I \subset \mathfrak m_R\). Set \(Z = V(I) \subset V = \Spec(R)\). Since \(R\) is \(\mathfrak m_R\)-adically complete it is a fortiori \(I\)-adically complete (Algebra, Lemma 090T). Moreover, we claim that for each \(n \geq 1\) the morphism \[\xi|_{\text{Spf}(R/I^n)} : \text{Spf}(R/I^n) \longrightarrow W\] actually comes from a morphism \[\xi'_n : \Spec(R/I^n) \longrightarrow W\] Namely, this follows from writing \(W = \colim W_n\) as in the proof of Lemma 0GI6, noticing that \(\xi|_{\text{Spf}(R/I^n)}\) maps into \(W_n\), and applying Formal Spaces, Lemma 0AQH to algebraize this to a morphism \(\Spec(R/I^n) \to W_n\) as desired. Let us denote \(\text{Spf}'(R) = V_{/Z}\) the formal spectrum of \(R\) endowed with the \(I\)-adic topology – equivalently the formal completion of \(V\) along \(Z\). Using the morphisms \(\xi'_n\) we obtain an adic morphism \[\hat x = (\xi'_n) : \text{Spf}'(R) \longrightarrow W\] of locally Noetherian formal algebraic spaces over \(S\). Consider the base change \[\text{Spf}'(R) \times_{\hat x, W, g} X'_{/T'} \longrightarrow \text{Spf}'(R)\] This is a formal modification by Algebraization of Formal Spaces, Lemma 0GDN. Hence by the main theorem on dilatations (Algebraization of Formal Spaces, Theorem 0GDU) we obtain a proper morphism \[V' \longrightarrow V = \Spec(R)\] which is an isomorphism over \(\Spec(R) \setminus V(I)\) and whose completion recovers the formal modification above, in other words \[V' \times_{\Spec(R)} \Spec(R/I^n) = \Spec(R/I^n) \times_{\xi'_n, W, g} X'_{/T'}\] This in particular tells us we have a compatible system of morphisms \[V' \times_{\Spec(R)} \Spec(R/I^n) \longrightarrow X' \times_S \Spec(R/I^n)\] Hence by Grothendieck’s algebraization theorem (in the form of More on Morphisms of Spaces, Lemma 0A4Z) we obtain a morphism \[x' : V' \to X'\] over \(S\) recovering the morphisms displayed above. Finally, set \(u' : V \setminus Z \to U'\) to be the restriction of \(x'\) to \(V \setminus Z \subset V'\); this gives the second component of our desired element \((Z, u', \hat x) \in F(V)\).

Lemma

Let \(S\) be a locally Noetherian scheme. Let \(V\) be a scheme locally of finite type over \(S\). Let \(Z \subset V\) be closed. Let \(W\) be a locally Noetherian formal algebraic space over \(S\) such that \(W_{red}\) is locally of finite type over \(S\). Let \(g : V_{/Z} \to W\) be an adic morphism of formal algebraic spaces over \(S\). Let \(v \in V\) be a closed point such that \(g\) is versal at \(v\) (as in Section 07XZ). Then after replacing \(V\) by an open neighbourhood of \(v\) the morphism \(g\) is smooth (see proof).

Proof

Since \(g\) is adic it is representable by algebraic spaces (Formal Spaces, Section 0AQ2). Thus by saying \(g\) is smooth we mean that \(g\) should be smooth in the sense of Bootstrap, Definition 03XZ.

Write \(W = \colim W_n\) as in Formal Spaces, Lemma 0GHM. Set \(V_n = V_{/Z} \times_{g, W} W_n\). Then \(V_n\) is a closed subscheme with underlying set \(Z\). Smoothness of \(g : V_{/Z} \to W\) is equivalent to the smoothness of all the morphisms \(V_n \to W_n\) (this holds because any morphism \(T \to W\) with \(T\) a quasi-compact scheme factors through \(W_n\) for some \(n\)). We know that the morphism \(V_n \to W_n\) is smooth at \(v\) by Lemma 0CXM¹⁶. Of course this means that given any \(n\) we can shrink \(V\) such that \(V_n \to W_n\) is smooth. The problem is to find an open which works for all \(n\) at the same time.

The question is local on \(V\), hence we may assume \(S = \Spec(R)\) and \(V = \Spec(A)\) are affine.

