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International audience; — We compute the p-adic geometric pro-étale cohomology of the rigid analytic affine space (in any dimension). This cohomology is non-zero, contrary to the étale cohomology, and can be described by means of differential forms.

We characterize primary operations in differential cohomology via stacks, and illustrate by differentially refining Steenrod squares and Steenrod powers explicitly. This requires a delicate interplay between integral, rational, and mod p cohomology, as well as cohomology with U(1) coefficients and differential forms. Along the way we develop computational techniques in differential cohomology, including a K\"unneth decomposition, that should also be useful in their own right, and point to applications to higher geometry and mathematical physics., Comment: 36 pages, minor corrections

The cohomology of the degree-$n$ general linear group over a finite field of characteristic $p$, with coefficients also in characteristic $p$, remains poorly understood. For example, the lowest degree previously known to contain nontrivial elements is exponential in $n$. In this paper, we introduce a new system of characteristic classes for representations over finite fields, and use it to construct a wealth of explicit nontrivial elements in these cohomology groups. In particular we obtain nontrivial elements in degrees linear in $n$. We also construct nontrivial elements in the mod $p$ homology and cohomology of the automorphism groups of free groups, and the general linear groups over the integers. These elements reside in the unstable range where the homology and cohomology remain poorly understood., Accepted to the Advances in Mathematics

We introduce a notion of crystalline cohomology for superschemes and show that it is isomorphic to the usual crystalline cohomology of the underlying commutative scheme if 2 is invertible on the base.

We consider the annihilator of certain local cohomology modules. Moreover, some results on vanishing of these modules will be considered.

In this text, we generalize Cech cohomology to sheaves F with values in blue B-modules where B is a blueprint with −1. If X is an object of the underlying site, then the cohomology sets H l ( X , F ) turn out to be blue B-modules. For a locally free O X -module F on a monoidal scheme X, we prove that H l ( X , F ) + = H l ( X + , F + ) where X + is the scheme associated with X and F + is the locally free O X + -module associated with F . In an appendix, we show that the naive generalization of cohomology as a right derived functor is infinite-dimensional for the projective line over F 1 .

We compare relative cohomology theories arising from using different proper resolutions of modules. Criteria for the vanishing of such distinctions are given in certain cases, and we show that this is related to the generalized Tate cohomology theory. We also demonstrate that the two balance properties admitted by the two different cohomology theories are actually equivalent in some cases. As applications, we recover many results obtained earlier in various contexts. At last we investigate derived functors with respect to the Auslander and Bass classes.

Necessary and sufficient isomorphism conditions for the second cohomology group of an algebraic group with an irreducible root system over an algebraically closed field of characteristic p ≥ 3h − 3, where h stands for the Coxeter number, and the corresponding second cohomology group of its Lie algebra with coefficients in simple modules are obtained, and also some nontrivial examples of isomorphisms of the second cohomology groups of simple modules are found. In particular, it follows from the results obtained here that, among the simple algebraic groups SL2(k), SL3(k), SL4(k), Sp4(k), and G 2, nontrivial isomorphisms of this kind exist for SL4(k) and G 2 only. For SL4(k), there are two simple modules with nontrivial second cohomology and, for G 2, there is one module of this kind. All nontrivial examples of second cohomology obtained here are one-dimensional.

In this paper, we investigate the properties of right orthogonal modules of $${{\mathscr {C}}}$$ , where $${{\mathscr {C}}}$$ is a class of left R-modules. As an application, we investigate the properties of right orthogonal modules of Ding injective left R-modules, and present various characterizations of semisimple and von Neumann regular rings and so on. Moreover, we also consider another cohomology, strong Tate cohomology, which connects the usual cohomology with the Ding cohomology.

Let f : X → Y be a proper and a flat morphism with fibers of dimension 1 and X regular of dimension 2. Suppose that the geometric fibers of f are connected, and the generic fiber is smooth. In this paper we consider a reductive group G on X and use results of Artin-Tate to find the obstruction for a G-gerb on X to come from an f * G-gerb on Y .

We recall some unusual features of the cohomology of arithmetic groups, and propose that they are explained by a hidden action of certain motivic cohomology groups.

