Flattening stratifications

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).

A sheaf can have fibres that look harmless while failing to form a flat family. Passing to residue fields erases the relations caused by nilpotent elements of the parameter ring. A flattening stratification records those relations in the scheme structures of its strata. Its universal property therefore concerns maps from arbitrary schemes, including nonreduced schemes.

We begin with relations between free modules and with the especially transparent case of an Artinian base. For a coherent sheaf on projective space, the problem then becomes a sequence of finite module problems, one for each sufficiently large graded degree. Regularity supplies a uniform comparison with the fibres. Noetherianity makes the accumulated equations finite. This proves Mumford's theorem without assuming the existence of a Hilbert or Quot scheme.

The prerequisites are flatness, tensor products, Nakayama's lemma, Fitting ideals, projective sheafification, generic flatness, and the regularity theorem proved in Regularity and bounded families. One cohomology input is used explicitly: a finitely presented sheaf flat over a base, with proper support on a morphism of finite presentation, has a perfect derived pushforward compatible with arbitrary base change [Stacks, Tag 0A1H]. Locally on the base, this means that a finite complex of finite free modules computes its cohomology after every base change. We explain the elementary consequences needed here, and list this prerequisite at the end.

1 The functor asks for flatness after pullback

Let \(f:X\to S\), and let \(F\) be a quasi-coherent sheaf on \(X\). Define a subfunctor of the functor of points of \(S\) by

Here \(F_T\) is the ordinary pullback to \(X_T\). Flatness is stable under base change, so this is indeed a contravariant functor. A universal flattening is a scheme \(S'\) over \(S\) representing it. Its structural map is a monomorphism: for a fixed map \(T\to S\), there is at most one factorization through \(S'\).

In this lesson a flattening stratification is a universal flattening of the form

where the \(S_\alpha\) are locally closed subschemes with disjoint underlying subsets whose union is \(|S|\). Every field-valued point belongs to the flattening functor, because every vector space is flat. Consequently any universal flattening is surjective on underlying points. The map need not be an isomorphism: a test scheme can meet a stratum set-theoretically and still fail to factor through its scheme structure.

For example, \(A/(a)\) becomes a free module of rank one over an \(A\)-algebra \(B\) precisely when the image of \(a\) is zero. Over a field this condition sees only the support. Over the dual numbers it distinguishes zero from a nonzero nilpotent.

2 Rank strata and their equations

For a finitely presented module \(M\), its Fitting ideals commute with every base change. With a presentation

\(\operatorname{Fitt}_r(M)\) is generated by the \((q-r)\)-minors of \(u\). We use the conventions that a minor of size zero is 1, impossible positive sizes give the zero ideal, and \(\operatorname{Fitt}_{-1}(M)=0\). The ideals are independent of the presentation. For the universal property below, the local matrix calculation also proves that independence on each rank chart.

Proposition 2.1. Let \(M\) be a finitely presented sheaf on a scheme \(S\). For \(r\geq0\), the locally closed subscheme

represents maps \(g:T\to S\) for which \(g^*M\) is locally free of constant rank \(r\). The notation \(D(I)\) means the complement of \(V(I)\), possibly a union of principal opens. Over a Noetherian \(S\), only finitely many nonempty rank strata occur, and their disjoint union is a flattening stratification of \(M\) on \(S\).

Proof. Work with the displayed matrix presentation. On an open where a \((q-r)\)-minor is invertible, invertible row and column operations put the matrix into block form

Its cokernel is the cokernel of \(C:A^{p-q+r}\to A^r\). On this open, vanishing of all \((q-r+1)\)-minors is equivalent to vanishing of all entries of \(C\): the minors containing the identity block give those entries, and if they vanish every larger minor vanishes. The cokernel is then \(A^r\).

Conversely, if this cokernel is free of rank \(r\), the surjection \(A^r\to\operatorname{coker}C\) is an isomorphism locally. Indeed, over a local ring its matrix modulo the maximal ideal is invertible, so its determinant is a unit. Thus \(C=0\). A free cokernel of rank \(r\) also forces some \((q-r)\)-minor of the original matrix to be invertible locally, by splitting off the kernel and choosing bases. Therefore the two Fitting conditions are necessary and sufficient, after any base change. They give exactly the claimed factorization property. These local descriptions agree on overlaps because the property is intrinsic.

