Duality and flat base change on formal schemes

Contemporary Mathematics, 1999

In §8.3 of our paper "Duality and Flat Base Change on Formal Schemes" [DFS] some important results concerning localization of, and preservation of coherence by, basic duality functors, were based on the false statement that any closed formal subscheme of an open subscheme of the completion P of a relative projective space is an open subscheme of a closed formal subscheme of P. In this note, the said results are provided with solid foundations. In Proposition 8.3.1 of our paper [DFS], the duality functors f ! and f # associated to a pseudo-proper map f : X → Y of noetherian formal schemes (i.e., right adjoints of suitable restrictions of the derived direct-image functor Rf *) are asserted to be local on X, as a consequence of flat base change. Moreover, in Proposition 8.3.2 it is asserted that (roughly speaking) f # preserves coherence. Brian Conrad pointed out that our justifications are deficient because they use the claim 8.3.1(c) that a map between noetherian formal schemes that can be factored as a closed immersion followed by an open one can also be factored as an open immersion followed by a closed one, which is not true in general. 1 Indeed, Conrad observed that for any (A, x, p) with A an adic domain, x ∈ A such that B := A {x} is a domain, and p a nonzero B-ideal contracting to (0) in A, the natural map Spf(B/p) → Spf(A) is a counterexample. Such a triple was provided to us by Bill Heinzer: With w, x, y, z indeterminates over a field k, set A := k[w, x, z][[y]] and B := A {x} = k[w, x, 1/x, z][[y]]. Let P be the prime ideal (w, z)A and R := A P ⊂ B P B = : S, so that R ⊂ S are 2-dimensional regular local domains such that the residue field of S (i.e., the fraction field of k[x, 1/x][[y]]) is transcendental over that of R (i.e., the fraction field of k[x][[y]]). Then [HR, p. 364, Theorem 1.12] says that there exist infinitely many height-one prime S-ideals in the generic fiber over R. Any of these contracts in B to a (prime) p as above.

Proceedings of the American Mathematical Society

Let Y be an integral Noetherian scheme. Let f:X→Y be a generically smooth, projective morphism, Cohen-Macaulay, equi-dimensional of relative dimension d and with geometrically connected fibres. Assume that X is reduced. Let ω X/Y d be the sheaf of relative regular differential forms. The aim of the paper is to study when the direct images of this sheaf are torsion-free. The question being local, it is assumed that Y=SpecD for some complete discrete valuation ring D with residue class field k algebraically closed and X=ProjS for some graded reduced D-algebra S. The authors give a set of equivalent conditions for R i f * ω X/Y d to be torsion-free. Using them the following results are proved. (1) Suppose that X/Y is arithmetically S k+2 (Serre’s condition). Then R d-i f * ω X/Y d is torsion-free and commutes with base change for all i≤k. (2) If R d-i f * ω X/Y d is torsion-free for all i≤k then R i f * O X is torsion-free for all i≤k. (3) Suppose that X/Y is globally a complete inters...

