Cousin Complex Research Articles

Cousin complexes via total fractions

With respect to the filtration provided by codimensions of points on a locally Noetherian scheme, we give a new approach to Cousin complexes of sheaves of modules using total fractions. Concrete examples are worked out to illustrate the computational aspect of Cousin complexes.

Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter $Q$-operators, arising from our previous study (arXiv:2001.04929, arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $\mathfrak{g}$-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety $G/P$ stratified by $B_{-}$-orbits.

Cousin complex on the complement to the strict normal-crossing divisor in a local essentially smooth scheme over a field

For any $\mathbb{A}^1$-invariant cohomology theory that satisfies Nisnevich excision on the category of smooth schemes over a field $k$ it is proved that the Cousin complex on the complement $U-D$ to the strict normal-crossing divisor $D$ in a local essentially smooth scheme $U$ is acyclic. This claim is also proved for the schemes $(X-D)\times(\mathbb{A}^1_k-Z_0)\times…\times(\mathbb{A}^1_k-Z_l)$, where $Z_0,…,Z_l$ is a finite family of closed subschemes in the affine line over $k$. Bibliography: 32 titles.

Modules with minimax Cousin cohomologies

Let R be a commutative Noetherian ring with non-zero identity and let X be an arbitrary R-module. In this paper, we show that if all the cohomology modules of the Cousin complex for X are minimax, then the following hold for any prime ideal p of R and for every integer n less than X, the height of p: (i) the nth Bass number of X with respect to p is finite; (ii) the nth local cohomology module of Xp with respect to pRp is Artinian.

Cohomology of line bundles on horospherical varieties

A horospherical variety is a normal algebraic variety where a connected reductive algebraic group acts with an open orbit isomorphic to a torus bundle over a flag variety. In this article we study the cohomology of line bundles on complete horospherical varieties. The main tool in this article is the machinery of Grothendieck–Cousin complexes, and we also prove a Kunneth-like formula for local cohomology.

Hochschild coniveau spectral sequence and the Beilinson residue

We develop the Hochschild analogue of the coniveau spectral sequence and the Gersten complex. Since Hochschild homology does not have devissage or A^1-invariance, this is a little different from the K-theory story. In fact, the rows of our spectral sequence look a lot like the Cousin complexes in Hartshorne's 1966 "Residues & Duality". Note that these are for coherent cohomology. We prove that they agree by an 'HKR isomorphism with supports'. Using the close ties of Hochschild homology to Lie algebra homology, this gives residue maps in Lie homology, which we show to agree with those \`a la Tate-Beilinson.

Extension Functors of Cousin Cohomology Modules

Let R be a commutative Noetherian ring with non-zero identity, $$\mathcal {F}$$ a filtration of $${{\mathrm{Spec}}}(R)$$ which admits an R-module X, and $${{\mathrm{C}}}_R(\mathcal {F},X)$$ the Cousin complex for X with respect to $$\mathcal {F}$$ . In this paper, we first introduce the Cousin functor and the Cousin spectral sequences. Then for non-negative integers s, t and a finite R-module N, we study the membership of R-modules $${{\mathrm{H}}}^{s-1}({{\mathrm{Ext}}}^t_R(N,{{\mathrm{C}}}_R(\mathcal {F},X)))$$ and $${{\mathrm{Ext}}}^{s}_R(N,{{\mathrm{H}}}^{t-1}({{\mathrm{C}}}_R(\mathcal {F},X)))$$ in Serre subcategories of the category of R-modules and find some conditions for validity of an isomorphism between them. Finally, we use these results to present some facts about the vanishing and finiteness of Cousin cohomology modules.

A Polar Complex for Locally Free Sheaves

We construct the so-called polar complex for an arbitrary locally free sheaf on a smooth variety over a field of characteristic zero. This complex is built from logarithmic forms on all irreducible subvarieties with values in a locally free sheaf. We prove that cohomology groups of the polar complex are canonically isomorphic to the cohomology groups of the locally free sheaf. Relations of the polar complex with Rost’s cycle modules, algebraic cycles, Hodge structures, Cousin complex, and adelic complex are discussed.