Every fibre of \(M\) has one finite dimension, so the rank strata partition \(|S|\). A finite affine cover bounds those dimensions on a Noetherian scheme. Finally, a finitely presented flat module is locally free, with locally constant rank. Decomposing a test scheme into its open and closed rank loci gives its unique factorization through the disjoint union. \(\square\)

The scheme structure in this proposition is essential. On the locus where all fibres have dimension \(r\), the ideal \(\operatorname{Fitt}_r(M)\) is the unit ideal, but \(\operatorname{Fitt}_{r-1}(M)\) can be a nonzero nilpotent ideal. It is this latter ideal that removes the residual relations.

3 All relations can be removed over an Artinian base

Theorem 3.1. Let \(A\) be an Artinian ring, \(X\) any \(A\)-scheme, and \(F\) any quasi-coherent sheaf on \(X\). There is an ideal \(J\subset A\) such that \(\operatorname{Spec}(A/J)\to\operatorname{Spec}A\) is the universal flattening of \(F\). In particular, a flattening stratification exists. Neither finite presentation of \(F\) nor properness of \(X\) is required.

Proof. First suppose \(A\) is local, with maximal ideal \(\mathfrak m\) and residue field \(k\). Its maximal ideal is nilpotent. For any \(A\)-module \(M\), choose lifts of a basis \((\bar e_i)_{i\in I}\) of \(M/\mathfrak mM\). They define

This is surjective even when \(I\) is infinite. Its cokernel \(C\) satisfies \(C=\mathfrak mC\); iteration and nilpotence force \(C=0\). Let \(R\) be its kernel. The basis condition gives \(R\subset\mathfrak mP\). Let \(J_M\) be the ideal generated by all coefficients of all finite-support vectors in \(R\).

For an \(A\)-algebra \(B\), right exactness gives

If \(J_MB=0\), the first map is zero and \(M\otimes_A B=B^{(I)}\), so it is flat. For the converse, suppose \(M_B=M\otimes_A B\) is flat, and put \(N=\ker(B^{(I)}\to M_B)\). The ideal \(\mathfrak mB\) is still nilpotent. Modulo this ideal the displayed surjection is an isomorphism, by the original choice of basis. Flatness makes

exact. Hence \(N=(\mathfrak mB)N\), and nilpotence gives \(N=0\). Every relation in \(R\) therefore has every coefficient zero in \(B\), which is exactly \(J_MB=0\). We have proved the universal property for an arbitrary module.

Choose affine opens \(U_\lambda\) covering \(X\), and set \(M_\lambda=\Gamma(U_\lambda,F)\). Pullback on \(U_{\lambda,B}\) is represented by \(M_\lambda\otimes_A B\). Flatness over \(B\) is equivalent to flatness of these modules: it can be checked locally on the affine source and the base. Thus the ideal

has the required property for every affine test scheme. General test schemes follow by gluing the local factorizations through a closed immersion. There is no finiteness difficulty with the sum: an Artinian ring is Noetherian, so it is generated by finitely many of the coefficients.

Finally, an Artinian ring is a finite product of local Artinian rings. Apply the construction to each open and closed factor and take the product of the resulting quotient rings. This is again a closed subscheme of \(\operatorname{Spec}A\), with the same universal property. \(\square\)

For \(A=k[\epsilon]/(\epsilon^2)\) and \(M=A/(\epsilon)\), the ideal is \((\epsilon)\). The universal flattening is the reduced point. The only underlying point of \(\operatorname{Spec}A\) is already flat as a fibre; the obstruction lies in its infinitesimal neighbourhood.

4 Two cohomology consequences

We isolate the cohomology input before using it in the projective construction.

Lemma 4.1. Let \(f:X\to T\) be of finite presentation and let \(G\) be a finitely presented sheaf, flat over \(T\), with proper support. If \(H^i(X_t,G_t)=0\) for every \(t\in T\) and every \(i>0\), then \(f_*G\) is finite locally free and commutes with arbitrary base change. Its rank at \(t\) is \(h^0(X_t,G_t)\).

