Étale neighbourhoods, henselization and quasi-finite morphisms

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Independent local AI review found corrections in the preceding version; this revision awaits independent correction verification. This lesson includes adapted Stacks proofs and is distributed under GFDL 1.2; original contributions remain CC0. See the source and licence notice below.

A Zariski neighbourhood may retain several algebraic branches that meet at a point. An étale neighbourhood permits a simple root to be chosen while preserving the local geometry. Taking all such choices produces the henselization. Allowing finite separable extensions of the residue field produces the strict henselization. These constructions also explain why a quasi-finite branch can be isolated as a finite scheme after an étale change of base. We then pass from a henselian local ring to a henselian pair and prove that every finite étale algebra, including all its morphisms, is determined by reduction along the pair ideal. Finally, integral descent extends the preceding thickening theorem to every universal homeomorphism.

We use Étale morphisms and their local structure and Infinitesimal lifting and the invariance of étale morphisms under thickenings. The algebraic prerequisites are Henselian local rings and henselization and Completion; the geometric prerequisite is Zariski's Main Theorem. We specify the imported statements below. No Noetherian, separatedness or finite-presentation hypothesis is imposed unless stated. A finite algebra means a finite module, whereas a finite type algebra means finitely many algebra generators.

1. Which neighbourhood categories are cofiltered?

Let \(s\in S\), and put \(k=\kappa(s)\). An étale neighbourhood is an étale morphism \(U\to S\) with a point \(u\) over \(s\). Morphisms of neighbourhoods are \(S\)-morphisms preserving the selected point. It is elementary if the canonical map \(k\to\kappa(u)\) is an isomorphism. Its residue extension is always finite separable [Stacks, Tag 02LE].

Fix a geometric point \(\bar s:\operatorname{Spec}\Omega\to S\), with \(\Omega\) algebraically closed. A neighbourhood of \(\bar s\) carries a lift \(\bar u:\operatorname{Spec}\Omega\to U\); its morphisms preserve this lift. Write \(k^{\mathrm{sep}}\subset\Omega\) for the elements separable algebraic over \(k\). Equivalently the marking specifies an embedding \(\kappa(u)\hookrightarrow k^{\mathrm{sep}}\).

Lemma 1.1 (prescribing the residue field). Every finite separable extension \(L/k\) occurs as the residue field of an étale neighbourhood of \(s\).

Proof. Work in an affine neighbourhood \(\operatorname{Spec}R\) of \(s\), represented by \(\mathfrak p\). Choose a primitive element of \(L/k\), with monic minimal polynomial \(\bar f\in k[T]\). Its finitely many coefficients belong to a localization \((R/\mathfrak p)_r\), for some \(r\notin\mathfrak p\). Lift them to \(R_r\), retaining the leading coefficient 1, and let

This is standard étale. The fibre contains the point with field \(k[T]/(\bar f)=L\), since separability makes \(\bar f'\) nonzero there. Its spectrum, followed by the open immersion \(\operatorname{Spec}R_r\to S\), is the required neighbourhood. ∎ [Stacks, Tag 02LF]

Proposition 1.2. The category of elementary étale neighbourhoods of \(s\), and the category of neighbourhoods of \(\bar s\), are cofiltered.

Proof. The identity neighbourhood makes both categories nonempty. For two elementary neighbourhoods, the product \(U_1\times_S U_2\) has a selected point corresponding to

it gives an elementary common refinement. For geometric neighbourhoods, the two lifts give a lift to the product, which is a common refinement with its marking.

Let \(a,b\colon U\rightrightarrows V\) be parallel arrows. Since \(V\to S\) is étale, its diagonal is an open immersion. The inverse image of that diagonal under \((a,b)\) is therefore an open subscheme \(E\subset U\), and on \(E\) the arrows agree as scheme morphisms. In the elementary case their residue-field maps are both the identity of \(k\), so \(u\in E\). In the geometric case they agree on the specified lift, so that lift factors through \(E\). The open inclusion \(E\to U\) equalizes the arrows and preserves the required marking. These are the three cofilteredness conditions. ∎ [Stacks, Tags 057A, 057B]

The residue marking matters. General pointed neighbourhoods need not be cofiltered: over \(\operatorname{Spec}\mathbf F_q\), take \(U=\operatorname{Spec}\mathbf F_{q^2}\). Its identity and its \(q\)-power automorphism preserve its single point. An equalizing nonempty refinement would induce an embedding \(\mathbf F_{q^2}\hookrightarrow\kappa(v)\) that equates this automorphism with the identity, which injectivity forbids. General pointed neighbourhoods have common refinements, and parallel arrows with the same residue-field map can be equalized by the proof above.

We may always refine to an affine neighbourhood of the selected point lying over a fixed affine open of \(S\). An affine open inclusion preserves the marking. Thus these affine neighbourhoods suffice for all the colimits used below.

2. Henselian rings through their sections

A local ring \((A,\mathfrak m,k)\) is henselian if every simple root \(\alpha\in k\) of the reduction of a monic polynomial \(F\in A[T]\) lifts to a root in \(A\). It is strictly henselian if in addition \(k\) is separably closed [Stacks, Tags 03QF, 03QK].

We import two equivalent algebraic descriptions from Henselian local rings and henselization. Henselianity is equivalent to lifting factorizations of monic polynomials into coprime monic factors, and to every finite \(A\)-algebra being a finite product of local rings. Each such local factor is henselian [Stacks, Tags 04GG, 04GH]. Complete local rings are henselian, as proved in Completion [Stacks, Tags 04GM, 03QG]. These algebraic results do not assert that a henselian ring is complete.

Theorem 2.1 (geometric criterion). The ring \(A\) is henselian if and only if every étale neighbourhood \((U,u)\) of its closed point with \(\kappa(u)=k\) has a section

through \(u\). The section through that specified point is unique. It is enough to test affine étale neighbourhoods.

Proof. Suppose first that \(A\) is henselian. The monic local structure theorem permits an affine open around \(u\) of the form

with \(F\) monic and \(F'\) invertible. An open neighbourhood of the closed point of \(\operatorname{Spec}A\) is the entire spectrum, so no shrinking of this base is needed. The point \(u\) gives a simple root \(\alpha\in k\) of \(\bar F\), with \(\bar g(\alpha)\ne0\). A lifted root \(a\in A\) satisfies \(g(a)\notin\mathfrak m\), hence \(g(a)\) is a unit. Evaluation at \(a\) defines (2.1).

Two sections through \(u\) coincide on the closed point, including its residue-field map. Their equalizer is open because the diagonal of \(U/A\) is open. It contains the closed point, hence is all of \(\operatorname{Spec}A\). This proves uniqueness even when \(U\) is not separated.

Conversely, given monic \(F\) and a simple residue root \(\alpha\), the algebra \((A[T]/(F))_{F'}\) is étale and has the rational point specified by \(\alpha\). A section through it is an evaluation homomorphism whose value on \(T\) is the required lifted root. Thus \(A\) is henselian. ∎ [Stacks, Tags 03QH, 04GG]

Sometimes the criterion says only that the étale algebra admits a section. Applied to every such algebra, this is equivalent: first localize to remove all the other points of its finite closed fibre. Any section then passes through the selected point. For an individual neighbourhood the chosen-point condition must still be kept.