In this note we give a generalization for the higher-order Hochschild cohomology and show that the secondary Hochschild cohomology is a particular case of this new construction.

Let C be a triangulated category with a proper class E of triangles. We prove that there exists an Avramov–Martsinkovsky type exact sequence in C, which connects E-cohomology, E-Tate cohomology and E-Gorenstein cohomology.

Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse-theoretic techniques. As a consequence, we derive efficient techniques for distributed computation of (ordinary) cohomology of a cell complex., 19 pages, 1 Figure. Added Section 5.3

De Rham cohomology it is very obvious that it relies heavily on both topology as well as analysis. We can say it creates a natural bridge between the two. To understand and be able to explain what exactly de Rham cohomology is to the world of mathematics we need to know de Rham groups. This is the reasons to calculate the de Rham cohomology of a manifold. This is usually quite difficult to do directly. We work with manifold. Manifold is a generalization of curves and surfaces to arbitrary dimension. A topological space M is called a manifold of dimension k if : · M is a topological Hausdorff space . · M has a countable topological base. · For all m∈M there is an open neighborhood U⊂M such that U is homeomorphic to an open subset V of ℝk. There are many different kinds of manifolds like topological manifolds, ℂk - manifolds, analytic manifolds, and complex manifolds, we concerned in smooth manifolds. A smooth manifold can described as a topological space that is locally like the Euclidian space of a dimension known. An important definion is homeomorphism. Let X, Y be topological spaces, and let f: X⟶Y e a bijection. If both f and the inverse function f−1: X⟶Y are continuous, then f is called a homeomorphism We introduce one of the useful tools for this calculating, the Mayer – Vietoris sequence. Another tool is the homotopy axiom. In this material I try to explain the Mayer – Vietoris sequence and give same examples. A short exact sequence of cochain complexes gives rise to a long exact sequence in cohomology, called the Mayer - Vietories sequence. Cohomology of the circle (S^1), cohomology of the spheres (S^2). Homeomorphism between vector spaces and an open cover of a manifold. we define de Rham cohomology and compute a few examples.

In Part 1 of the paper we give a new construction of equivariant Alexander–Spanier cohomology for actions of a finite group G, which is simpler and more direct than the original one given in [2] . In Part 2 we prove some useful results concerning equivariant Alexander–Spanier cohomology theory. In Part 3 we use equivariant Alexander–Spanier cohomology and the results proved in Part 2 , in the case where G = C p is a cyclic group of prime order p, to develop P.A. Smith theory from scratch, under minimal, in fact necessary, assumptions.

We present a survey of results concerning the use of Hochschild cohomology in representation theory of algebras.

For a simply connected (non-nilpotent) solvable Lie group $G$ with a lattice $\Gamma$ the de Rham and Dolbeault cohomologies of the solvmanifold $G/\Gamma$ are not in general isomorphic to the cohomologies of the Lie algebra $\mathfrak g$ of $G$. In this paper we construct, up to a finite group, a new Lie algebra $\tilde{\mathfrak g}$ whose cohomology is isomorphic to the de Rham cohomology of $G/\Gamma$ by using a modification of $G$ associated with a algebraic sub-torus of the Zariski-closure of the image of the adjoint representation. This technique includes the construction due to Guan and developed by the first two authors. In this paper, we also give a Dolbeault version of such technique for complex solvmanifolds, i.e. for solvmanifolds endowed with an invariant complex structure. We construct a finite dimensional cochain complex which computes the Dolbeault cohomology of a complex solvmanifold $G/\Gamma$ with holomorphic Mostow bundle and we give a construction of a new Lie algebra $\breve {\mathfrak g}$ with a complex structure whose cohomology is isomorphic to the Dolbeault cohomology of $G/\Gamma$., Comment: 23 pages; to appear in Transformation Groups

To a system of closed one-forms on the torus, we associate a differential complex and compute the induced cohomology groups provided that a related matrix satisfies a Diophantine condition.

We compute the group cohomology of 32 Γ 3 f , a certain group of order 32. For this we construct explicit cocycle representatives of the cohomology generators. We thus lay to rest a discrepancy between several published computations of this cohomology ring.