Proof from the stated prerequisite. Locally on \(T\), take a finite complex of finite free modules computing cohomology after arbitrary base change, as supplied by [Stacks, Tag 0A1H]. At a point \(t\), its tensor product with \(\kappa(t)\) has cohomology only in degree zero. A bounded complex of vector spaces is the direct sum of its cohomology and two-term identity complexes. Lift the invertible minors used for the latter decomposition to a neighbourhood of \(t\). Row and column operations split off the corresponding two-term identity complexes there. The remaining complex has only a free term in degree zero: the other remaining terms have rank zero, as seen at \(t\). Consequently the original complex is represented there by a finite free module in degree zero. Its tensor product with any algebra has the same description. This proves local freeness, the rank formula, and arbitrary base change. \(\square\)

Lemma 4.2. In a flat projective family of finitely presented sheaves, the Hilbert polynomial is locally constant.

Proof from the same prerequisite. Near a given point, for each of the finitely many twists \(0,1,\ldots,n\) on \(\mathbb P^n_T\), choose its finite free cohomology complex. Its Euler characteristic on a fibre is the alternating sum of the ranks of its terms. It is therefore locally constant. Each fibre Hilbert polynomial has degree at most \(n\); its values at these \(n+1\) integers determine it. Shrink simultaneously for these twists. The polynomial is constant there. For a projective closed subscheme use its pushforward to projective space, which preserves fibre cohomology. \(\square\)

These proofs separate cohomology and base change from flattening. In particular, they use flatness of the sheaf in advance; they do not apply the flat-family conclusion to an arbitrary sheaf on the original base.

5 A graded comparison uniform over all fibres

Let \(S=\operatorname{Spec}A\) with \(A\) Noetherian, and put \(R=A[x_0,\ldots,x_n]\). A coherent sheaf \(F\) on \(\mathbb P^n_A\) has a presentation by finite sums of line bundles

To obtain it, use Serre generation first on \(F\) and then on the coherent kernel of the first surjection. The morphism between the line bundles is a matrix of homogeneous polynomials. Let \(M\) be the cokernel of the corresponding map between finite graded free \(R\)-modules. Then \(\widetilde M=F\). Sheafification commutes with arbitrary base change and cokernels, so

for every \(A\)-algebra \(B\). Each graded piece \(M_m\) is a finite, hence finitely presented, \(A\)-module.

Lemma 5.1. Only finitely many polynomials \(P_{F_s}\) occur for \(s\in S\). There is an integer \(N\) such that for every \(s\in S\) and \(m\geq N\),

The isomorphism is the natural map from a graded piece to sections.

Proof. Generic flatness, applied on the reduced irreducible components of the base, gives a finite decomposition into reduced locally closed schemes over which the restricted sheaf is flat. Here is why the decomposition is finite: remove a dense open from every irreducible component on which generic flatness holds, and perform Noetherian induction on the remaining proper closed subset. No bound on the Krull dimension of \(A\) is needed. Each resulting stratum is Noetherian, so Lemma 4.2 and quasi-compactness give only finitely many polynomials there. This proves the first assertion. This preliminary decomposition is used only to establish numerical finiteness; its reduced scheme structures are not asserted to represent the flattening functor.

On a fibre, write \(K_s=\ker(L_{0,s}\to F_s)\) and \(K_{1,s}=\ker(L_{1,s}\to K_s)\). Pullback preserves the right exact presentation, so \(L_{1,s}\to K_s\) is surjective. Their polynomials are

Both range over finite sets. The uniform regularity theorem in Regularity and bounded families applies to quotients of the fixed split sheaves \(L_0\) and \(L_1\), and thus uniformly bounds these kernels and \(F_s\). Its bound is independent of the field. For completeness, split sheaves with several fixed twists cause no problem: after one fixed twist each admits a surjection from a trivial bundle by monomials. The number of monomials and the polynomial of its kernel depend only on those twists and \(n\). The uniform kernel theorem therefore gives the same reduction over every field.

Choose \(N\) larger than these bounds and than all shifts in the graded presentation. For \(m\geq N\), the two kernel sequences give an exact sequence

because \(H^1(K_s(m))=H^1(K_{1,s}(m))=0\). The first two terms are exactly the graded free modules in degree \(m\), tensored with \(\kappa(s)\). Right exactness identifies their cokernel with \(M_m\otimes_A\kappa(s)\). This proves the natural isomorphism. The uniform bound on \(F_s\) gives the higher vanishing. \(\square\)

Notice that the lemma compares with fields. It has not asserted that direct images of the original nonflat sheaf commute with every base change. The graded module will let us avoid that assertion.