For example, \(\mathbf Z_p\) is henselian. A simple root modulo \(p\) has a unique lift there. The algebraic criterion makes the same conclusion available over henselian rings that are not complete and over rings that are not Noetherian.

3. The two colimits of local choices

For a point \(s\) of an arbitrary scheme the colimit maps come from restriction of functions. Therefore they run over the opposite of the neighbourhood category:

In the second formula the separable closure is chosen as above. The algebraic local colimits and their universal properties are imported from Henselian local rings and henselization, Theorems 4.2 and 5.1. We state them with our markings and prove their comparison with geometric neighbourhoods.

Theorem 3.1 (elementary colimit). For a local ring \((A,\mathfrak m,k)\), the filtered colimit \(H\) of étale \(A\)-algebras \(B\) marked by a prime \(\mathfrak q\) over \(\mathfrak m\) with residue field \(k\) is local and henselian. Its maximal ideal is \(\mathfrak mH\), its residue field is \(k\), and every local map from \(A\) to a henselian local ring factors uniquely through a local map from \(H\). Consequently \(H=A^h\).

Algebraic import. A marking with residue field \(k\) is precisely the evaluation \(B\to k\) used in Henselian local rings and henselization, Theorem 4.2. That theorem proves locality, the maximal ideal and residue-field assertions, henselianity and the stated universal property over any local base ring. Thus its algebraic colimit is the \(H\) in this statement. The geometric identification remains to be proved below. [Stacks, Tags 04GN, 05KS]

To pass from this local-ring result to the first formula in (3.1), put \(A=R_{\mathfrak p}=\mathcal O_{S,s}\) in an affine neighbourhood \(\operatorname{Spec}R\). After a principal refinement at its selected point, a marked étale algebra over \(A\) has a square Jacobian chart and spreads to an étale algebra over \(R_r\) for some \(r\notin\mathfrak p\): take a finite square Jacobian presentation from the preceding lesson, then clear the finitely many denominators in its equations and the identity expressing its determinant as a unit. Its marked fibre point spreads as well because the fibre remains over \(k\). Thus every local stage occurs geometrically. Conversely a geometric affine stage gives an étale algebra after tensoring with \(A\), and each denominator outside its selected prime can be inverted by a principal open refinement. Every function germ needed for the local construction occurs after such a refinement. The two systems therefore give inverse maps on their colimits. Affine refinements from Section 1 give the same answer if non-affine neighbourhoods are included.

Theorem 3.2 (geometric colimit). Using markings \(\kappa(\mathfrak q)\hookrightarrow k^{\mathrm{sep}}\) instead, the colimit \(H_{\mathrm{geom}}\) is \(A^{sh}\) with this chosen separable closure. It is henselian, local with maximal ideal \(\mathfrak mH_{\mathrm{geom}}\), and has residue field \(k^{\mathrm{sep}}\). A local map \(A\to C\) into a strictly henselian ring factors uniquely through it after a compatible embedding \(k^{\mathrm{sep}}\hookrightarrow\kappa(C)\) has been specified.

Algebraic import and geometric comparison. A prime together with a residue embedding into \(k^{\mathrm{sep}}\) is exactly an evaluation \(B\to k^{\mathrm{sep}}\). The algebraic colimit and all stated local-ring properties, including uniqueness with the specified embedding, are Henselian local rings and henselization, Theorem 5.1.

For the second formula in (3.1), work on an affine neighbourhood \(\operatorname{Spec}R\) of \(s\), with \(A=R_{\mathfrak p}\). After a principal refinement at its selected point, a marked étale \(A\)-algebra has a finite square Jacobian presentation and a finite identity certifying that its determinant is invertible. Clearing their denominators gives an étale algebra over some \(R_r\), \(r\notin\mathfrak p\). Base change back to \(A\) recovers the algebra, and its selected point and residue embedding determine a geometric marking on this neighbourhood. Conversely, tensoring the coordinate ring of a geometric affine neighbourhood with \(A\) gives an algebraic marked stage; each denominator outside its selected prime is supplied by a principal open refinement. Finitely many elements and algebra relations can be put at a common stage, by Proposition 1.2, and their finitely many denominators can be inverted together. These constructions therefore induce inverse maps on the colimits, respecting the residue embeddings. Affine cofinality from Section 1 permits non-affine neighbourhoods as well. This proves the second geometric identification in (3.1). ∎ [Stacks, Tags 04GP, 04GW, 03QL]

If \(A=k\) is a field, a marked étale algebra is a product of finite separable fields and its selected field factor is a cofinal refinement. It follows directly that

With a fixed marking the universal properties give canonical uniqueness. After the marking is forgotten, different identifications of a separable closure can give different isomorphisms. For a finite field these already include its nontrivial power automorphisms.

We import the remaining algebraic properties with their exact scope: \(A\to A^h\to A^{sh}\) are faithfully flat; \(A\) is Noetherian if and only if either henselization is Noetherian; and for Noetherian \(A\), the completion of \(A^h\) is canonically the completion of \(A\) [Stacks, Tags 07QM, 06LJ]. The last statement concerns ordinary henselization. A strict henselization can enlarge the residue field and its completion accordingly.

Henselianity still does not imply completeness. For instance, \(\mathbf Q[x]_{(x)}^h\) is countable: up to isomorphism there are countably many finite presentations over the countable base and countably many marked stages, since each stage has finitely many closed-fibre points. Each stage is countable, so their colimit is countable. Its completion is \(\mathbf Q[[x]]\), which is uncountable by the binary coefficient sequences. Thus this henselian ring is not complete.

4. Extracting the finite branches

We use the algebraic statement of Zariski's Main Theorem. If \(A\to B\) is finite type, \(B'\subset B\) is the integral closure of \(A\) in \(B\), and \(\mathfrak q\) is a quasi-finite point, then there is \(g\in B'\setminus\mathfrak q\) such that \(B'_g=B_g\) [Stacks, Tag 00Q9]. This statement allows arbitrary rings and does not require finite presentation.

Lemma 4.1 (finite part over a henselian base). Let \(A\) be henselian local with residue field \(k\), and let \(B\) be a finite type \(A\)-algebra. Then

where each \(B_i\) is finite and local over \(A\), with a single point in its closed fibre, and \(C\) has no point in its closed fibre at which the morphism is quasi-finite. Equivalently every irreducible component of \(\operatorname{Spec}(C\otimes_A k)\) has positive dimension. The \(B_i\) correspond to all the isolated points of \(\operatorname{Spec}(B\otimes_A k)\).

Proof. The fibre is a finite type \(k\)-scheme, hence Noetherian, so it has only finitely many isolated points \(\mathfrak q_1,\ldots,\mathfrak q_r\). They are exactly its quasi-finite points. Algebraic Zariski's Main Theorem gives \(g_i\in B'\setminus\mathfrak q_i\) with \(B'_{g_i}=B_{g_i}\).