Let $X_1$ and $X_2$ be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann (CR) manifolds of dimensions $2m-1$ and $2n-1$ in $\mathbb{C}^{m+1}$ and $\mathbb{C}^{n+1}$, respectively. We introduce the Thom-Sebastiani sum $X=X_1\oplus X_2$ which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension $2m+2n+1$ in $\mathbb{C}^{m+n+2}$. Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in $\mathbb{C}^{n+1}$ for all $n\geq 2$ forms a semigroup. $X$ is said to be an irreducible element in this semigroup if $X$ cannot be written in the form $X_1\oplus X_2$. It is a natural question to determine when $X$ is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly, we show that if $X=X_1\oplus X_2$, then the Kohn-Rossi cohomology of the $X$ is the product of those Kohn-Rossi cohomology coming from $X_1$ and $X_2$ provided that $X_2$ admits a transversal holomorphic $S^1$-action.

We construct a realization functor from Levine's triangulated category of motives, for any cohomology theory that takes values in a tensor category that is not necessarily a category of vector spaces, on smooth quasi-projective schemes over a reduced separated noetherian scheme. As a sample application, we construct a realization functor associated with the triple that consists of the rigid cohomology, the de Rham cohomology and the specialization map between them, whose domain is Levine's motivic category for the case in which the base scheme is the spectrum of a ring of p-adic integers.

Extending Eilenberg–Mac Lane’s cohomology of abelian groups, a cohomology theory is introduced for commutative monoids. The cohomology groups in this theory agree with the pre-existing ones by Grillet in low dimensions, but they differ beyond dimension two. A natural interpretation is given for the three-cohomology classes in terms of braided monoidal groupoids.

For an arrangement with complement X and fundamental group G, we relate the truncated cohomology ring, H^{, LaTeX2e, 22 pages, to appear in Singularities and Arrangements, Sapporo-Tokyo 1998, Advanced Studies in Pure Mathematics

This is a survey on the equivariant cohomology of Lie group actions on manifolds, from the point of view of de Rham theory. Emphasis is put on the notion of equivariant formality, as well as on applications to ordinary cohomology and to fixed points., Comment: 40 pages. Final version, to appear in Sao Paulo J. Math. Sci

In this note, we examine the bundle picture of the pullback construction of Lie algebroids. The notion of submersions by Lie algebroids is introduced, which leads to a new proof of the local normal form for lie algebroid transversals of [Bursztyn et al., Crelle, 2017], and which we use to deduce that Lie algebroids transversals concentrate all local cohomology. The locally trivial version of submersions by Lie algebroids $\mathfrak{S}$ is then discussed, and we show that this notion is equivalent to the existence of a complete Ehresmann connection for $\mathfrak{S}$, extending the main result in [del Hoyo, Indag. Math. 2016]. Finally, we show that locally trivial version of submersions by Lie algebroids gives rise to a system of local coefficients, which is an integral part of a version of the homotopy invariance of de Rham cohomology in the context of Lie algebroids, and we apply such local systems to extend the localization theorem of [Chen et al. 2006]., Comment: 10 pages. Comments are welcome

For a discrete group G, we represent the Bredon cohomology with local coefficients as the homotopy classes of maps in the category of equivaraint crossed complexes. Subsequently, we construct a naive parametrized G-spectrum, such that the cohomology theory defined by it reduces to the Bredon cohomology with local coefficients when restricted to suspension spectra., Comment: 29 pages, Accepted version JPAA

For a B-algebra A we introduce a Hochschild-like cohomology and use it to describe simultaneous deformations of the product and of the B-algebra structure on A[[t]]. These deformations have the property that the natural projection A[[t]]→A is a morphism of B-algebras.

On a smoothly stratified space, we identify intersection cohomology of any given perversity with an associated weighted \(L^2\) cohomology for iterated fibred cusp metrics on the smooth stratum.

We construct sheaf-cohomological analogues of Mathai-Quillen forms, that is, holomorphic bundle-valued differential forms whose cohomology classes are independent of certain deformations, and which are believed to possess Thom-like properties. Ordinary Mathai-Quillen forms are special cases of these constructions, as we discuss. These sheaf-theoretic variations arise physically in A/2 and B/2 pseudo-topological field theories, and we comment on their origin and role., Comment: 52 pages, LaTeX

We investigate the relative and Tate cohomology theories with respect to Ding modules and complexes, consider their relations with classical and Gorenstein cohomology theories. As an application, the Avramov-Martsinkovsky type exact sequence of Ding modules is obtained.