Lemma 5.2. For any \(A\)-algebra \(B\), if \(M_m\otimes_A B\) is flat over \(B\) for every \(m\geq N\), then \(F_B\) is flat over \(B\). Conversely, if \(F_B\) is flat over \(B\), then for every \(m\geq N\) the natural map

is an isomorphism, and these modules are finite locally free.

Proof. For the forward direction, on \(D_+(x_i)\) the module of sections of \(\widetilde{M\otimes_A B}\) is

with transition maps multiplication by \(x_i\). Starting the sequence at any larger index leaves the colimit unchanged. A filtered colimit of flat modules is flat. Thus the sheaf is flat over \(B\) on each standard affine open, proving the assertion.

For the converse, every fibre over \(\operatorname{Spec}B\) is a field extension of a fibre in Lemma 5.1. Its higher cohomology at \(m\geq N\) is zero. Lemma 4.1 gives a finite locally free module \(G_m=H^0(F_B(m))\), with formation compatible with field-valued base change. The natural map \(M_m\otimes_A B\to G_m\) is an isomorphism on every residue field, by Lemma 5.1 and field extension. It is therefore surjective by Nakayama, since its cokernel is finite. Locally \(G_m\) is free, so the surjection splits. Its kernel is then a direct summand of a finite module and is finite. Its fibre at every prime is zero, because the target is flat and the original fibre map is an isomorphism. Nakayama makes this kernel zero. This proves the isomorphism. \(\square\)

This argument applies to arbitrary \(B\), with no Noetherian assumption on it.

6 Mumford's theorem and the finite sum of equations

Theorem 6.1 (Mumford). Let \(S\) be Noetherian and \(F\) coherent on \(\mathbb P^n_S\). There is a finite set \(\mathcal P\) of numerical polynomials and, for each \(P\in\mathcal P\), a locally closed subscheme \(S_P\subset S\) such that:

1. its underlying subset consists exactly of points \(s\) with \(P_{F_s}=P\);
2. for every \(g:T\to S\), the sheaf \(F_T\) is flat over \(T\) if and only if \(g\) factors uniquely through \(\coprod_{P\in\mathcal P}S_P\);
3. a factorization through \(S_P\) means that the flat family has constant fibre polynomial \(P\).

The theorem also holds for a coherent sheaf on a projective closed subscheme of \(\mathbb P^n_S\).

Proof. Begin over an affine open of \(S\), using \(M\) and \(N\) from Section 5. Apply Proposition 2.1 successively to

restricting to each preceding rank stratum before the next step. This produces finitely many locally closed subschemes, with the universal property that these \(n+1\) modules become locally free of prescribed ranks. Lemma 5.1 says that at a point these ranks are

A polynomial of degree at most \(n\) is determined by these values. Thus discard empty strata and label the others \(W_P\), for \(P\in\mathcal P\). Their underlying subsets already have the right fibre polynomials.

For each \(m\geq N\), the fibres of \(M_m|_{W_P}\) have constant dimension \(P(m)\). Therefore its \(P(m)\)-th Fitting ideal is the unit ideal. Proposition 2.1 says that local freeness of rank \(P(m)\) is now represented by a closed subscheme of \(W_P\), defined by

Every \(J_m\) is contained in the nilradical: its zero scheme has all points of \(W_P\). Equivalently, its radical is the radical of the zero ideal. Let

Since \(W_P\) is Noetherian, this sum stabilizes as a finite sum of coherent ideals. To see this globally, take a finite affine cover and use the ascending chain condition on each affine coordinate ring; the maximum of the finitely many stopping indices works on the whole cover. Thus \(J\) is coherent. Define \(S_P=V(J)\subset W_P\). Its underlying subset is still all of \(|W_P|\), while its scheme structure enforces local freeness of every \(M_m\) for \(m\geq N\).