Choose algebra generators \(b_1,\ldots,b_n\) of \(B/A\). In \(B'_{g_i}\), each \(b_j\) can be written using a numerator in \(B'\) and a power of \(g_i\) as denominator. Let \(C\subset B'\) be the \(A\)-subalgebra generated by all the \(g_i\) and these finitely many numerators. These elements are integral over \(A\), so \(C\) is finite over \(A\). Since \(C_{g_i}\) contains all \(b_j\) and \(g_i^{-1}\), it equals \(B_{g_i}\).

The maximal ideals \(\mathfrak n_i=\mathfrak q_i\cap C\) are distinct. If \(\mathfrak n_i=\mathfrak n_j\), then \(g_i\) avoids both, so both points occur in \(B_{g_i}=C_{g_i}\) over that same prime and must be equal. Henselianity decomposes \(C\) into local factors. Let \(e_i\in C\) be the idempotent of the factor with maximal ideal \(\mathfrak n_i\). The element \(g_i\) is a unit in \(e_iC\), and hence in \(e_iB\). Consequently

Thus these orthogonal idempotents give finite local factors \(B_i=e_iB\), each with its single closed-fibre point \(\mathfrak q_i\). The residual factor is \((1-\sum e_i)B\), and its closed fibre has no isolated point. If \(r=0\), simply take this residual factor to be \(B\). In a finite type scheme over a field, a zero-dimensional irreducible component is an isolated point: it is a closed point and cannot lie in a different maximal irreducible component. This gives the equivalent dimension statement. ∎ [Stacks, Tags 04GJ, 03QE]

Theorem 4.2 (one quasi-finite point). Let \(f:X\to S\) be locally of finite type and quasi-finite at \(x\), with image \(s\). There is an elementary étale neighbourhood \((U,u)\) of \(s\) and an open \(V\subset X_U\) finite over \(U\), whose fibre over \(u\) consists of a single point \(v\) above \(x\). Moreover \(\kappa(v)=\kappa(x)\) under the canonical identification \(\kappa(u)=\kappa(s)\).

Proof. Choose affine neighbourhoods \(\operatorname{Spec}R\subset S\) and \(Y=\operatorname{Spec}B\subset X\) around \(s,x\) with \(B\) finite type over \(R\). Put \(A=R_{\mathfrak p}\) and \(H=A^h\). The closed fibre after base change to \(H\) is unchanged. Lemma 4.1 provides a finite local factor of \(B\otimes_R H\) corresponding to \(x\). Thus an idempotent \(e\) defines this factor as \(e(B\otimes_R H)\).

Theorem 3.1 expresses \(H\) as the colimit of elementary affine étale neighbourhood algebras \(R_\lambda\) of \(s\). The element \(e\), and the equality \(e^2=e\), occur at some stage. Suppose \(b_1,\ldots,b_n\) generate \(B\) as an \(R\)-algebra. The finite \(H\)-algebra \(e(B\otimes_R H)\) has these \(eb_i\) as algebra generators, and each satisfies a monic integrality equation. Their finitely many coefficients occur at a later stage. Their equations also hold at a later stage: equality of two elements in a filtered colimit means they become equal in some stage. There are finitely many equations, so a common stage suffices. Constants in these equations are multiplied by the factor's unit \(e\).

At that stage, the factor

is generated by finitely many integral elements. The monic equations bound the needed powers of each generator, proving directly that \(C_\lambda\) is a finite \(R_\lambda\)-module. Its spectrum is clopen in \(Y_{R_\lambda}\), hence open in \(X_{R_\lambda}\). Over the marked point \(u\), the fibre is the same selected factor as over \(H\), since its residue field is still \(\kappa(s)\). It has the required single point and unchanged residue field. Take \(U=\operatorname{Spec}R_\lambda\) and \(V=\operatorname{Spec}C_\lambda\). ∎ [Stacks, Tag 02LK]

This descent used finite algebra generators and finitely many equations for their integrality. It did not assume that the relations defining \(B\) were finitely generated.

Theorem 4.3 (several points and separatedness). Given finitely many distinct quasi-finite points \(x_1,\ldots,x_n\) over \(s\), one elementary neighbourhood works simultaneously, with finite opens \(V_i\) and single fibre points \(v_i\) above \(x_i\). If \(f\) is separated, they can be chosen disjoint and clopen, giving

The fibre of \(W\) avoids the selected \(x_i\). It is empty if these are all the points of \(X_s\).

Proof. Apply Theorem 4.2 to each point and take a common elementary refinement. Base change preserves finiteness and the single selected fibre points. In the separated case each finite open is closed: its graph in \(V_i\times_U X_U\) is closed, and projection to \(X_U\) is proper as a base change of the finite map \(V_i\to U\). Thus \(V_i\to X_U\) is proper and has closed image. Each intersection \(V_i\cap V_j\) is therefore finite over \(U\), and its image is closed and misses \(u\), since its fibre there is empty. Shrink \(U\) away from these finitely many images. The remaining opens are disjoint and clopen. Their complement gives (4.3) and the asserted fibre statement. ∎ [Stacks, Tags 02LL, 02LN]

Selecting only one point does not remove other fibre points. For the disjoint union of two identity maps \(S\amalg S\to S\), selecting the first point leaves the second point in \(W_s\).

There is also a residue-splitting version [Stacks, Tags 02LM, 02LO, 02LP]. After an étale neighbourhood that need not be elementary, the finite parts can be required to have purely inseparable residue extensions at their single selected fibre points. Here is the construction. For the finitely many selected finite residue extensions \(K_i/k\), let \(L_i\subset K_i\) be their maximal separable subextensions. The extensions \(K_i/L_i\) are purely inseparable. Choose a finite Galois extension \(L/k\) containing all embeddings of the \(L_i\), and realize it using Lemma 1.1. Each

is a product of fields purely inseparable over \(L\): split \(L_i\otimes_k L\) into its embedding factors, then extend the purely inseparable extension to each factor. Apply Theorem 4.3 to all these lifted points. The subsequent elementary refinement keeps their residue extensions unchanged. For an affine finite type \(X\), select every isolated fibre point; the remaining part has no quasi-finite point in the distinguished fibre. A finite field extension preserves positive-dimensional fibre components, so none are missed by this selection. This proves the finite type algebra variant as well as the selected-point scheme versions.

5. Splitting unramified schemes and étale coverings

Theorem 5.1. Let \(f:X\to S\) be finite and unramified. For every \(s\in S\), an étale neighbourhood \((U,u)\) makes \(X_U\) a finite disjoint union of closed subschemes of \(U\). If \(f\) is finite étale, this disjoint union consists of copies of \(U\).