Journal of Algebra, 2016

We introduce new enhancements for the bounded derived category D b (Coh(X)) of coherent sheaves on a suitable scheme X and for its subcategory Perf(X) of perfect complexes. They are used for translating Fourier-Mukai functors to functors between derived categories of dg algebras, for relating homological smoothness of Perf(X) to geometric smoothness of X, and for proving homological smoothness of D b (Coh(X)). Moreover, we characterize properness of Perf(X) and D b (Coh(X)) geometrically. Contents 1. Introduction 1 2. Derived categories of sheaves and subcategories 3 3. Enhancements 7 4. Smoothness of categories and schemes 19 5. Properness of categories and schemes 31 6. Fourier-Mukai functors 34 Appendix A. Sheaf homomorphisms and external tensor products 41 Appendix B. Triangulated categories in terms of dg endomorphism algebras 49 Appendix C.Čech enhancements for locally integral schemes 50 References 59 ENHANCEMENTS OF DERIVED CATEGORIES OF COHERENT SHEAVES AND APPLICATIONS 5 Assume that Perf(X) = D b (Coh(X)). Since any point of X contains a closed point in its closure and the localization of a regular local ring is regular it is enough to show that the local ring of each closed point is regular. Let x ∈ X be a closed point. Equip {x} = {x} with the induced reduced scheme structure and let i : {x} → X be the closed embedding. View κ(x) = O X,x /m x as a coherent sheaf on {x}. Then i * (κ(x)) ∈ Coh(X) ⊂ D b (Coh(X)) = Perf(X). This implies that the restriction of i * (κ(x)) to an affine open neighborhood U of x has a finite resolution by finitely generated projective O X (U)-modules. Taking the stalk at x shows that the O X,x-module (i * (κ(x))) x = κ(x) has finite projective dimension. Hence O X,x is regular. Assume that X is regular and of finite dimension. By intelligent truncation it is sufficient to show that any F ∈ Coh(X) is in Perf(X). Let U = Spec R ⊂ X be an affine open subset and d = dim R < ∞. Choose an exact sequence 0 → K → P −d → • • • → P 0 → F | U → 0 where all P i are finitely generated projective R-modules. Localizing at an arbitrary p ∈ Spec R shows that K p is a projective R p-module (here we use that gldim R p = dim R p ≤ d). Hence K is a projective R-module. This shows that F | U ∈ Perf(U) and hence F ∈ Perf(X). 2.2. Resolution property. A Noetherian scheme is said to have the resolution property if any coherent sheaf is a quotient of a vector bundle. For example, any Noetherian separated scheme that is integral and locally factorial (for example regular) has the resolution property, by a theorem of Kleiman [Har77, Ex. III.6.8]; any Noetherian scheme with an ample family of line bundles has the resolution property, by [TT90, Lemma. 2.1.3(b)]. We say that a scheme X satisfies condition (RES) or that X is a (RES)-scheme (for "resolution") if (RES) X is a Noetherian separated scheme of finite dimension that has the resolution property. Proposition 2.2. If X satisfies condition (RES) then any object of D − Coh (Sh(X)) is isomorphic in D(Sh(X)) to a bounded above complex of vector bundles. If on a Noetherian scheme X any coherent sheaf is isomorphic in D(Sh(X)) to a bounded above complex of vector bundles, then it is easy to see (using intelligent truncation) that X has the resolution property. Proof. The objects of D − Coh (Sh(X)) are precisely the pseudo-coherent complexes, by [TT90, Example 2.2.8]. Now observe that the proof of [TT90, Prop. 2.3.1.(e)] works without the assumption that X has an ample family. Its important ingredient [TT90, Lemma 2.1.3(c)] is true in our setting. Namely, if G → F is an epimorphism of quasi-coherent sheaves with F coherent, then there is a vector bundle E and a morphism E → G such that the composition E → G → F is an epimorphism onto F. This follows from [Har77, Exercise II.5.15] and the resolution property. 2.3. Strictly perfect complexes. We say that a scheme X satisfies condition (GSP) or that X is a (GSP)-scheme (for "globally strictly perfect") if

These are notes from Mathematics 233br, an advanced graduate seminar on schemes, taught by Dr. Junecue Suh during the Spring of 2014 at Harvard University. Please excuse the roughness and brevity of some of the sections. Any errors found in these notes should be attributed to the scribe. The first half of the course was concerned with derived categories, primarily following the text of Gelfand and Manin ([2]). Since our lectures did not differ in any significant way from their treatment, I did not deem it necessary to write up those notes. Before proceeding, the reader should be acquainted with the content of this text, the first chapters of which are crucial to understanding the following lectures.

Publications Mathematiques De L Ihes, 2008

In this paper we develop a theory of Grothendieck's six operations of lisse-étale constructible sheaves on Artin stacks locally of finite type over an affine regular noetherian scheme of dimension ≤ 1. We also generalize the classical base change theorems and Kunneth formula to stacks, and prove new results about cohomological descent for unbounded complexes.