By Lemma 5.2, the family over \(S_P\) is flat. Any map factoring through its disjoint union therefore gives a flat family. Conversely, suppose \(F_T\) is flat. Lemma 4.2 decomposes \(T\) into open and closed polynomial loci \(T_P\). On \(T_P\), Lemma 5.2 makes every \(g^*M_m\), \(m\geq N\), locally free of rank \(P(m)\). The first \(n+1\) ranks force the map through \(W_P\). The remaining ranks annihilate every \(J_m\), hence their sum \(J\), so it factors through \(S_P\). This holds locally on arbitrary affine test schemes and glues. The factorization is unique, since each stratum is an immersion and different strata have disjoint underlying subsets. Their cross fibre products are empty, so their disjoint union is a monomorphism.

For a general Noetherian base, use a finite affine open cover. On overlaps, the constructed fixed-polynomial strata represent the same functor, so Yoneda gives unique identifications respecting their maps to the base. These identifications satisfy the cocycle condition and glue the strata. Locally closed immersions are local on the target, so the glued strata are locally closed subschemes of \(S\). The union of the finitely many sets of polynomials from the cover is finite. This gives the global theorem.

Finally, for a closed immersion \(i:X\hookrightarrow\mathbb P^n_S\), use \(i_*F\). Closed pushforward commutes with base change, preserves cohomology and fibre Hilbert polynomials, and is flat over the base exactly when \(F\) is. The theorem for \(i_*F\) is therefore the desired theorem on \(X\). \(\square\)

The proof does not say that the first \(n+1\) twists alone suffice for flatness. They label the polynomial strata. The higher twists can impose additional nilpotent equations. Noetherianity shows that finitely many of these additional equations suffice, without specifying their number in advance.

Corollary 6.2. Order the finite set \(\mathcal P\) by eventual values: \(P<Q\) when \(P(m)<Q(m)\) for all sufficiently large \(m\). Then

Proof. Choose one \(m\geq N\) large enough that evaluation at \(m\) realizes this order on the finite set. The dimension of the fibre of the finite module \(M_m\) is upper semicontinuous: a matrix presentation makes the condition that it be at least an integer a closed determinantal condition. It equals \(P(m)\) on \(S_P\), by Lemma 5.1. At a point in its closure the dimension is therefore at least \(P(m)\). The choice of \(m\) identifies its polynomial as some \(Q\geq P\). \(\square\)

7 Examples: fibre changes and actual failures of flatness

A coordinate cross is flat under the sum map

Take \(X=\operatorname{Spec}k[x,y]/(xy)\), with \(t=x+y\) giving a map to \(\mathbb A^1_k\). Eliminate \(y=t-x\):

The polynomial in \(x\) is monic, so this is a free \(k[t]\)-module with basis \(1,x\). The morphism is finite flat of degree two, and its flattening stratification has one stratum, all of \(\mathbb A^1\). For \(t\ne0\) the fibre is two distinct points; at zero it is a double point. This change in reducedness does not change length and does not obstruct flatness. The calculation works in every characteristic.

Likewise, in \(\mathbb P^1\times\mathbb A^1\) with coordinates \([X:Y]\), the equation \(X(X-tY)=0\) gives a family entirely contained in \(Y\ne0\). Its affine ring is the same free rank-two ring. Its fibre Hilbert polynomial is the constant 2, including at zero.

An additional special point changes the polynomial

In \(\mathbb P^2\), let \(L\) be a line and let \(p\) lie outside it. Over \(S=\mathbb A^1=\operatorname{Spec}k[t]\), form the disjoint closed union

The two components have disjoint supports, so

Its fibre polynomial is \(m+1\) away from zero and \(m+2\) at zero. The universal flattening is

with the reduced scheme structure \((t)\) on the closed stratum. To check the scheme structure, after base change to \(B\) the second summand is \(B/tB\), supported on the constant point \(p\). It is flat precisely when it is locally free. Its rank is either zero, which means \(t\) is a unit locally, or one, which means \(t=0\) locally. Thus a test scheme splits into the two corresponding open and closed pieces. A nonzero nilpotent value of \(t\) fails this test.

A distinction about the flat locus

For a finitely presented module on a Noetherian base itself, the locus where its stalk is flat is open. Indeed, a flat finitely presented module over a local ring is free. A basis at that stalk extends to a map from a finite free module on a neighbourhood; its kernel and cokernel vanish after shrinking, by finite presentation. Thus it remains free on a neighbourhood.