Proof. Work over an affine neighbourhood of \(s\). The fibre algebra of a finite unramified morphism is a finite product of finite separable residue fields, by Unramified morphisms. Apply the residue-splitting version of Section 4 to all of its points. Its purely inseparable residue extensions are also separable, hence trivial. Since a finite morphism is separated, we obtain disjoint clopen finite parts \(V_i\), each with a single distinguished fibre point of residue \(\kappa(u)\), and a finite complement \(W\) with empty distinguished fibre. Its image is closed, so shrinking \(U\) removes \(W\).

Write \(U=\operatorname{Spec}R\) and \(V_i=\operatorname{Spec}C_i\). Unramifiedness says that the entire fibre algebra \(C_i\otimes_R\kappa(u)\), not just its residue field, equals \(\kappa(u)\). The cokernel of the map

is a finite \(R\)-module whose reduction at \(u\) is zero. Nakayama makes its localization at \(u\) zero. Finitely many generators then vanish on a common open neighbourhood of \(u\). Shrink to this open for every \(i\); (5.1) becomes surjective. Thus each \(V_i\to U\) is a closed immersion. This argument also covers finite unramified morphisms that are not finitely presented.

If \(f\) is étale, each closed immersion \(V_i\to U\) is also étale. By the open immersion criterion from Étale morphisms and their local structure, it is an open immersion. Its image is an open neighbourhood of \(u\). Shrink \(U\) to the intersection of these finitely many images. Each \(V_i\to U\) is now an isomorphism. The number of copies may be zero. ∎ [Stacks, Tags 04HJ, 04HN]

In particular an étale morphism is locally on its source an open piece of a finite étale morphism after a suitable étale neighbourhood of the base: apply Theorem 4.2 and retain its finite étale open. Theorem 5.1 then splits this piece into local copies of the base [Stacks, Tags 04HL, 04HM]. Finiteness is what allows a residual piece with empty fibre to be removed by deleting its closed image.

Theorem 5.2 (henselian finite étale equivalence). For henselian local \((A,\mathfrak m,k)\), reduction is an equivalence between finite étale \(A\)-algebras and finite étale \(k\)-algebras:

It preserves all morphisms and automorphisms. More generally, if \(B\) is finite étale and \(C\) is any finite \(A\)-algebra, then

Proof. A finite étale \(k\)-algebra is a product of finite separable fields. For a field factor choose a monic separable minimal polynomial \(\bar F\) and lift its coefficients to a monic \(F\in A[T]\). The algebra \(B=A[T]/(F)\) is finite free. The derivative is invertible in \(B/\mathfrak mB\). Every maximal ideal of the finite algebra \(B\) lies over \(\mathfrak m\), so \(F'\) is in no maximal ideal of \(B\) and is a unit. Thus \(B\) is étale and lifts the chosen field. Products prove essential surjectivity.

For (5.3), split \(C\) into its henselian local factors \(C_j\), using the imported finite-algebra criterion. A specified map on reductions gives, for each \(j\), a map

This selects a rational closed-fibre point of the étale \(C_j\)-algebra \(B\otimes_A C_j\). Theorem 2.1 gives its unique section, hence a map \(B\to C_j\). Its reduction is the prescribed map: the local finite \(k\)-algebra \(C_j/\mathfrak mC_j\) has nilpotent radical, and maps from the étale \(k\)-algebra \(B/\mathfrak mB\) agreeing after quotienting by that radical agree by infinitesimal uniqueness from the preceding lesson. The same section uniqueness shows that every lift is this one. Combining the factors proves (5.3). Taking \(C\) finite étale proves full faithfulness in (5.2), completing the equivalence. ∎ [Stacks, Tag 04GK]

The uniqueness here is relative to a prescribed reduction map. A lifted finite separable field extension retains its field automorphisms; the theorem does not discard them. Over a strictly henselian ring the finite étale algebras are therefore exactly finite products of the base ring, with the maps corresponding to maps between the finite sets of components.

Henselian pairs and finite étale algebras

The local result reduces modulo one maximal ideal. For a closed subscheme of a nonlocal affine scheme, the right condition is instead a henselian pair. A pair \((A,I)\) is henselian when \(I\subset\operatorname{Jac}(A)\) and every coprime monic factorization modulo \(I\) of a monic polynomial over \(A\) lifts to a monic factorization over \(A\). Here coprime means that the two polynomials generate the unit ideal. There is no Noetherian or completeness condition. For \(I=\mathfrak m\) in a local ring this is exactly the factorization criterion already used in Section 2.

We now prove the general equivalence. Its proof is adapted from the Stacks Project's More on Algebra, Tags 09XI and 09ZL, through the pinned AI Integrated Stacks Project edition identified in the source notice below. The following lemmas explain the lifting and finite-component steps needed beyond the local case.

Lemma 5.3 (idempotents lift in integral algebras). If \((A,I)\) is henselian and \(D\) is an integral \(A\)-algebra, reduction gives a bijection

Proof. First suppose \(D\) is finite over \(A\). Every maximal ideal of \(D\) contracts to a maximal ideal of \(A\), so \(ID\subset\operatorname{Jac}(D)\). Choose \(b\in D\) lifting an idempotent \(\bar e\). Decompose \(D/ID\) into its \(\bar e\)-part and its \(1-\bar e\)-part, and choose finitely many generators in each part. Their lifts generate \(D\) over \(A\), by Nakayama's lemma. Multiplication by \(b\) on these generators can be expressed by a matrix \(M\) whose reduction is diagonal with entries 1 on the first list and 0 on the second. Indeed, the differences from those scalar actions lie in \(ID\), and can be expressed using coefficients in \(I\).

The adjugate identity gives a monic annihilator \(f(T)=\det(T-M)\) of \(b\). If the two lists have lengths \(r,s\), then

Henselian factorization gives \(f=gh\), with \(g,h\) monic and reductions \((T-1)^r,T^s\). Their resultant is a unit: its reduction is a unit, and \(I\) is in the Jacobson radical. The Sylvester-matrix adjugate therefore supplies a Bézout identity for \(g,h\). The Chinese remainder theorem gives an idempotent of \(A[T]/(f)\) equal to 1 on its \(g\)-factor and 0 on its \(h\)-factor. Send \(T\) to \(b\). The resulting idempotent of \(D\) reduces to \(\bar e\). Empty factors cause no difficulty: their polynomial is 1 and their component is zero.

For integral \(D\), write \(b^2-b=\sum_{j=1}^m i_jd_j\), with \(i_j\in I\). The subalgebra generated by \(b,d_1,\ldots,d_m\) is finite over \(A\), since each generator is integral. In its reduction \(b\) is idempotent, so the finite case supplies the required lift in \(D\).

Finally \(ID\subset\operatorname{Jac}(D)\) also holds for every integral \(D\), by contraction of maximal ideals. If idempotents \(e,e'\) have the same reduction, then \(e(1-e')\) and \(e'(1-e)\) are idempotents in that radical. Such an idempotent is zero, since its complementary element is a unit. Thus \(e=ee'=e'\), proving uniqueness. ∎

The determinant trick in this proof uses a generating list, not a free basis. This is why neither \(D\) nor \(A\) needs to be Noetherian. It proves the precise integral-algebra consequence of the henselian-pair criterion [Stacks, Tag 09XI], rather than assuming that the finite algebra is a product of local rings.

