Duality in the cohomology of crystalline local systems

Let k be a perfect field of a positive characteristic p, K-the fraction field of the ring of Witt vectors W(k) Let X be a smooth and proper scheme over W(k). We present a candidate for a cohomology theory with coefficients in crystalline local systems: p -adic étale local systems on X_K characterized by associating to them so called Fontaine-crystals on the crystalline site of the special fiber X _k. We show that this cohomology satysfies a duality theorem.     

On the action of the Iwahori–Hecke algebra on modular invariants

We compute the action of the modular Iwahori–Hecke algebra on the ring of invariants of the mod p cohomology of elementary p-groups under Borel subgroup of the general linear group. Applications include a direct proof of the structure of the universal Steenrod algebra and a new proof of a key result on the structure of the Takayasu modules.     

A note on cohomology with limited torsion

In this note we prove that if a classx in a torsion free (ordinary) cohomology ring of a topological space satisfiesx ^p =py (p-a prime) thany ^p is divisible byp. The proof and the applications of this statement are related to the theory of secondary operations.     

Morita Theory for Comodules Over Corings

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule Σ of an A-coring 𝒞. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring 𝒞 or the comodule Σ is finitely generated and projective as an A-module. That is, we obtain relations between the category of 𝒞-comodules and the category of firm modules for a firm ring R, which is an ideal of the endomorphism algebra End ^𝒞(Σ). For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map.     

Finiteness Properties of Duals of Local Cohomology Modules

We investigate Matlis duals of local cohomology modules and prove that, in general, their zeroth Bass number with respect to the zero ideal is not finite. We also prove that, somewhat surprisingly, if we apply local cohomology again (i.e., to the Matlis dual of the local cohomology module), we get (under certain hypotheses) either zero or E, an R-injective hull of the residue field of the local ring R.     

ON THE STRUCTURE OF COHOMOLOGY RINGS OF p-NILPOTENT LIE ALGEBRAS

In this paper the authors investigate the structure of the restricted Lie algebra cohomology of p-nilpotent Lie algebras with trivial p-power operation. Our study is facilitated by a spectral sequence whose E ₂-term is the tensor product of the symmetric algebra on the dual of the Lie algebra with the ordinary Lie algebra cohomology and converges to the restricted cohomology ring. In many cases this spectral sequence collapses, and thus, the restricted Lie algebra cohomology is Cohen–Macaulay. A stronger result involves the collapsing of the spectral sequence and the cohomology ring identifying as a ring with the E ₂-term. We present criteria for the collapsing of this spectral sequence and provide some examples where the ring isomorphism fails. Furthermore, we show that there are instances when the spectral sequence does not collapse and yields cohomology rings which are not Cohen-Macaulay.     

Group cohomology and control of p-fusion

We show that if an inclusion of finite groups H≤G of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F-isomorphism in mod p cohomology, then H controls p-fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p-subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p=2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories. The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p-fusion systems over the same finite p-group are detected on elementary abelian p-groups if p odd and abelian 2-groups of exponent at most 4 if p=2.     

Purity for Hodge-Tate representations

We prove that a kind of purity holds for Hodge-Tate representations of the fundamental group of the generic fiber of a semi-stable scheme over a complete discrete valuation ring of mixed characteristic with perfect residue field. As an application, we see that the relative p-adic étale cohomology with proper support of a scheme separated of finite type over the generic fiber is Hodge-Tate if it is locally constant.     

The cohomology of the moduli space of controllable linear systems

In this paper we explicitly describe, by generators and relations, the cohomology ring of the manifold _n,m (F) of controllable linear systems having m inputs and state-space dimension n. It is shown that the cohomology ring of _n,m (F) is isomorphic to the invariant cohomology ring of a product of projective spaces. Estimates for the cup length of the cohomology ring are obtained.     

Hodge Structures Associated to SU(p, 1)

Let A be an abelian variety over ? such that the semisimple part of the Hodge group of A is a product of copies of SU(p, 1) for some p > 1. We show that any effective Tate twist of a Hodge structure occurring in the cohomology of A is isomorphic to a Hodge structure in the cohomology of some abelian variety.     

Syntomic regulators andp-adic integration I: Rigid syntomic regulators

We construct a new version of syntomic cohomology, called rigid syntomic cohomology, for smooth schemes over the ring of integers of ap-adic field. This version is more refined than previous constructions and naturally maps to most of them. We construct regulators fromK-theory into rigid syntomic cohomology. We also define a “modified” syntomic cohomology, which is better behaved in explicit computations yet is isomorphic to rigid syntomic cohomology in most cases of interest.     

On the first cohomology of arithmetic groups

We study the restriction to smaller subgroups, of cohomology classes on arithmetic groups (possibly after moving the class by Hecke correspondences), especially in the context of first cohomology of arithmetic groups. We obtain vanishing results for the first cohomology of cocompact arithmetic lattices in SU(n,1) which arise from hermitian forms over division algebras D of degree p ², p an odd prime, equipped with an involution of the second kind. We show that it is not possible for a ‘naive’ restriction of cohomology to be injective in general. We also establish that the restriction map is injective at the level of first cohomology for non co-compact lattices, extending a result of Raghunathan and Venkataramana for co-compact lattices. Received: 14 September 2000 / Accepted: 6 June 2001     

Face rings of simplicial complexes with singularities

The face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is nonsingular, i.e., has the homology of a wedge of spheres of the expected dimension. This is derived from an enumerative result for local cohomology of face rings modulo generic linear forms, as compared with local cohomology of the face ring itself. The enumerative result is generalized to squarefree modules. A concept of Cohen–Macaulay in codimension c is defined and characterized for arbitrary finitely generated modules and coherent sheaves. For the face ring of an r-dimensional complex Δ, it is equivalent to nonsingularity of Δ in dimension r − c; for a coherent sheaf on projective space, this condition is shown to be equivalent to the same condition on any single generic hyperplane section. The characterization of nonsingularity in dimension m via finite local cohomology thus generalizes from face rings to arbitrary graded modules.     

