Infinitesimal local study of formal schemes

Marta Pérez Rodríguez

Abstract

We make a systematic study of the infinitesimal lifting conditions of a pseudo finite type map of noetherian formal schemes. We recover the usual general properties in this context, and, more importantly, we uncover some new phenomena. We define a completion map of formal schemes as the one that arises canonically by performing the completion of a noetherian formal scheme along a subscheme, following the well-known pattern of ordinary schemes. These maps are etale in the sense of this work (but not adic). They allow us to give a local description of smooth morphisms. This morphisms can be factored locally as a completion map followed from a smooth adic morphism. The latter kind of morphisms can be factored locally as an etale adic morphism followed by a (formal) affine space. We also characterize etale adic morphisms giving an equivalence of categories between the category of etale adic formal schemes over a noetherian formal scheme (X, O_X) and the category of etale schemes over th...

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Pseudo finite type maps arise in several significant contexts. To compute residues in a point of a variety over a perfect field we make the completion of the local ring at the given point. The geometric interpretation of this construction is to make a formal completion of the variety at the point and consider it a formal scheme over the base field. The resulting formal scheme is not adic but it is indeed of pseudo finite type. In the context of the extension of Grothendieck duality to formal schemes (see [AJL99]) one generalizes the construction of relative residues and arrives to the following situation. We have a commutative diagram where the horizontal map f is a finite type map of formal schemes and the vertical maps, 71,72 are closed immersions. We obtain a completion mor- phism f: X;z — Y;w that is of pseudo finite type but it is adic only if f-'(W) = Z, an important case but that does not always hold in practice. Recently Lipman, Nayak and Sastry in [LNS05] made a canonical construc- tion of Cousin complexes and its associated pseudofunctors for pseudo finite type maps of noetherian formal schemes.

where, for each n € N, Xp := (X,Ox/F"') and Y;, := (Y,Oy/K"t"). For each n € N,, there exists a unique morphism of schemes fn : Xn > Yn such that, form >n > 0 the diagrams: commute. The morphism f is the direct limit of the system {fn,tmn;Jmn} associated to the Ideals of definition 7 C Ox and K C Og and we will write

ProoF. The proof of (3) = (4) is the same as in the case of schemes (cf. [GD61, (5.5.12)]). The implication (1), (3) > (5) is analogous to that of schemes using 1.3.4.(1), 1.3.4.(3) and [EGA I, (10.15.4)]. In order to prove (1), (3) = (7) consider the following commutative diagrams:

for all m > n > 0. Applying the direct limit over n € N we obtain a morphism f: Xx —+ Yay in NFS, that we call completion of f along a and 3)’, such that the following diagram is commutative: Then the morphism f induces the following commutative diagram of locally noetherian schemes:

where the horizontal sequences correspond with the First Fundamental E

where the horizontal sequences are precisely the Second Fundamental Exact Sequences of the homomorphisms of discrete rings B; — A; —» Aj;/I; for j € {n,m} and the vertical maps are surjective. Since Imo is a finite type A/I-module there results that

such that the diagram, where the composition of the middle horizontal maps is JIn+1) 2.4.3. In definition 2.4.1 the test morphisms for the lifting condition are closed subschemes of affine 2)-schemes given by square-zero ideals. An easy patching argument gives that the uniqueness of lifting conditions holds for closed subschemes of arbitrary 2)-schemes given by square-zero ideals. This applies to formally unramified and formally étale morphisms.