Lemma 5.4 (an étale lift before imposing finiteness). For any ring \(A\), ideal \(I\), and étale \(A/I\)-algebra \(C\), there is an étale \(A\)-algebra \(D\) with a specified isomorphism \(D/ID\cong C\).

Proof. Lemma 6.1 of Infinitesimal lifting and the invariance of étale morphisms under thickenings supplies one square presentation of \(C\), with equations \(\bar f_1,\ldots,\bar f_n\) and invertible Jacobian determinant \(\bar\Delta\). Lift the coefficients to equations \(f_i\) over \(A\), let \(\Delta\) be their Jacobian determinant, and set

Its reduction is \(C\), because \(\bar\Delta\) already has an inverse there. Its square Jacobian determinant is \(\Delta^2\): the last column is zero above its last entry \(\Delta\). Thus it is a unit. The square Jacobian criterion proves that \(D\) is étale. We invert the determinant explicitly; \(I\) need not be nil. ∎ [Stacks, Tag 00U9.]

This lift need not yet be finite. The next lemma extracts exactly the desired closed-fibre component without assuming that the entire integral closure is a finite module.

Lemma 5.5 (the integral component around a finite fibre). Let \(D\) be a finite type \(A\)-algebra and suppose

Let \(R\) be the integral closure of the image of \(A\) in \(D\). There is a decomposition

compatible with the map to \(D/ID\), and an element \(g\in R\) reducing to \((1,0)\), such that \(R_g\to D_g\) is an isomorphism.

Proof. Put \(T=\operatorname{Spec}C\subset\operatorname{Spec}D\) and write \(\phi:\operatorname{Spec}D\to\operatorname{Spec}R\). At every point of \(T\), the map to \(\operatorname{Spec}A\) is quasi-finite, since its fibre lies in a finite algebra over a residue field. The algebraic Zariski's Main Theorem, Zariski's Main Theorem, Theorem 1.1, in the exact arbitrary-base form used in Section 4, gives elements \(h_t\in R\), nonvanishing at \(t\), for which \(R_{h_t}=D_{h_t}\). Let \(U\) be the union of these principal opens in \(\operatorname{Spec}R\). On \(U\), the map \(\phi\) is an isomorphism.

The subscheme \(T\) is open and closed in \(\operatorname{Spec}(D/ID)\), and lies over \(U\). Its image is consequently open in \(\operatorname{Spec}(R/IR)\). It is also closed in \(\operatorname{Spec}R\): \(C\) is finite over \(R\), because any finite list generating it over \(A/I\) also generates it over \(R\). Finite maps have closed image. Thus its image defines an open and closed component \(W\) of \(\operatorname{Spec}(R/IR)\). This component is contained in \(U\), where \(\phi\) is an isomorphism, so \(W\cong T\) as schemes. This proves (5.7), including its scheme structure and compatibility.

Here is why one may choose a single \(g\) with both required properties. Let \(J\subset R\) define the closed complement of \(U\), and let \(\bar e=(1,0)\in R/IR\). On the \(\bar e\)-component the ideal \(J\) generates the unit ideal, because this component is contained in \(U\). Hence \(\bar e\in J(R/IR)\): multiply a unit-ideal expression on that component by \(\bar e\). Choose \(g\in J\) lifting \(\bar e\). Then \(D(g)\subset U\). Restricting the isomorphism on \(U\) to this principal open gives \(R_g=D_g\). ∎ [Stacks, Tag 09XH, More on Algebra, finite-type component lemma.]

Theorem 5.6 (finite étale equivalence for every henselian pair). For a henselian pair \((A,I)\), reduction induces an equivalence

It lifts every object and every morphism, with uniqueness for a prescribed reduction. Equivalently, base change gives an equivalence of categories of finite étale schemes over \(\operatorname{Spec}A\) and \(\operatorname{Spec}(A/I)\).

Proof of full faithfulness. Let \(B,B'\) be finite étale over \(A\), and put \(E=B\otimes_A B'\). An \(A\)-algebra map \(B\to B'\) is the same thing as a section \(E\to B'\) of \(B'\to E\). The corresponding scheme section of a finite étale map is open and closed: it is open because the map is étale and closed because it is separated. Thus sections correspond exactly to idempotents \(e\in E\) for which \(B'\to eE\) is an isomorphism. The same correspondence holds after reduction modulo \(I\).

Since \(E\) is finite over \(A\), Lemma 5.3 lifts a reduction idempotent uniquely. It remains to check that a reduction isomorphism

lifts to an isomorphism \(B'\to eE\). Its cokernel is a finite \(B'\)-module, killed by Nakayama, since \(IB'\subset\operatorname{Jac}(B')\). Thus the map is surjective. The target \(eE\) is finite projective over \(B'\), being a direct factor of a finite étale algebra. The module criterion is proved in Tor and flat modules, Theorem 5.3: a finitely presented flat module is finite projective. The surjection splits as a module map. Its kernel is therefore finite; reduction of the split sequence identifies its reduction with the zero kernel of the displayed isomorphism. Nakayama kills that kernel as well. We have an isomorphism. This proves existence and uniqueness of the section lifting each reduction section, hence the required bijection on all Hom sets.

Proof of essential surjectivity. Let \(C\) be finite étale over \(A/I\). Lemma 5.4 lifts it to an étale algebra \(D\) over \(A\). Apply Lemma 5.5 to \(D/ID=C\), with zero complementary factor, and let \(R\) be the integral closure used there. Lift the idempotent \((1,0)\in R/IR\) by Lemma 5.3, obtaining

The component of \(g\) in \(R_1\) is congruent to 1 modulo \(IR_1\). As \(R_1\) is integral over \(A\), this ideal is in its Jacobson radical, and that component of \(g\) is a unit. The isomorphism \(R_g=D_g\) therefore exhibits \(R_1\) as an idempotent factor of the étale \(A\)-algebra \(D_g\). It follows that \(R_1\) is étale and in particular of finite type over \(A\). It is also integral over \(A\). Finitely many integral algebra generators give a finite module: bound the exponent of each generator using its monic equation, and use the resulting finitely many monomials to span. Consequently \(R_1\) is finite étale over \(A\) and has reduction \(C\). This proves essential surjectivity and completes the equivalence. ∎ [Stacks, Tag 09ZL; the scheme form is Tag 09ZS.]

No part of this proof replaces \(I\) by a maximal ideal, makes \(R\) finite, or assumes that \(A\) is Noetherian. The local equivalence (5.2) is the special case \(I=\mathfrak m\); its additional assertion (5.3), allowing any finite target algebra, remains useful in its own right.

For a nonlocal example, let \(A=A_1\times A_2\), where each \((A_i,\mathfrak m_i)\) is henselian local, and take \(I=\mathfrak m_1\times\mathfrak m_2\). Factor monic polynomials componentwise to check that \((A,I)\) is henselian. Formula (5.8) classifies finite étale \(A\)-algebras by independent finite étale algebras over the two residue fields. Their ranks can differ on the two components. More generally, the theorem applies along any ideal satisfying the pair conditions, without requiring that \(A\) be a product of local rings.

