Cohomology Functors Research Articles

On endomorphisms of the de Rham cohomology functor

On endomorphisms of the de Rham cohomology functor

Generalized periodicity in group cohomology

Given a finite group G, we introduce “encoding pairs,” which are a pair of G-modules M and M ′ equipped with a shifted natural isomorphism between the cohomological functors H • ( G , Hom Z ( M , − ) ) and H • ( G , Hom Z ( M ′ , − ) ) . Studying these encoding pairs generalizes the theory of periodic cohomology for finite groups, allowing us to generalize the cohomological input of a theorem due to Swan that roughly says that a finite group with periodic cohomology acts feely on some sphere.

The left adjoint of derived parabolic induction

We prove that the derived parabolic induction functor, defined on the unbounded derived category of smooth mod p representations of a p-adic reductive group, admits a left adjoint textrm{L}(U,-). We study the cohomology functors textrm{H}^icirc textrm{L}(U,-) in some detail and deduce that textrm{L}(U,-) preserves bounded complexes and global admissibility in the sense of Schneider–Sorensen. Using textrm{L}(U,-) we define a derived Satake homomorphism and prove that it encodes the mod p Satake homomorphisms defined explicitly by Herzig.

Stable categories of spherical modules and torsionfree modules

Auslander and Bridger [Stable module theory, Memoirs of the American Mathematical Society, No. 94, American Mathematical Society, Providence, R.I., 1969] introduced the notions of n n -spherical modules and n n -torsionfree modules. In this paper, we construct an equivalence between the stable category of n n -spherical modules and the category of modules of grade at least n n , and provide its Gorenstein analogue. As an application, we prove that if R R is a Gorenstein local ring of Krull dimension d > 0 d>0 , then there exists a stable equivalence between the category of ( d − 1 ) (d-1) -torsionfree R R -modules and the category of d d -spherical modules relative to the local cohomology functor.

Gruson–Serganova character formulas and the Duflo–Serganova cohomology functor

Abstract We establish an explicit formula for the character of an irreducible finite-dimensional representation of g ⁢ l ⁢ ( m | n ) \mathfrak{gl}(m|n) . The formula is a finite sum with integer coefficients in terms of a basis E μ \mathcal{E}_{\mu} (Euler characters) of the character ring. We prove a simple formula for the behavior of the “superversion” of E μ \mathcal{E}_{\mu} in the g ⁢ l ⁢ ( m | n ) \mathfrak{gl}(m|n) and o ⁢ s ⁢ p ⁢ ( m | 2 ⁢ n ) \mathfrak{osp}(m|2n) -case under the map ds \mathrm{ds} on the supercharacter ring induced by the Duflo–Serganova cohomology functor DS \mathrm{DS} . As an application, we get combinatorial formulas for superdimensions, dimensions and g 0 \mathfrak{g}_{0} -decompositions for g ⁢ l ⁢ ( m | n ) \mathfrak{gl}(m|n) and o ⁢ s ⁢ p ⁢ ( m | 2 ⁢ n ) \mathfrak{osp}(m|2n) .

Local cohomology for Gorenstein homologically smooth DG algebras

In this paper, we introduce the theory of local cohomology and local duality to Notherian connected cochain DG algebras. We show that the notion of the local cohomology functor can be used to detect the Gorensteinness of a homologically smooth DG algebra. For any Gorenstein homologically smooth locally finite DG algebra $${\cal A}$$ , we define a group homomorphism $${\rm{Hdet}}:{\rm{Au}}{{\rm{t}}_{dg}}\left( {\cal A} \right) \to {k^ \times }$$ , called the homological determinant. As applications, we present a sufficient condition for the invariant DG subalgebra $${{\cal A}^G}$$ to be Gorenstein, where $${\cal A}$$ is a homologically smooth DG algebra such that $$H\left( {\cal A} \right)$$ is a Noetherian AS-Gorenstein graded algebra and G is a finite subgroup of $${\rm{Au}}{{\rm{t}}_{dg}}\left( {\cal A} \right)$$ . Especially, we can apply this result to DG down-up algebras and non-trivial DG free algebras generated in two degree-one elements.

Dimension of complexes related to special Gorenstein projective precovers(II)

Let R be a ring and X a complex of R-modules. We give some characterizations of the dimension of X related to special Gorenstein projective precovers. Let R be a ring such that forms a cotorsion pair cogenerated by a set, where denotes the category consisting of all Gorenstein projective R-modules. Then the dimension related to special Gorenstein projective precovers is equal to the infimum of the set there exists a complete Gorenstein projective resolution of X such that σi is bijective for each Forthmore, if is finite and n an integer then if and only if for any i > n and any projective R-module Q, where is the relative cohomology functor.