By the universal property of the fiber product there exists a Q)/-morphism Z—> X Xy Y! that agrees with u in T, so f’ is formally smooth. Besides, applying Proposition 1.3.4.(3) there results that f’ is of pseudo finite type and therefore, f’ is smooth. The proof of the unramified case is similar and we leave it to the patient reader. I a Now, let us show that given a smooth morphism f : X + Y) and another morphism 3)’ — Y) in NFS, then the morphism f’ : X x6 — YJ’ is smooth. Let Z be an affine 2)/-scheme, T © Z a closed subscheme defined by a square zero Ideal Z C Oz andu: T > X xq Y’ a Y’-morphism. Since f is smooth, we have that the 2)-morphism T 4 ¥ Xg YY’ > & lifts toa N-morphism Z > X¥ that makes the following diagram commutative

where the vertical arrows are morphisms of completion that, being slightly imprecise, we denote both by «. Let us prove (1). If f is smooth (unrami- fied or étale) by the last Corollary and Proposition 2.4.10.(1) we have that for= Ko f is also smooth (unramified or étale). Since « is étale from Propo- sition 2.4.14 we deduce that fis smooth (unramified or étale, respectively). Assertion (2) is consequence of (1) and of Proposition 1.4.15. PRooFr. By the diagram (1.4.12.1) we have the following commutative diagram in NFS

B-algebra. Let E be a discrete ring , J C E a square zero ideal and consider the commutative diagram of continuous homomorphisms of topological rings

PRooF. By Proposition 2.4.18 we may assume that f is in NFS. Let Z be an affine a scheme, w : Z > Y) amorphism, T — Z a closed 2)-subscheme given by a square zero Ideal and consider u : T + X a Y)-morphism. Since f and w are morphisms of affine formal schemes we find an integer k > 0 such that w*(K*+1)Oz = 0 and u*(J*t!)Or = 0 and therefore u and w factors as T —*> X, “> X¥ and Z —S Y, “> 2), respectively. Since f; is formally smooth, there exists a Y,-morphism vz : Z — Xx such that the following diagram is commutative Thus the Q)-morphism v := ix 0 vp satisfies that v|7 = u and then, f is formally smooth. Moreover, since fo is a finite type morphism, it holds that f is of pseudo finite type and therefore, f is smooth.

PRooF. By Proposition 3.4.1 there exists a unique morphism g : 2) > X such that the following diagram is commutative REMARK. The last result is also true if f is formally smooth.

In Sch open immersions are characterized as being the étale morphisms which are radical (see [EGA IV4, (17.9.1)]). In the following theorem we extend this characterization and relate open immersions in formal schemes (Definition 1.2.24) with their counterparts in schemes.

is commutative. On the other hand, since the morphisms f, are étale, for all n € N, Proposition 3.3.1 implies that f is étale. Then by Corollary 2.4.14 we have that f’ is étale and applying Corollary 3.4.2, f’ is an isomorphism and therefore, f is an open immersion.

From Proposition 2.4.14 there results that y is étale and then by Corollary 3.4.2 we get that y is an isomorphism. L REMARK. In implication (3) => (1) we have proved that given 2) in NFS, a closed formal subscheme J’ C Q) defined by the Ideal Z C Oy, then for every Ideal of definition K C Og of 2), it holds that

is commutative. Applying Proposition 2.4.14 we have that g is étale and from Corollary 3.4.2 we deduce that g is an isomorphism and therefore, f is adic étale.

Since f is unramified, by Proposition 2.4.14 it holds that « is unramified. On the other hand, Kg is a closed immersion and applying Corollary 3.1.13 we have that « is a pseudo closed immersion. where ko is a closed immersion and fj is an étale morphism. Theorem 3.4.6 implies that there exists an étale adic morphism f’ : X’ > 2) in NFS such that X! xg Yo = XG. Then if U& C X is the open formal scheme with underlying topological space Up, by Proposition 3.4.1 there exists a morphism « : > X’ such that the following diagram is commutative: As a consequence of the last result we obtain the following local descrip- ion for étale morphisms.

4.1.1. Given f : X — XY a morphism of pseudo finite type consider a mmutative diagram in NFS where 39 © 3 is a closed formal subscheme given by a square zero Ideal ZI Cc O3. A morphism u: 3 > & is a lifting of uo over Y if it makes this diagram commutative. For instance, if f is étale for all morphisms uo under the last conditions there always exists a unique lifting (see Corollary 2.4.4).