The Recillas’s Conjecture on Szegö Kernels Associated to Harish-Chandra Modules

The solution of the field equations that involves non-flat differential operators (curved case) can be obtained as the extensions Φ+Szego operators in G/K with G, a non-compact Lie group with K, compact. This could be equivalent in the context of the Harish-Chandra modules category to the obtaining of extensions in certain sense (Cousin complexes of sheaves of differential operators to their classification) of Verma modules as classifying spaces of these differential operators and their corresponding integrals through of geometrical integral transforms.

G-Gorenstein Modules

Let R be a commutative Noetherian ring. In this paper, we study those finitely generated R-modules whose Cousin complexes provide Gorenstein injective resolutions. We call such a module a G-Gorenstein module. Characterizations of G-Gorenstein modules are given and a class of such modules is determined. It is shown that the class of G-Gorenstein modules strictly contains the class of Gorenstein modules. Also, we provide a Gorenstein injective resolution for a balanced big Cohen-Macaulay R-module. Finally, using the notion of a G-Gorenstein module, we obtain characterizations of Gorenstein and regular local rings.

Gorenstein Injective Resolutions, Cousin Complexes and Top Local Cohomology Modules

Let R be a Gorenstein local ring. We show that for a balanced big Cohen–Macaulay module M over R, the Cousin complex [Formula: see text] provides a Gorenstein injective resolution of M. Also, over a d-dimensional Gorenstein local ring R with maximal ideal 𝔪, we show that [Formula: see text], the dth local cohomology module of M with respect to 𝔪, is Gorenstein injective if (a) M is a balanced big Cohen–Macaulay R-module, or (b) M ∈ G(R), where G(R) is the Auslander's G-class of R.

Modules with finite Cousin cohomologies have uniform local cohomological annihilators

Let A be a Noetherian ring. It is shown that any finite A-module M of finite Krull dimension with finite Cousin complex cohomologies has a uniform local cohomological annihilator. The converse is also true for a finite module M satisfying ( S 2 ) which is over a local ring with Cohen–Macaulay formal fibers.

Finiteness of cousin cohomologies

The notion of the Cousin complex of a module was given by Sharp in 1969. It wasn’t known whether its cohomologies are finitely generated until recently. In 2001, Dibaei and Tousi showed that the Cousin cohomologies of a finitely generated A A -module M M are finitely generated if the base ring A A is local, has a dualizing complex, M M satisfies Serre’s ( S 2 ) (S_2) -condition and is equidimensional. In the present article, the author improves their result. He shows that the Cousin cohomologies of M M are finitely generated if A A is universally catenary, all the formal fibers of all the localizations of A A are Cohen-Macaulay, the Cohen-Macaulay locus of each finitely generated A A -algebra is open and all the localizations of M M are equidimensional. As a consequence of this, he gives a necessary and sufficient condition for a Noetherian ring to have an arithmetic Macaulayfication.

Applications of duality theory to Cousin complexes

We use the anti-equivalence between Cohen–Macaulay complexes and coherent sheaves on formal schemes to shed light on some older results and prove new results. We bring out the relations between a coherent sheaf M satisfying an S 2 condition and the lowest cohomology N of its “dual” complex. We show that if a scheme has a Gorenstein complex satisfying certain coherence conditions, then in a finite étale neighborhood of each point, it has a dualizing complex. If the scheme already has a dualizing complex, then we show that the Gorenstein complex must be a tensor product of a dualizing complex and a vector bundle of finite rank. We relate the various results in [P. Sastry, Duality for Cousin complexes, in: Contemp. Math., vol. 375, Amer. Math. Soc., Providence, RI, 2005, pp. 137–192] on Cousin complexes to dual results on coherent sheaves on formal schemes.

A STUDY OF COUSIN COMPLEXES THROUGH THE DUALIZING COMPLEXES#

ABSTRACT For the Cousin complex of certain modules, we investigate the finiteness of cohomology modules, the local duality property, and the injectivity of its terms. The existence of canonical modules of Noetherian non-local rings and the Cousin complexes of them with respect to the height filtration are discussed.