Universal homeomorphisms preserve étale objects

The thickening theorem in the preceding lesson says that nilpotent structure does not change étale objects. Purely inseparable identifications do not change them either. We can now prove this stronger statement using integral descent. The order matters: the integral-descent proof uses the general henselian-pair equivalence just proved, whereas our proof of that equivalence used no integral descent.

We use two exact internal results of Descending properties of schemes and morphisms: Theorem 5.2 gives full faithfulness for étale descent along a universally submersive map, and Theorem 6.4 gives effective descent of quasi-compact separated étale objects along an arbitrary surjective integral map. Both proofs are already written. A universally submersive map gives the quotient topology after every base change; integral surjections have this property because they are closed after every base change. The integral-descent proof's finite étale henselian-pair input is supplied here by Theorem 5.6.

Theorem 5.7 (topological invariance of étale objects). Let \(p:S'\to S\) be a universal homeomorphism: equivalently, it is integral, surjective and universally injective. Pullback gives an equivalence

The objects may be nonseparated and need not be quasi-compact. This equivalence preserves and reflects jointly surjective coverings, and therefore gives an equivalence of the small étale sites and their sheaf topoi.

Proof. First we show that every étale \(S'\)-scheme has a unique descent datum. The diagonal

is a closed immersion because \(p\) is integral and hence separated. It is surjective on points because \(p\) is universally injective. Thus it is a nil thickening. The same statements hold for the small diagonal into the triple product. The thickening equivalence of the preceding lesson, Theorem 5.3, now gives a unique isomorphism between the two pullbacks of any étale \(U'\to S'\) to \(S'\times_S S'\), restricting along \(\delta\) to the identity of \(U'\). On the triple product its two composites restrict to the identity along the small diagonal. Full faithfulness for that thickening makes the composites equal, proving the cocycle equation. A morphism of étale \(S'\)-schemes respects these data: after restriction along \(\delta\) the compatibility equation is the original morphism on both sides, and thickening full faithfulness proves it everywhere.

Consequently forgetting the datum is an equivalence between étale descent data for \(S'/S\) and étale \(S'\)-schemes. Since \(p\) is universally submersive, the internal descent Theorem 5.2 gives full faithfulness from étale \(S\)-schemes to these data. Combining the two assertions proves full faithfulness in (5.10).

For essential surjectivity, first suppose \(S\) and \(U'\) are affine. Then \(S'\) is affine because \(p\) is integral. The map \(U'\to S'\) is affine, quasi-compact, separated and étale. Its canonical datum is therefore effective by the internal integral-descent Theorem 6.4. It yields a quasi-compact separated étale \(U\to S\), with a specified isomorphism \(U\times_S S'\cong U'\).

Now let \(U'\) be arbitrary. Cover it by affine opens \(U_i'\), each mapping into the inverse image of an affine open of \(S\). The affine case gives étale \(U_i\to S\) with pullback \(U_i'\). The map \(U_i'\to U_i\) is a universal homeomorphism, so the open intersection \(U_i'\cap U_j'\) corresponds to a unique open \(U_{ij}\subset U_i\). Its base change is that intersection as a scheme, since restriction to an open commutes with base change. Full faithfulness in (5.10) descends the identity on the intersection to an isomorphism \(U_{ij}\to U_{ji}\). Its inverse also descends, and faithfulness on triple intersections proves the cocycle equation. Gluing the \(U_i\) along these open isomorphisms gives an étale \(U\to S\) with pullback \(U'\). This gluing imposes no separatedness or quasi-compactness on the final object.

Finally an étale family is jointly surjective precisely when its underlying images cover. Pullback along the universal homeomorphism on each target object preserves and reflects that condition. The category equivalence therefore preserves and reflects the étale coverings, proving the site and topos assertions. ∎

This proof adapts the canonical-datum and gluing argument of the Stacks Project, Étale Cohomology, Tags 04DY–04DZ, in the pinned AI Integrated Stacks Project edition. Its effectivity input has an exact internal proof rather than an external proof link. For example, the absolute Frobenius of any scheme of characteristic \(p>0\) is a universal homeomorphism: on each affine chart its map is \(a\mapsto a^p\), every target element is integral over the image, primes have unique inverse images, and residue extensions are purely inseparable. Theorem 5.7 thus applies even when Frobenius is not finite. The corresponding cohomological interpretation is developed in Pushforward, pullback and finite morphisms, Section 6, Theorem 6.2 (the étale-cohomology course, lesson 7).

6. Roots, branches and a finite local piece

Assume \(\operatorname{char}k\ne2\) and put \(R=k[x]_{(x)}\). The polynomial \(T^2-(1+x)\) has the simple residue root 1. Hence \(R^h\) contains a root \(s\) with

It is not a rational function in \(k(x)\): the valuation at the irreducible polynomial \(x+1\) of a square is even, whereas that of \(1+x\) is 1. In particular it is not in \(R\). The other root \(-s\) has residue \(-1\), so the marking distinguishes the two choices.

Proposition 6.1 (the node). The local ring at the origin of

is a domain, but its henselization is reduced with exactly two minimal primes. More precisely, with \(D=R^h\) and \(s\) as in (6.1), its henselization is

Proof. The polynomial \(y^2-x^2(1+x)\) is irreducible over \(k(x)\), by the preceding square obstruction, and the quotient embeds into that field extension because the polynomial is monic. Thus \(R[y]/(y^2-x^2(1+x))\) is a domain. It is finite over the local ring \(R\); its closed fibre is \(k[y]/(y^2)\), which has one maximal ideal. Hence this finite algebra is local and is exactly the local ring \(A\) of the node.

The imported Noetherian henselization and completion results identify \(\widehat D\) with \(k[[x]]\). The Noetherian local completion map is faithfully flat and therefore injective [Stacks, Tag 00MC]. Consequently \(D\) is a domain and \(x\ne0\) in \(D\).

Set \(C=A\otimes_R D\). It is finite over henselian \(D\), and its closed fibre is again \(k[y]/(y^2)\). It is therefore a local henselian ring. The local map \(A\to C\) induces \(A^h\to C\). Conversely the local map \(R\to A^h\) induces \(D\to A^h\), hence \(C\to A^h\). Their composite on \(A^h\) is the identity by its universal property. The other composite fixes \(A\), and fixes \(D\) by the uniqueness of the local map extending \(R\to C\); these generate \(C\). Thus \(C=A^h\). Equation (6.1) factors its defining polynomial as in (6.3).

Evaluation at \(y=xs\) and at \(y=-xs\) maps \(C\) to \(D\times D\). To check injectivity, work in \(D[Y]\). If a polynomial is divisible by both \(Y-xs\) and \(Y+xs\), write it as \((Y-xs)q(Y)\) and evaluate at \(-xs\). Then \(-2xs\,q(-xs)=0\). The domain property and \(2xs\ne0\) imply \(q(-xs)=0\), so monic division shows \(Y+xs\) divides \(q\). The intersection of the two linear ideals is therefore their product, the defining ideal of \(C\).