For a sheaf on a nonproper \(X\to S\), the set of \(s\) where it is flat at every point of \(X_s\) need not be open. Let \(S=\operatorname{Spec}k[x,y]\), let \(X=D(x)\subset S\), and put \(F=\mathcal O_X/(y)\). This sheaf is finitely presented on \(X\). It fails to be flat at every point with \(x\ne0\) and \(y=0\), because multiplication by the nonzero element \(y\) of the base domain becomes zero on a nonzero quotient. Elsewhere on \(X\) it is zero and flat. Fibres above \(V(x)\) are empty. The set of base points where it is flat throughout the fibre is therefore

It contains the origin but no neighbourhood of it, so it is not open. Properness would make the image of a closed nonflat locus closed; its absence is precisely relevant in this example.

8 Beyond projective stratifications

There is a broader theorem, but its conclusion should be distinguished from a finite decomposition into locally closed subschemes.

For a finite type \(X\to S\) and a finite type sheaf \(F\), purity relative to \(S\) has the following formulation. For each \(s\), pass to the henselization \(S^h=\operatorname{Spec}\mathcal O_{S,s}^h\). Every associated point of the restriction of \(F\) to any fibre over \(S^h\) must have a specialization in its closed fibre \(X_s\). Equivalently, associated points cannot escape the closed fibre in an elementary étale neighbourhood. This is [Stacks, Tag 05J4]; it is relative purity, distinct from the usual assertion that a single sheaf has all associated points of one fixed dimension.

Theorem 8.1 (Raynaud–Gruson, stated). If \(X\to S\) is of finite presentation and \(F\) is finitely presented and pure relative to \(S\), then \(\operatorname{Flat}_F\) is represented by a monomorphism of finite presentation \(S'\to S\). In particular this conclusion holds when \(X\to S\) is proper and of finite presentation. The same assertions for flattening \(X\) itself use \(F=\mathcal O_X\).

The precise locators are [Stacks, Tags 05UG and 05UH], with purity of sheaves with proper support in Tag 05K3. We do not prove the general theorem here. Its monomorphism conclusion alone does not assert that it is the disjoint union of a finite collection of locally closed immersions. Theorem 6.1 proves that stronger description in the projective Noetherian situation.

There is also a local theorem that isolates flatness near a chosen point. Its condition concerns specified points after base change, rather than every point of the base-changed scheme.

Theorem 8.2 (henselian local flattening, stated). Let \(A\to B\) be a local homomorphism of local rings, with \(A\) henselian and \(B\) essentially of finite type over \(A\). Let \(M\) be a finite \(B\)-module. There is an ideal \(I\subset A\) such that, for every local homomorphism \(A\to A'\) of local rings, setting \(B'=B\otimes_A A'\) and \(M'=M\otimes_A A'\), one has

Equivalently, \(A/I\) corepresents this flatness functor on local \(A\)-algebras with local structure maps. If \(B\) is essentially of finite presentation over \(A\) and \(M\) is finitely presented over \(B\), then \(I\) is finitely generated. This is [Stacks, Tag 05PG]. The two finiteness assertions have different hypotheses; neither the closed-fibre point condition nor the henselian assumption may be dropped from this statement.

Without the finiteness and purity hypotheses, a universal flattening can fail to exist. Here is the mechanism of [Stacks, Tag 0FJ1], with a proof for schemes. Let \(A=k[x,y]\), \(X=S=\operatorname{Spec}A\), and let the quasi-coherent sheaf correspond to

It is not finitely presented as an \(A\)-module. Put \(A_j=A/(x,y)^j\). Since \(x\) is nilpotent in \(A_j\), localizing it makes the ring zero; hence \(M\otimes_A A_j=0\), which is flat. But for \(\widehat A=k[[x,y]]\), the module

is nonzero and is killed by \(y\). It is not flat: tensoring the injection \(\widehat A\xrightarrow{y}\widehat A\) would give a zero map on this nonzero module.

If a scheme represented the flattening functor, the compatible maps \(\operatorname{Spec}A_j\to S'\) would all take their sole point to one point of \(S'\). They lie in a common affine neighbourhood \(U\) of that point. Compatible homomorphisms \(\Gamma(U,\mathcal O_U)\to A_j\) give a homomorphism to \(\lim A_j=\widehat A\), hence an \(S\)-map \(\operatorname{Spec}\widehat A\to S'\). That would make the last module flat, a contradiction. This rules out a representing scheme. The stronger exclusion of algebraic spaces is part of the cited example, using the corresponding formal lifting property.