Torus Actions, Equivariant Moment-Angle Complexes, and Coordinate Subspace Arrangements

We show that the cohomology algebra of the complement of a coordinate subspace arrangement in the m-dimensional complex space is isomorphic to the cohomology algebra of the StanleyReisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) After that we calculate the latter cohomology algebra by means of the standard Koszul resolution of a polynomial ring. To prove these facts, we construct a homotopy equivalence (equivariant with respect to the torus action) between the complement of a coordinate subspace arrangement and the moment-angle complex defined by a simplicial complex. The moment-angle complex is a certain subset of the unit polydisk in the m-dimensional complex space invariant with respect to the action of the m-dimensional torus. This complex is a smooth manifold provided that the simplicial complex is a simplicial sphere; otherwise, the complex has a more complicated structure. Then we investigate the equivariant topology of the moment-angle complex and apply the EilenbergMoore spectral sequence. We also relate our results with well-known facts in the theory of toric varieties and symplectic geometry. Bibliography: 23 titles.     

A note on the algebra of operations for Hopf cohomology at odd primes

Let p be any prime, and let \({\mathcal B}(p)\) be the algebra of operations on the cohomology ring of any cocommutative \(\mathbb {F}_p\)-Hopf algebra. In this paper we show that when p is odd (and unlike the \(p=2\) case), \({\mathcal B}(p)\) cannot become an object in the Singer category of \(\mathbb {F}_p\)-algebras with coproducts, if we require that coproducts act on the generators of \({\mathcal B}(p)\) coherently with their nature of cohomology operations.

     

The Auslander–Reiten translate on monomial rings

For t in Nn, E. Miller has defined a category of positively t-determined modules over the polynomial ring S in n variables. We consider the Auslander–Reiten translate, Nt, on the (derived) category of such modules. A monomial ideal I is t-determined if each generator xa has a?t. We compute the multigraded cohomology and Betti spaces of for every iterate k and also the S-module structure of these cohomology modules. This comprehensively generalizes results of Hochster and Gräbe on local cohomology of Stanley–Reisner rings.     

Coinduction functor and simple comodules

Consider a coring with exact rational functor, and a finitely generated and projective right comodule. We construct a functor (coinduction functor) which is right adjoint to the hom-functor represented by this comodule. Using the coinduction functor, we establish a bijective map between the set of representative classes of torsion simple right comodules and the set of representative classes of simple right modules over the endomorphism ring. A detailed application to group-graded modules is also given.     

Intersection formulae in flag manifolds

Summary Two sets of generators of the cohomology ring of a complex (incomplete) flag manifold are obtained in terms of Ehresmann classes. Intersection formulae of the bases elements with any Ehresmann class are then given, thus determining the ring structure of the cohomology ring.     

Fonctions L associées aux F-isocristaux surconvergents II: Zéros et pôles unités

Résumé Nous démontrons la conjecture de Katz concernant la méromorphie et la caractérisation des zéros et p?les unités des fonctions L associées aux représentations p-adiques lorsque celles-ci se prolongent sur une compactification du schéma de base. Comme cas particuliers importants, on obtient celui de la fonction zêta d’un schéma quelconque et celui d’une représentation p-adique quelconque sur un schéma propre.

---

If X is a smooth variety over a finite field ? _q of characteristic p > 0 and ℱ is a p-adic sheaf associated to a representation of the fundamental group of X, N. Katz conjectures, in his Bourbaki talk 409, that the L function L (X, ℱ, t) has its p-adic unit roots and poles given in terms of p-adic étale cohomology. We prove this conjecture in the case of the structure sheaf ℱ = ℤ _p , that is for the Zeta function, and also more generally when the p-adic sheaf ℱ extends to a smooth sheaf on a compactification of X: as a consequence we get the Unit-Root Zeta function of Dwork and Sperber as an L function. The idea of the proof is to get the p-adic étale cohomology with coefficients and compact support as the fixed points of Frobenius acting on rigid cohomology with compact support. For this purpose, we first build a crystalline Artin–Schreier short exact sequence on the syntomic site of a scheme which is separated of finite type over a perfect field k: this naturally generalizes the work of J.M. Fontaine and W. Messing in the proper smooth case. Then getting rigid cohomology with coefficients as a limit of crystalline cohomologies of variable level we deduce a long exact sequence connecting p-adic étale cohomology (with compact support) to rigid cohomology (with compact support). When X is smooth and affine over an algebraically closed field, the former exact sequence splits into short exact sequences that identify the p-adic étale cohomology with support of X to the part of its rigid cohomology invariant under Frobenius. We can then describe the p-adic unit roots and poles of the Zeta function of X; as a corallary we get the Unit-Root Zeta function of Dwork and Sperber as an L function. In the appendix we show that the characteristic spaces of Frobenius in rigid cohomology commute with isometric extensions of the base, and that isocrystals associated to p-adic sheaves with finite monodromy are overconvergent: we thus obtain a p-adic proof of the rationality of the corresponding L-function.

---

---

Oblatum 8-XII-1994 & 30-IV-1996