Every class has the form \(a+by\); its values are \((a+bxs,a-bxs)\). Their difference is in \(xD\). Conversely for \(u-v\in xD\), take

since \(2s\) is a unit. This proves the pair-ring description. The ideals

have domain quotient \(D\), and are incomparable: the other linear factor maps to \(\pm2xs\ne0\). Their intersection is zero. Every prime contains one of them because their product is zero. Thus they are exactly the two minimal primes and the ring is reduced. ∎

The two branches in (6.3) meet at their common closed point: the pair coordinates have the same residue modulo \(x\). The henselization is local, so it is not the product \(D\times D\). Splitting local factors of finite algebras does not require each local factor to be a domain.

For the quasi-finite example, let

It is quasi-finite. It is not finite: \(1/(x-1)\) cannot be integral over \(k[t]\). The integrally closed discrete valuation ring \(k[x]_{(x-1)}\) contains \(k[t]\), so such integrality would put this element in the valuation ring, contradicting its negative valuation.

Near \(t=1\), use the elementary étale neighbourhood

It is étale because \(2a\) is invertible, and \(\kappa(u)=k\). The two factors \(x-a\) and \(x+a\) are comaximal because \(2a\) is a unit. The Chinese remainder theorem and then inversion of \(x-1\) give

The second factor is the finite branch \(x=-a\) through \(x=-1\), isomorphic to \(U\). The first is the positive branch \(x=a\), whose fibre over \(u\) is empty. Its image is the open \(D(a-1)\), which cannot be removed by shrinking around \(u\). This is why a finite branch and a residual branch are separate parts of the localization theorem.

7. Exercises

1. Easy. For \(\operatorname{char}k\ne2\), prove that the residue-1 root of \(T^2-(1+x)\) belongs to \(k[x]_{(x)}^h\) and does not belong to \(k[x]_{(x)}\).
2. Medium. Compute the henselization of the local node ring (6.2). Identify its minimal primes and explain why it is local although it has two branches.
3. Medium. Let \((A,\mathfrak m,k)\) be henselian. Prove that finite étale \(A\)-algebras, including their morphisms, are determined by their reductions modulo \(\mathfrak m\). Explain the meaning of uniqueness for a chosen lift.
4. Medium. Perform the étale localization of (6.5) at \(t=1\). Identify the finite branch and calculate the fibre of the other branch.
5. Hard. Prove the geometric section characterization of henselianity for arbitrary local rings. Include uniqueness through the specified rational point and explain why existence without specifying the point must be tested on every étale neighbourhood.

Exercise 6 (medium — a finite cover which does not split globally). Let \(k\) be a field of characteristic different from 2, \(A=k[t,t^{-1}]\), \(I=(t-1)\), and \(B=A[u]/(u^2-t)\). Prove that \(B\) is finite étale and connected, whereas \(B/IB\cong k\times k\). Show directly that an idempotent of the reduction does not lift and that the coprime residue factorization of \(T^2-t\) does not lift. Explain which henselian-pair conditions fail.

8. Solutions

Solution 1. The derivative at the residue root 1 is 2, which is nonzero. In the henselian local ring \(R^h\), Hensel's simple-root property gives \(s^2=1+x\) and \(s\equiv1\pmod{x}\). Equivalently, the marked étale algebra \((R[T]/(T^2-(1+x)))_{2T}\), with selected point \(x=0,T=1\), puts this element directly into the colimit. If \(s\) belonged to \(R\), it would belong to \(k(x)\). Taking the valuation at \(x+1\) in \(s^2=1+x\) would give an even integer equal to 1. This contradiction proves the claim.

Solution 2. Put \(R=k[x]_{(x)}\), \(D=R^h\), and let \(s\) be the root from Solution 1. The node ring is \(A=R[y]/(y^2-x^2(1+x))\): it is finite local over \(R\) because its closed fibre is the local algebra \(k[y]/(y^2)\). The finite algebra

has that same local closed fibre, so it is local henselian. The universal properties of \(R^h\) and \(A^h\) give inverse maps \(C\leftrightarrows A^h\): one extends \(A\to C\), the other combines \(A\to A^h\) and the unique extension \(D\to A^h\); uniqueness verifies both composites on the generators.

The map to \(D^2\) by the two evaluations is injective. Indeed, in the domain \(D\), a polynomial divisible by both linear factors is divisible by their product, as evaluation of the quotient at the other root and cancellation of \(2xs\) show. Its image is exactly the pairs with difference in \(xD\): the inverse formula is \(a=(u+v)/2\), \(b=(u-v)/(2xs)\), giving \(a+by\). The kernels of the two projections are the two ideals \((y-xs)\) and \((y+xs)\); their quotients are \(D\), their intersection is zero, and every prime contains one since their product is zero. They are distinct and incomparable, hence exactly the minimal primes. The pair ring has a single maximal ideal, consisting of pairs with common residue zero, because it is the finite local ring just constructed. The congruence condition joins the two branches at that closed point.

Solution 3. Every finite étale \(k\)-algebra is a product of finite separable fields \(k[T]/(\bar F_i)\). Lift each monic polynomial to \(F_i\in A[T]\). The finite free algebra \(A[T]/(F_i)\) has derivative invertible modulo \(\mathfrak m\); all its maximal ideals lie over \(\mathfrak m\), so the derivative is a unit everywhere. Its product with the other lifts is finite étale and has the desired reduction.

For a prescribed residue map \(B/\mathfrak mB\to C/\mathfrak mC\), decompose finite étale \(C\) into henselian local factors \(C_j\). Their reductions are fields. In each factor the map selects a rational point of \(\operatorname{Spec}(B\otimes_A C_j)\) over the closed point. Its unique section lifts the map to \(B\to C_j\). Combining the factors gives a lift to \(C\). Any other lift selects the same point in every factor and therefore is the same map by section uniqueness. This proves full faithfulness as well as existence of every object. Two lifts with a chosen identification of their reductions have one isomorphism respecting that identification. Without prescribing it, residue automorphisms lift to automorphisms and can give several isomorphisms.

Solution 4. Let \(Q=k[a,a^{-1},(a+1)^{-1}]\) and take \(t=a^2\) and \(u=(a-1)\). The derivative \(2a\) is a unit, and the residue map at \(u\) is the identity of \(k\), so this is an elementary étale neighbourhood of \(t=1\). Factoring \(x^2-a^2\), the difference of its roots is \(2a\), a unit, so the Chinese remainder theorem gives \(Q[x]/(x^2-a^2)=Q\times Q\), with evaluations at \(x=a\) and \(x=-a\). Inverting \(x-1\) inverts \(a-1\) in the first factor and \(-a-1\), already a unit, in the second. Hence (6.7) follows. The negative branch is \(\operatorname{Spec}Q\), finite and isomorphic to the base, with its selected point through \(-1\). The positive branch is \(\operatorname{Spec}Q[(a-1)^{-1}]\). Tensoring its ring with \(Q/(a-1)=k\) gives the zero ring, so its selected fibre is empty.