Correspondences of Categories for Subregular $${{\mathcal {W}}}$$-Algebras and Principal $${\mathcal {W}}$$-Superalgebras

Based on the Kazama-Suzuki type coset construction and its inverse coset between the subregular $\mathcal{W}$-algebras for $\mathfrak{sl}_n$ and the principal $\mathcal{W}$-superalgebras for $\mathfrak{sl}_{1|n}$, we prove weight-wise linear equivalences of their representation categories. Our main results are then improvements of these correspondences incorporating the monoidal structures. Firstly, in the rational case, we obtain the classification of simple modules and their fusion rules via simple current extensions from their Heisenberg cosets. Secondly, beyond the rational case, we use certain kernel VOAs together with relative semi-infinite cohomology functors to get functors from categories of modules for the subregular $\mathcal{W}$-algebras for $\mathfrak{sl}_n$ to categories of modules for the principal $\mathcal{W}$-superalgebras for $\mathfrak{sl}_{1|n}$ and vice versa. We study these functors and in particular prove isomorphisms between the superspaces of logarithmic intertwining operators. As a corollary, we obtain correspondences of representation categories in the monoidal sense beyond the rational case as well.

Equivariant uniform Alexander-Spanier cohomology theory

In this paper we construct an equivariant cohomology functor with a contravariant coefficient system from the category of pair of uniform G spaces and G maps to the category of abelian groups, where G is a finite group. We call this Equivariant Uniform Alexander-Spanier Cohomology functor as our construction generalizes the construction of Uniform Alexander-Spanier Cohomology functor [1] on uniform spaces. We show that on the category of pair of precompact uniform G spaces, this cohomology functor satisfies all the Eilenberg-Steenrod axioms for an equivariant cohomology theory including the dimension axiom.

On weight complexes, pure functors, and detecting weights

This paper is dedicated to the study of weight complex functors (defined on triangulated categories endowed with weight structures) and their applications. We introduce pure (co)homological functors that “ignore all non-zero weights”; these have a nice description in terms of weight complexes.An important example is the weight structure wG generated by the orbit category in the G-equivariant stable homotopy category SH(G); the corresponding pure cohomological functors into abelian groups are the Bredon cohomology associated to Mackey functors ones. Pure functors related to “motivic” weight structures are also quite useful.Our results also give some (more) new weight structures. Moreover, we prove that certain exact functors are conservative and “detect weights”.

Representability and autoequivalence groups

AbstractFor a finite dimensional algebra A, the bounded homotopy category of projective A-modules and the bounded derived category of A-modules are dual to each other via certain categories of locally-finite cohomological functors. We prove that the duality gives rise to a 2-categorical duality between certain strict 2-categories involving bounded homotopy categories and bounded derived categories, respectively. We apply the 2-categorical duality to the study of triangle autoequivalence groups.

Local cohomology and local homology complexes with respect to a pair of ideals

We introduce local cohomology complexes and local homology complexes with respect to a pair of ideals $(I,J)$ of a commutative noetherian ring $R$, characterize them using Čech complexes and establish "MGM" equivalence between the local cohomology functor $\mathrm{R}\Gamma_{I,J}$ and local homology functor $\mathrm{L}\Lambda_{I,J}$ on $\mathrm{D}(R)$.

Relative cohomology of complexes II: Vanishing of relative cohomology

Let R be a ring such that (GP,GP⊥) forms a cotorsion pair cogenerated by a set, where GP denotes the category of all Gorenstein projective R-modules. Recently, the first author defined for any complex X the relative cohomology functors ExtGP⁎(X,−) as H−⁎(Hom(G,−)) in which G is a special Gorenstein projective precover of X. In the present paper, we introduce the dimension of X related to Gorenstein projective precovers, and show that such a dimension of X is equal to the least integer n for which ExtGPi(X,Q)=0 for all i>n and all R-modules Q∈GP⊥. This result gives a “Gorenstein” version of the relationship between the projective dimension of complexes introduced by Avramov and Foxby and the absolutely cohomology functors ExtR⁎(−,−).

On Weakly Negative Subcategories, Weight Structures, and (Weakly) Approximable Triangulated Categories

In this note we prove that certain triangulated categories are (weakly) approximable in the sense of A. Neeman. We prove that a triangulated category $$\underline{C}$$ that is compactly generated by a single object $$G$$ is weakly approximable if $$\underline{C}(G,G[i])=\{0\}$$ for $$i>1$$ (we say that $$G$$ is weakly negative if this assumption is fulfilled; the case where the equality $$\underline{C}(G,G[1])=\{0\}$$ is fulfilled as well was mentioned by Neeman himself). Moreover, if $$G\cong\bigoplus_{0\leq i\leq n}G_{i}$$ and $$\underline{C}(G_{i},G_{j}[1])=\{0\}$$ whenever $$i\leq j$$ then $$\underline{C}$$ is also approximable. The latter result can be useful since (under a few more additional assumptions) it allows to characterize a certain explicit subcategory of $$\underline{C}$$ as the category of finite cohomological functors from the subcategory $$\underline{C}^{c}$$ of compact objects of $$\underline{C}$$ into $$R$$ -modules (for a noetherian commutative ring $$R$$ such that $$\underline{C}$$ is $$R$$ -linear). One may apply this statement to the construction of certain adjoint functors and $$t$$ -structures. Our proof of (weak) approximability of $$\underline{C}$$ under the aforementioned assumptions is closely related to (weight decompositions for) certain (weak) weight structures, and we discuss this relationship in detail