In this paragraph we reduce to the case where \(W\) is an affine formal algebraic space. Choose an affine formal scheme \(W'\) and an étale morphism \(W' \to W\) such that the image of \(v\) in \(W_{red}\) is in the image of \(W'_{red} \to W_{red}\). Then \(V_{/Z} \times_{g, W} W' \to V_{/Z}\) is an adic étale morphism of formal algebraic spaces over \(S\) and \(V_{/Z} \times_{g, W} W'\) is an affine formal algebraic space. By Algebraization of Formal Spaces, Lemma 0AR4 there exists an étale morphism \(\varphi : V' \to V\) of affine schemes such that the completion of \(V'\) along \(Z' = \varphi^{-1}(Z)\) is isomorphic to \(V_{/Z} \times_{g, W} W'\) over \(V_{/Z}\). Observe that \(v\) is the image of some \(v' \in V'\). Since smoothness is preserved under base change we see that \(V'_n \to W'_n\) is smooth for all \(n\). In the next paragraph we show that after replacing \(V'\) by an open neighbourhood of \(v'\) the morphisms \(V'_n \to W'_n\) are smooth for all \(n\). Then, after we replace \(V\) by the open image of \(V' \to V\), we obtain that \(V_n \to W_n\) is smooth by étale descent of smoothness. Some details omitted.

Assume \(S = \Spec(R)\), \(V = \Spec(A)\), \(Z = V(I)\), and \(W = \text{Spf}(B)\). Let \(v\) correspond to the maximal ideal \(I \subset \mathfrak m \subset A\). We are given an adic continuous \(R\)-algebra homomorphism \[B \longrightarrow A^\wedge\] Let \(\mathfrak b \subset B\) be the ideal of topologically nilpotent elements (this is the maximal ideal of definition of the Noetherian adic topological ring \(B\)). Observe that \(\mathfrak b A^\wedge\) and \(IA^\wedge\) are both ideals of definition of the Noetherian adic ring \(A^\wedge\). Also, \(\mathfrak m A^\wedge\) is a maximal ideal of \(A^\wedge\) containing both \(\mathfrak b A^\wedge\) and \(IA^\wedge\). We are given that \[B_n = B/\mathfrak b^n \to A^\wedge/\mathfrak b^n A^\wedge = A_n\] is smooth at \(\mathfrak m\) for all \(n\). By the discussion above we may and do assume that \(B_1 \to A_1\) is a smooth ring map. Let \(\mathfrak m_1 \subset A_1\) denote the maximal ideal corresponding to \(\mathfrak m\). Since smoothness implies flatness, we see that: for all \(n \geq 1\) the map \[\mathfrak b^n/\mathfrak b^{n + 1} \otimes_{B_1} (A_1)_{\mathfrak m_1} \longrightarrow \left(\mathfrak b^nA^\wedge/\mathfrak b^{n + 1}A^\wedge\right)_{\mathfrak m_1}\] is an isomorphism (see Algebra, Lemma 0AS8). Consider the Rees algebra \[B' = \bigoplus\nolimits_{n \geq 0} \mathfrak b^n/\mathfrak b^{n + 1}\] which is a finite type graded algebra over the Noetherian ring \(B_1\) and the Rees algebra \[A' = \bigoplus\nolimits_{n \geq 0} \mathfrak b^nA^\wedge/\mathfrak b^{n + 1}A^\wedge\] which is a finite type graded algebra over the Noetherian ring \(A_1\). Consider the homomorphism of graded \(A_1\)-algebras \[\Psi : B' \otimes_{B_1} A_1 \longrightarrow A'\] By the above this map is an isomorphism after localizing at the maximal ideal \(\mathfrak m_1\) of \(A_1\). Hence \(\Ker(\Psi)\), resp. \(\Coker(\Psi)\) is a finite module over \(B' \otimes_{B_1} A_1\), resp. \(A'\) whose localization at \(\mathfrak m_1\) is zero. It follows that after replacing \(A_1\) (and correspondingly \(A\)) by a principal localization we may assume \(\Psi\) is an isomorphism. (This is the key step of the proof.) Then working backwards we see that \(B_n \to A_n\) is flat, see Algebra, Lemma 0AS8. Hence \(B_n \to A_n\) is smooth (as a flat ring map with smooth fibres, see Algebra, Lemma 00TF) and the proof is complete.

Lemma

In Situation 0GH8 the functor \(F\) satisfies openness of versality.

Proof

We have to show the following. Given a scheme \(V\) locally of finite type over \(S\), given \(\xi \in F(V)\), and given a finite type point \(v_0 \in V\) such that \(\xi\) is versal at \(v_0\), after replacing \(V\) by an open neighbourhood of \(v_0\) we have that \(\xi\) is versal at every finite type point of \(V\). Write \(\xi = (Z, u', \hat x)\).