9 Exercises

1. Basic. Over the dual numbers \(A=k[\epsilon]/(\epsilon^2)\), find the universal flattening of \(A\oplus A/(\epsilon)\). Explain why all residue-field information misses the answer.

2. Intermediate. For a map \(A^p\to A^q\), prove that its cokernel becomes locally free of rank \(r\) exactly on the Fitting stratum in Proposition 2.1, over arbitrary \(A\)-algebras. Include ranks zero and \(q\).

3. Intermediate. Compute the flattening stratification of \(X(X-tY)=0\) in \(\mathbb P^1\times\mathbb A^1\). Contrast it with the line plus special point of Section 7, including maps from nonreduced schemes.

4. Intermediate. Prove that the flat locus of a finitely presented module on a Noetherian base is open. Then verify that \(D(y)\cup V(x)\) in Section 7 is the whole-fibre flat locus of the specified sheaf and is not open.

5. Advanced. Reconstruct Theorem 6.1 from its graded presentation. Identify separately the uses of generic flatness, uniform regularity, the first \(n+1\) rank conditions, and Noetherianity. Prove the universal property for arbitrary test schemes, not only for residue fields.

6. Advanced. Let \(T\) be any scheme and let \(G\) be a flat finitely presented sheaf on \(\mathbb P^n_T\). Deduce local constancy of its Hilbert polynomial from the cohomology-complex prerequisite. Explain why a polynomial jump in the line plus point family proves nonflatness, whereas a jump in the number of distinct geometric points in the degree-two family does not.

10 Solutions

1. Lift the residue-field basis represented by \((1,0)\) and \((0,1)\). The only relation module is generated by \((0,\epsilon)\). The coefficient ideal of Theorem 3.1 is \((\epsilon)\), so the universal flattening is \(\operatorname{Spec}k\). Directly, after base change to \(B\), the second summand is \(B/\epsilon B\). If the direct sum is flat, its direct summand is flat. Reduction modulo the nilpotent ideal \(\epsilon B\) has rank one, so the nilpotent relation argument forces \(\epsilon B=0\). Conversely, that condition makes the entire module \(B^2\). Every residue field already kills \(\epsilon\), so it sees rank two at the sole point, with no information about the nonzero infinitesimal relation.

2. On an invertible \((q-r)\)-minor, elementary operations split off an identity block. The remaining matrix has \(r\) rows. Its entries vanish precisely when all \((q-r+1)\)-minors vanish, and in that case the remaining cokernel is free of rank \(r\). Conversely, a free cokernel makes the surjection from those \(r\) generators an isomorphism, so all remaining relations vanish. The conditions are unchanged by arbitrary tensor product, since determinants are polynomials in the entries. For \(r=0\), an invertible \(q\)-minor makes the matrix surjective and its cokernel zero; there is no further equation. For \(r=q\), the invertible minor is the size-zero minor 1 and the condition is that every entry of the original matrix vanish. Impossible ranks give the empty stratum. These observations also handle a presentation with redundant relations.

3. The family lies in \(Y\ne0\) and has ring \(k[t,x]/(x^2-tx)\). Division by a monic polynomial gives the basis \(1,x\) over every base algebra, so all pullbacks are free of rank two. Its universal flattening is the identity of \(\mathbb A^1\). This includes sending \(t\) to a nonzero nilpotent. In the second family, after pullback the point summand is \(B/tB\). Where it has rank zero, \(t\) is invertible; where it has rank one, the surjection \(B\to B/tB\) between free rank-one modules is an isomorphism, so \(t=0\). Thus the map must factor through \(D(t)\amalg V(t)\). In a local algebra where \(t\) is a nonzero nilpotent it does not factor, and the second family is not flat.