Cohomology Operations andπ-Strongly Homotopy Commutative Hopf Algebra

Steenrod operations are cohomology operations that are themselves natural transformations between cohomology functors. There are two distinct types of steenrod operations initially constructed by Norman Steenrod and called Steenrod squares and reduced p-th power operations usually denoted <i>Sq</i> and <i>pⁱ</i> respectively. Since their creation, it has been proved that these operations can be constructed in the cohomology of many algebraic structures, for instance in the cohomology of simplicial restricted Lie algebras, the cohomology of cocommutative Hopf algebras and the homology of infinite loop space. Later on J. P. May developped a general algebraic setting in which all the above cases can be studied. In this work we consider a cyclic group π of oder a fixed prime p and combine theπ-strongly homotopy commutative Hopf algebra structure to the May’s approach with the aim to build these natural transformations on the Hochschild cohomology groups. Moreover we give under some conditions a link of these natural transformations with the Gerstenhaber algebra structure.

Some splitting theorems for local cohomology modules with respect to a pair of ideals

Suppose that [Formula: see text] is a commutative ring with identity, [Formula: see text], [Formula: see text] are ideals of [Formula: see text], and let [Formula: see text] be a finitely generated [Formula: see text]-module. Let [Formula: see text] be the [Formula: see text]th local cohomology functor with respect to [Formula: see text]. In this paper, for fixed integers [Formula: see text] and [Formula: see text], we study the existence of the following isomorphisms of local cohomology modules: (i) [Formula: see text]; (ii) [Formula: see text]; and, (iii) [Formula: see text] for some filter regular element [Formula: see text] on [Formula: see text]. Moreover, we provide some applications of the above isomorphisms.

Tensor product of motives via Künneth formula

Following Nori's original idea we here provide certain motivic categories with a canonical tensor structure. These motivic categories are associated to a cohomological functor on a suitable base category and the tensor structure is induced by the cartesian tensor structure on the base category via a cohomological Künneth formula.

Local cohomology in Grothendieck categories

Let [Formula: see text] be a locally noetherian Grothendieck category. In this paper, we define and study the section functor on [Formula: see text] with respect to an open subset of [Formula: see text]. Next, we define and study local cohomology theory in [Formula: see text] in terms of the section functors. Finally, we study abstract local cohomology functor on the derived category [Formula: see text].

Hochschild cohomology of algebras arising from categories and from bounded quivers

The main objective of this paper is to provide a theory for computing the Hochschild cohomology of algebras arising from a linear category with finitely many objects and zero compositions. For this purpose, we consider such a category using an ad hoc quiver Q , with an algebra associated to each vertex and a bimodule to each arrow. The computation relies on cohomological functors that we introduce, and on the combinatorics of the quiver. One point extensions are occurrences of this situation, and Happel’s long exact sequence is a particular case of the long exact sequence of cohomology that we obtain via the study of trajectories of the quiver. We introduce cohomology along paths, and we compute it under suitable Tor vanishing hypotheses. The cup product on Hochschild cohomology enables us to describe the connecting homomorphism of the long exact sequence. Algebras arising from a linear category where the quiver is the round trip one, provide square matrix algebras which have two algebras on the diagonal and two bimodules on the corners. If the bimodules are projective, we show that five-terms exact sequences arise. If the bimodules are free of rank one, we provide a complete computation of the Hochschild cohomology. On the other hand, if the corner bimodules are projective without producing new cycles, Hochschild cohomology in large enough degrees is that of the product of the algebras on the diagonal. As a by-product, we obtain some families of bound quiver algebras which are of infinite global dimension, and have Hochschild cohomology zero in large enough degrees.

Hearts and towers in stable infty -categories

We exploit the equivalence between t-structures and normal torsion theories on a stable infty -category to show how a few classical topics in the theory of triangulated categories, i.e., the characterization of bounded t-structures in terms of their hearts, their associated cohomology functors, semiorthogonal decompositions, and the theory of tiltings, as well as the more recent notion of Bridgeland’s slicings, are all particular instances of a single construction, namely, the tower of a morphism associated with a J-slicing of a stable infty -category , where J is a totally ordered set equipped with a monotone mathbb {Z}-action.