Solution 5. Suppose \(A\) henselian and take a rational selected point of an étale \(A\)-scheme. On a monic standard étale chart through it the point is a simple root \(\alpha\) of a monic \(F\), with the inverted chart element nonzero at \(\alpha\). Lift \(\alpha\) using Hensel's property. The inverted element evaluates to a unit of the local ring, so evaluation defines a section into the chart. The base need not shrink: every open containing its closed point is the entire local spectrum. Two sections through the point have an open equalizer because the étale diagonal is open; their equalizer contains the closed point, hence is the whole base.

Conversely, given a simple root \(\alpha\) of any monic residue polynomial, consider \((A[T]/(F))_{F'}\) and its rational point \(T=\alpha\). A section through this point sends \(T\) to a root of \(F\) reducing to \(\alpha\), proving henselianity. If only existence of some section is assumed for every étale algebra, first invert an element whose reduction is 1 at this selected fibre point and 0 at all other fibre points. The resulting étale algebra still has the chosen point, and any section must pass through it. Thus the universal existence condition also suffices. A section of just one unlocalized algebra might pass through a different root and would not establish the required lifting property.

Solution 6. The monic equation makes \(B\) free of rank two over \(A\). The element \(u\) is a unit since \(u^2=t\), and its derivative \(2u\) is a unit, so \(B\) is finite étale. Eliminating \(t\) identifies it with \(k[u,u^{-1}]\), a domain, hence a connected ring with only the idempotents 0 and 1. Reduction sends \(u^2-t\) to \(u^2-1\); its roots differ by the unit 2, so the Chinese remainder theorem gives \(B/IB=k\times k\). The idempotent \((1,0)\) cannot lift to \(B\).

The residue factorization is \((T-1)(T+1)\). A monic lift of these two degree-one factors would give a root of \(T^2-t\) in \(k[t,t^{-1}]\). Its units are \(ct^n\), with \(c\in k^*\) and \(n\in\mathbf Z\): compare the lowest and highest exponents of a Laurent polynomial and its inverse. A root would be a unit whose square is \(t\), requiring \(2n=1\), which is impossible. Thus the henselian factorization condition fails. The Jacobson-radical condition fails too: \((t+1)\) is a maximal ideal not containing \(t-1\), since 2 is nonzero in \(k\). The theorem's pair conditions are therefore substantive even though the covering is finite étale.

What this lesson does not prove

The algebraic equivalences for henselian rings, the permanence of henselianity under finite local extensions, complete local rings being henselian, faithful flatness and Noetherianity of the two henselizations, and unchanged completion for ordinary henselization are imported from the named algebraic prerequisites [Stacks, Tags 04GG, 04GH, 04GM, 07QM, 06LJ]. Noetherian local completion is faithfully flat [Stacks, Tag 00MC]. The algebraic Zariski's Main Theorem is imported in the form specified in Section 4 [Stacks, Tag 00Q9]. The algebraic local colimit constructions and their universal properties are imported from Henselian local rings and henselization, Theorems 4.2 and 5.1. Sections 1–5 prove geometric cofilteredness, affine cofinality, spreading, the neighbourhood colimit comparisons, and the section and localization applications.

Theorem 5.6 proves finite étale equivalence for every henselian pair, with the idempotent and integral-component steps in Lemmas 5.3–5.5. Its one-square-presentation prerequisite is Lemma 6.1 of the preceding infinitesimal-lifting lesson, which is already written. Algebraic Zariski's Main Theorem is used at its full arbitrary-base, finite-type generality: Zariski's Main Theorem, Theorem 1.1, is the internal provider. Its main induction is written in Sections 1–2, and its complete supporting proofs are in the programme’s AI Integrated Stacks Project algebra chapter: Coefficients in the radical of a conductor (Tag 00PY) and Strong transcendence excludes quasi-finite points (Tag 00Q2). The former includes the conductor setup and leading-coefficient argument; the latter retains reducedness and finiteness over the algebra generated by the strongly transcendental element. Both are also included in the portable programme reader, with their original Stacks Project attribution and GNU FDL 1.2 licence in the pinned edition identified below. This establishes the named proof-provider correspondence; it does not claim recursive prerequisite closure. Theorem 5.7 uses the written integral-descent proof of Descending properties of schemes and morphisms, Theorem 6.4, after the pair-equivalence input has been established above. These dependencies neither assume the pair equivalence nor use it to prove the local structure theorem. The interpretation of strict henselizations through sheaves on the étale site is taught in The étale site and its points, Section 4, Theorem 4.1 (the étale-cohomology course, lesson 6).

Sources, History and GNU Free Documentation License

The proof of the general henselian-pair equivalence and its integral-component construction are adapted from The Stacks Project, More on Algebra, Tags 09XI, 09XH and 09ZL, as retained in AI Integrated Stacks Project, commit 565b10e987aba5969b21145a0833f42d69f96790. Pinned source, labels lemma-characterize-henselian-pair, lemma-helper-finite-type and lemma-finite-etale-equivalence. The scheme statement is also Tag 09ZS. The universal-homeomorphism proof adapts Étale Cohomology, Tags 04DY–04DZ, labels theorem-etale-topological and theorem-topological-invariance, from the same pinned edition. The AI Integrated Stacks Project edition includes AI-written additions and corrections which have not been reviewed by the Stacks Project maintainers. The adapted passages used here are the stated Stacks proofs; no paid-book proof is substituted.

History. The Stacks Project, copyright (C) 2005–2025 Johan de Jong, supplies the original openly licensed proof sources. This lesson, Étale neighbourhoods, henselization and quasi-finite morphisms, was written by GPT-6.1 Sol (OpenAI) in Codex at Ultra and published by Open Mathematics Courses in October 2026. The 2 October 2026 edition integrates the general pair theorem into Section 5, expands the idempotent and integral-component arguments, adds a nonlocal example and Exercise 6, integrates the canonical descent and gluing proof for general universal homeomorphisms, and preserves the earlier local proof and five exercises. On 3 October 2026, the prerequisite note was refreshed to point to the complete conductor-coefficient and strongly-transcendental proofs already supplied in the programme’s AI Integrated Stacks Project chapter. Source edition and original notice.

Licence. Copyright (C) 2005–2025 Johan de Jong for the retained Stacks material. Permission is granted to copy, distribute and/or modify this lesson under the terms of the GNU Free Documentation License, Version 1.2, with no Invariant Sections, no Front-Cover Texts and no Back-Cover Texts. The source permits version 1.2 or later; this modified lesson is offered under version 1.2. A complete unaltered copy is provided as GNU Free Documentation License. Original contributions are also dedicated under CC0; that dedication does not remove the GFDL obligations of this lesson containing the adaptations. The other six original lessons and the course-local reader scripts retain their stated CC0 licence. The Markdown source is the editable, transparent form of this lesson.