First case: \(v_0 \not \in Z\). Then we can first replace \(V\) by \(V \setminus Z\). Hence we see that \(\xi = (\emptyset, u', \emptyset)\) and the morphism \(u' : V \to X'\) is versal at \(v_0\). By More on Morphisms of Spaces, Lemma 0APN this means that \(u' : V \to X'\) is smooth at \(v_0\). Since the set of points where a morphism is smooth is open, we can after shrinking \(V\) assume \(u'\) is smooth. Then the same lemma tells us that \(\xi\) is versal at every point as desired.

Second case: \(v_0 \in Z\). Write \(W = \colim W_n\) as in Formal Spaces, Lemma 0GHM. By Lemma 0GI9 we may assume \(\hat x : V_{/Z} \to W\) is a smooth morphism of formal algebraic spaces. It follows immediately that \(\xi = (Z, u', \hat x)\) is versal at all finite type points of \(Z\). Let \(V' \to V\), \(\hat x'\), and \(x'\) witness the compatibility between \(u'\) and \(\hat x\). We see that \(\hat x' : V'_{/Z} \to X'_{/T'}\) is smooth as a base change of \(\hat x\). Since \(\hat x'\) is the completion of \(x' : V' \to X'\) this implies that \(x' : V' \to X'\) is smooth at all points of \((V' \to V)^{-1}(Z) = |x'|^{-1}(T') \subset |V'|\) by the already used More on Morphisms of Spaces, Lemma 0APN. Since the set of smooth points of a morphism is open, we see that the closed set of points \(B \subset |V'|\) where \(x'\) is not smooth does not meet \((V' \to V)^{-1}(Z)\). Since \(V' \to V\) is proper and hence closed, we see that \((V' \to V)(B) \subset V\) is a closed subset not meeting \(Z\). Hence after shrinking \(V\) we may assume \(B = \emptyset\), i.e., \(x'\) is smooth. By the discussion in the previous paragraph this exactly means that \(\xi\) is versal at all finite type points of \(V\) not contained in \(Z\) and the proof is complete.

Here is the final result.

Theorem

Let \(S\) be a locally Noetherian scheme such that \(\mathcal{O}_{S, s}\) is a G-ring for all finite type points \(s \in S\). Let \(X'\) be an algebraic space locally of finite type over \(S\). Let \(T' \subset |X'|\) be a closed subset. Let \(W\) be a locally Noetherian formal algebraic space over \(S\) with \(W_{red}\) locally of finite type over \(S\). Finally, we let \[g : X'_{/T'} \longrightarrow W\] be a formal modification, see Algebraization of Formal Spaces, Definition 0GDK. If \(X'\) and \(W\) are separated¹⁷ over \(S\), then there exists a proper morphism \(f : X' \to X\) of algebraic spaces over \(S\), a closed subset \(T \subset |X|\), and an isomorphism \(a : X_{/T} \to W\) of formal algebraic spaces such that

1. \(T'\) is the inverse image of \(T\) by \(|f| : |X'| \to |X|\),

2. \(f : X' \to X\) maps \(X' \setminus T'\) isomorphically to \(X \setminus T\), and

3. \(g = a \circ f_{/T}\) where \(f_{/T} : X'_{/T'} \to X_{/T}\) is the induced morphism.

In other words, \((f : X' \to X, T, a)\) is a solution as defined earlier in this section.

Proof

Let \(F\) be the functor constructed using \(X'\), \(T'\), \(W\), \(g\) in this section. By Lemma 0GHC it suffices to show that \(F\) corresponds to an algebraic space \(X\) locally of finite type over \(S\). In order to do this, we will apply Proposition 0GE7. Namely, by Lemma 0GHF the diagonal of \(F\) is representable by closed immersions and by Lemmas 0GHG, 0GI5, 0GI6, 0GI7, 0GI8, and 0GIA we have axioms [0], [1], [2], [3], [4], and [5].

Remark

The proof of Theorem 0GIB uses that \(X'\) and \(W\) are separated over \(S\) in two places. First, the proof uses this in showing \(\Delta : F \to F \times F\) is representable by algebraic spaces. This use of the assumption can be entirely avoided by proving that \(\Delta\) is representable by applying the theorem in the separated case to the triple \(E'\), \((E' \to V)^{-1}Z\), and \(E'_{/Z} \to E_W\) found in Remark 0GHD (this is the usual bootstrap procedure for the diagonal). Thus the proof of Lemma 0GI8 is the only place in our proof of Theorem 0GIB where we really need to use that \(X' \to S\) is separated. The reader checks that we use the assumption only to obtain the morphism \(x' : V' \to X'\). The existence of \(x'\) can be shown, using results in the literature, if \(X' \to S\) is quasi-separated, see More on Morphisms of Spaces, Remark 0GHK. We conclude the theorem holds as stated with “separated” replaced by “quasi-separated”. If we ever need this we will precisely state and carefully prove this here.