Cohomologie des fibrés en droites sur les compactifications des groupes réductifs

We consider equivariant compactifications of some reductive complex and connected group, G, considered as an homogeneous space for G× G. For a finite covering G ̃ of G, the cohomology groups of line bundles over those compactifications are naturally finite dimensional G ̃ × G ̃ -modules. We determine, in this article, all those G ̃ × G ̃ -modules by calculating their multiplicities along each simple G ̃ × G ̃ -module. The achieved formula is in particular valid for the wonderful compactification of adjoint groups and also generalizes the well known description of the cohomology of line bundles over complete toric varieties. Our method is based upon the Grothendieck–Cousin complex which, if g is the Lie algebra of G, is a g× g -module complex. We analyse those g× g -modules by giving some filtrations in the associated graduate of which, some “generalized Verma modules” occur.

Towards a theory of Bass numbers with application to Gorenstein algebras

The notion of Gorenstein rings in the commutative ring theory is general- ized to that of Noetherian algebras which are not necessarily commutative. We faithfully follow in the steps of the commutative case: Gorenstein algebras will be defined using the notion of Cousin complexes developed by R. Y. Sharp (Sh1). One of the goals of the present paper is the characterization of Gorenstein algebras in terms of Bass numbers. The commutative theory of Bass numbers turns out to carry over with no extra changes. Certain algebras having locally finite global dimension are also characterized. The special case where the algebras are free modules over base rings is explored. Thanks to these observations, it is clarified how the Gorensteinness is inherited under flat base changes. In conclusion, a characterization for local algebras to be Gorenstein is given, account- ing for the reason why the theory behaves so well in the commutative case. Examples are explored and open problems are given. See (GN2) and (GN3) for further develop- ments.

COMPARISON OF MULTIGRADED AND UNGRADED COUSIN COMPLEXES

AbstractThis paper generalizes, in two senses, work of Petzl and Sharp, who showed that, for a $\mathbb{Z}$-graded module $M$ over a $\mathbb{Z}$-graded commutative Noetherian ring $R$, the graded Cousin complex for $M$ introduced by Goto and Watanabe can be regarded as a subcomplex of the ordinary Cousin complex studied by Sharp, and that the resulting quotient complex is always exact. The generalizations considered in this paper are, firstly, to multigraded situations and, secondly, to Cousin complexes with respect to more general filtrations than the basic ones considered by Petzl and Sharp. New arguments are presented to provide a sufficient condition for the exactness of the quotient complex in this generality, as the arguments of Petzl and Sharp will not work for this situation without additional input.AMS 2000 Mathematics subject classification: Primary 13A02; 13E05; 13D25; 13D45

A generalization of the dualizing complex structure and its applications

The structural property of dualizing complex is generalized and its applications are given to show that (1) over a ring possessing a dualizing complex, the conditions of Serre on the canonical module are connected with the local cohomologies of the ring itself; (2) the Cousin complexes of certain modules over a ring possessing a dualizing complex have finitely generated cohomologies; (3) a quotient of a Cohen–Macaulay local ring which satisfies (S2) and admits canonical module, also possesses a dualizing complex.

Minimal injective resolutions of Cohen-Macaulay isolated singularities

Let $(R, \frak {m})$ be a commutative Gorenstein complete local ring with dim R = d and let $\Lambda $ be an R-algebra which is not necessarily commutative but finitely generated as an R-module. In this paper the structure of minimal injective resolutions $E^\bullet _\Lambda (M)$ for the $\Lambda $ -lattices M is explored, in terms of the Cousin complexes $C^{\bullet }_R(M)$ for M and the minimal projective resolutions of the $\Lambda ^{op}$ -modules $M^* = \hbox {Hom}_R(M,R)$ as well, under the assumption that $\Lambda $ is a Cohen-Macaulay isolated singularity. As a consequence we get the following. Assume that R is a regular local ring and let $k \in \Bbb Z$ . Then the ring $\Lambda $ is k-Gorenstein if and only if the ring $\Delta = (R/\frak {m})\otimes _R \Lambda $ is (k - d)-Gorenstein.