We establish several close links between the Galois structures of a range of arithmetic modules including certain natural families of ray class groups, the values at strictly positive integers of p-adic Artin L-series, the Shafarevich–Weil Theorem and the conjectural surjectivity of certain norm maps in cyclotomic ℤ p {\mathbb{Z}_{p}} -extensions. Non-commutative Iwasawa theory and the theory of organising matrices play a key role in our approach.

Let $G$ be a connected and non-necessarily compact Lie group acting on a connected manifold $M$. In this short note we announce the following result: for a $G$-invariant closed differential form on $M$, the existence of a closed equivariant extension in the Cartan model for equivariant cohomology is equivalent to the existence of an extension in the homotopy quotient., Comment: 5 pages. Minor corrections. To appear in DGA

We use a differential form cohomology theory on transitive digraphs to give a new proof of a theorem of Gerstenhaber and Schack about isomorphism between simplicial cohomology and Hochschild cohomology of a certain algebra associated with the simplicial complex.

In this paper, we consider the path semigroup l 1 -algebra for a quiver and the inverse semigroup l 1 -algebra of a quiver, the latter of which can be used in the construction of Cuntz–Krieger algebras. The main objectives of the paper are to determine the simplicial and cyclic cohomology groups of these algebras. First, we determine the simplicial and cyclic cohomology of the path algebra of the quiver, showing the simplicial cohomology groups of dimension n vanish for n > 1 . We then determine the simplicial and cyclic cohomology of the inverse semigroup algebra. The work uses the Connes–Tzygan long exact sequence.

The paper discusses the structure of the Hopf cyclic homology and cohomology of the algebra of smooth functions on a manifold provided that the algebra is endowed with an action or a coaction of the algebra of Hopf functions on a finite or compact group or of theHopf algebra dual to it. In both cases, an analog of the Connes-Hochschild-Kostant-Rosenberg theorem describing the structure of Hopf cyclic cohomology in terms of equivariant cohomology and other more geometric cohomology groups is proved.

One of the main questions in the study of local cohomology of normal domains that are not Cohen–Macaulay is whether the local cohomology of the ring in degrees less than the dimension of the ring is almost zero in an integral extension of the ring. Many examples of non-Cohen–Macaulay normal domains are of Segre product type. In this paper we show that the local cohomology in degrees less that the dimension of the ring for these rings is indeed almost zero in an integral extension.

In this paper we provide a detailed calculation of the Sweedler cohomology of the algebra of functions on (Z/2)^r, in all degrees, over a field of characteristic 2. The result is strikingly different from the characteristic 0 analog. Then we show that there is a variant in characteristic p of the result obtained by Kassel and the author in characteristic zero, which provides a near-complete calculation of the second lazy cohomology group in the case of function algebras over a finite group., With this update we pretty much revert to version 2 (including the old title), for the good reason that this version has been accepted for publication in Comm. Alg. (without part II which appeared in version 3). So this final paper is identical with the published one

We study the Bredon-Illman cohomology with local coefficients for a G-space X in the case of G being a totally disconnected, locally compact group. We prove that any short exact sequence of equivariant local coefficients systems on X gives a long exact sequence of the associated Bredon-Illman cohomology groups with local coefficients.

Using Morel's weight truncations in categories of mixed sheaves, we attach to varieties over finite fields or the complex numbers a series of groups called the weightless cohomology groups. These lie between the usual cohomology and intersection cohomology, have natural ring structures, and are functorial for certain morphisms.

In this paper, by using the de Rham model of Chen-Ruan cohomology, we define the relative Chen-Ruan cohomology ring for a pair of almost complex orbifold (G,H) with H being an almost sub-orbifold of G. Then we use the Gromov-Witten invariants of Ĝ, the blow-up of G along H, to give a quantum modification of the relative Chen-Ruan cohomology ring H*CR(G,H) when H is a compact symplectic sub-orbifold of the compact symplectic orbifold G.