On the decomposition of the De Rham complex on formal schemes

We show that if X is a pseudo-proper smooth noetherian formal scheme over a positive characteristic p field k then its De Rham complex τ ≤p (FX/k ∗Ωb• X/k) is decomposable. Along the way we establish the Cartier isomorphism Ωbi X(p)/Y γ→ Hi (FX/Y ∗Ωb•X/Y) associated to a map f : X → Y of positive characteristic p noetherian formal schemes where X(p) denotes the base change of X along the Frobenius morphism of Y and FX/Y denotes the relative Frobenius of X over Y.Ministerio de Economía y Competitividad | Ref. MTM2014-59456Agencia Estatal de Investigación | Ref. MTM2017-89830-PXunta de Galicia | Ref. ED431C 2019/1

The derived category of quasi-coherent sheaves and axiomatic stable homotopy

We prove in this paper that for a quasi-compact and semi-separated (non necessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(A_qc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies D_qct(X) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it is not unital, therefore its abstract nature differs essentially from that of the derived category of a usual scheme.Comment: v2: 31 pages, some improvements in exposition; v3 updated bibliography, to appear Adv. Mat

Classifying Compactly generated t-structures on the derived category of a Noetherian ring

We classify complactly generated t-structures on the derived category of modules over a commutative Noetherian ring R in terms of decreasing filtrations by supports on Spec(R). A decreasing filtration by supports \phi : Z -> Spec(R) satisfies the weak Cousin condition if for any integer i \in Z, the set \phi(i) contains all the inmediate generalizations of each point in \phi(i+1). Every t-structure on D^b_fg(R) (equivalently, on D^-_fg(R)) is induced by complactly generated t-structures on D(R) whose associated filtrations by supports satisfy the weak Cousin condition. If the ring R has dualizing complex we prove that these are exactly the t-structures on D^b_fg(R). More generally, if R has a pointwise dualizing complex we classify all compactly generated t-structures on D_fg(R).Comment: v2 41 pages, improved exposition

A functorial formalism for quasi-coherent sheaves on a geometric stack

A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-coherent sheaves on the small flat topology, Cartesian presheaves on the underlying category, and comodules over a Hopf algebroid associated to a presentation of a geometric stack are equivalent categories. As a consequence, we show that the category of quasi-coherent sheaves on a geometric stack is a Grothendieck category. We also associate, in a 2-functorial way, to a 1-morphism of geometric stacks, an adjunction f^∗⊣f_∗ for the corresponding categories of quasi-coherent sheaves that agrees with the classical one defined for schemes. This construction is described both geometrically in terms of the small flat site and algebraically in terms of comodules over the Hopf algebroid.Ministerio de Ciencia e Innovación | Ref. MTM2008-03465Ministerio de Ciencia e Innovación | Ref. MTM2011-26088Xunta de Galicia | Ref. GRC2013-04

On the derived category of quasi-coherent sheaves on an Adams geometric stack

Let X be an Adams geometric stack. We show that D(Aqc(X)), its derived category of quasi-coherent sheaves, satisfies the axioms of a stable homotopy category defined by Hovey, Palmieri and Strickland in [13]. Moreover we show how this structure relates to the derived category of comodules over a Hopf algebroid that determines X.Ministerio de Ciencia e Innovación | Ref. MTM2011-26088Ministerio de Economía y Competitividad | Ref. MTM2014-59456Xunta de Galicia | Ref. GRC2013-04

On the existence of a compact generator on the derived category of a noetherian formal scheme

In this paper, we prove that for a noetherian formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies D_qct(X) is generated by a single compact object. In an appendix we prove that the category of compact objects in D_qct(X) is skeletally small.Comment: 13 page

Local homology and cohomology on schemes

Abstract. We prove a sheaf-theoretic derived-category generalization of Greenlees-May duality (a far-reaching generalization of Grothendieck’s local duality theorem): for a quasi-compact separated scheme X and a “proregular ” subscheme Z—for example, any separated noetherian scheme and any closed subscheme—there is a sort of sheafified adjointness between local cohomology supported in Z and left-derived completion along Z. In particular, left-derived completion can be identified with local homology, i.e., the homology of RHom • (RΓ Z OX, −). Sheafified generalizations of a number of duality theorems scattered about the literature result: the Peskine-Szpiro duality sequence (generalizing local duality), the Warwick Duality theorem of Greenlees, the Affine Duality theorem of Hartshorne. Using Grothendieck Duality, we also get a generalization of a Formal Duality theorem of Hartshorne, and of a related local-global duality theorem. In a sequel we will develop the latter results further, to study Grothendieck dualit