4. At a flat stalk, finite presentation gives freeness. Choose local sections mapping to a basis and form a map from a finite free sheaf. Its cokernel vanishes after shrinking by Nakayama. Its kernel is finite over the Noetherian base and vanishes at that stalk, so it too vanishes after shrinking. This proves openness. For the specified open immersion, the sheaf vanishes where \(y\ne0\), fails flatness where \(x\ne0,y=0\), and has no source fibre where \(x=0\). This gives exactly \(D(y)\cup V(x)\). Any neighbourhood of the origin intersects the punctured \(x\)-axis: its localizations on that irreducible line contain a nonempty open neighbourhood of the origin and hence its generic point. Those points are excluded from the locus, so the locus is not open.

5. A finite graded presentation gives \(M\) with \(\widetilde M=F\) and finite graded pieces. Generic flatness produces finitely many reduced parameter strata; flat-family Euler characteristics give finitely many fibre polynomials. Uniform regularity of the two successive fibre kernels then yields one \(N\) for which \(M_m\otimes\kappa(s)=H^0(F_s(m))\) and higher cohomology vanishes for every \(m\geq N\). The first \(n+1\) Fitting rank conditions create \(W_P\) and identify its polynomial. On \(W_P\), each further rank condition is closed, with ideal \(J_m\); their sum stabilizes by Noetherianity and defines \(S_P\). All graded pieces are flat there, so standard affine localizations are colimits of flat modules and the sheaf is flat. For a flat pullback to any test algebra, the proper flat cohomology prerequisite gives locally free sections compatible with residue fields. The natural map from each pulled-back \(M_m\) is an isomorphism by the split-surjection and Nakayama argument of Lemma 5.2. All Fitting equations therefore vanish. Polynomial loci on the test scheme are open and closed, so these unique local factorizations glue. This proves the universal property, including nilpotent tests.

6. For each of the twists \(0,\ldots,n\), choose near a given parameter point a finite free complex computing fibre cohomology. Its alternating sum of term ranks is constant after shrinking. Those sums equal the \(n+1\) values of the fibre Hilbert polynomial and determine it, so the polynomial is locally constant. The line plus special point has polynomials \(m+1\) and \(m+2\) arbitrarily close to zero, hence cannot be flat there. The degree-two family has polynomial 2 throughout. Its direct free-module calculation establishes flatness even though its special fibre is nonreduced. The Hilbert polynomial counts length, not the number of distinct points.

What this lesson does not prove

We use generic flatness [Stacks, Tag 051R], Serre generation and vanishing for ample twists [Stacks, Tag 01XO and Lemmas in that section], and the projective sheafification correspondence. We also use the exact proper flat cohomology theorem in Tag 0A1H: for a morphism of finite presentation, a bounded complex of finitely presented sheaves flat over the base with proper support, tensored with a perfect sheaf complex, has perfect derived pushforward compatible with arbitrary base change. Here the sheaf complex is one sheaf and the perfect factor is \(\mathcal O\); Lemmas 4.1 and 4.2 prove the consequences used in the construction. The regularity input is proved in Regularity and bounded families.

The general Raynaud–Gruson representability theorem is stated with its hypotheses in Theorem 8.1, using Tags 05UG–05UH and the purity definition in Tag 05J4. Theorem 8.2 is the henselian local theorem [Stacks, Tag 05PG]; its proof is outside this lesson. We prove the nonrepresentability example for schemes; its algebraic-space extension is imported from Tag 0FJ1. These inputs do not replace the Artinian, determinantal, or projective flattening proofs, which are given here in full.

References

• [Stacks] The Stacks project, More on Flatness, flattening functors and stratifications, Tags 05MG, 05P6–05PB; universal flattening, Tags 05UG–05UH, retained in AI Integrated Stacks Project Stacks. AI Integrated Stacks Project includes AI-proposed corrections and AI-written additions not reviewed by the Stacks project maintainers.
• [Stacks] Divisors, Fitting stratification, Tag 05P9; Derived Categories of Schemes, proper flat cohomology and arbitrary base change, Tag 0A1H.
• [Nitsure] N. Nitsure, Construction of Hilbert and Quot Schemes, 2005, Sections 3–4, especially Lemmas 3.1–3.2 and Theorem 4.3.
• [Raynaud–Gruson] M. Raynaud and L. Gruson, Critères de platitude et de projectivité. Techniques de « platification » d'un module, Inventiones Mathematicae 13 (1971), 1–89. The relative-purity formulation used here is the one in the cited Stacks tags.