Comparison of relative cohomology theories with respect to semidualizing modules

We introduce and investigate in this paper a kind of Tate homology of modules over a commutative coherent ring based on Tate ℱC-resolutions, where C is a semidualizing module. We show firstly that the class of modules admitting a Tate ℱC-resolution is equal to the class of modules of finite 𝒢(ℱC)-projective dimension. Then an Avramov–Martsinkovsky type exact sequence is constructed to connect such Tate homology functors and relative homology functors. Finally, motivated by the idea of Sather–Wagstaff et al. [Comparison of relative cohomology theories with respect to semidualizing modules, Math. Z. 264 (2010) 571–600], we establish a balance result for such Tate homology over a Cohen–Macaulay ring with a dualizing module by using a good conclusion provided in [E. E. Enochs, S. E. Estrada and A. C. Iacob, Balance with unbounded complexes, Bull. London Math. Soc. 44 (2012) 439–442].

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Homological aspects of semidualizing modules

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15121 ◽

2010 ◽

Vol 106 (1) ◽

pp. 5 ◽

Cited By ~ 46

Author(s):

Ryo Takahashi ◽

Diana White

Keyword(s):

Projective Dimension ◽

Dual Theory ◽

Injective Dimension ◽

Relative Cohomology ◽

Semidualizing Module ◽

Finite Projective Dimension ◽

Semidualizing Modules ◽

The Relationship ◽

Cohomology Modules

We investigate the notion of the $C$-projective dimension of a module, where $C$ is a semidualizing module. When $C=R$, this recovers the standard projective dimension. We show that three natural definitions of finite $C$-projective dimension agree, and investigate the relationship between relative cohomology modules and absolute cohomology modules in this setting. Finally, we prove several results that demonstrate the deep connections between modules of finite projective dimension and modules of finite $C$-projective dimension. In parallel, we develop the dual theory for injective dimension and $C$-injective dimension.

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Balance for relative cohomology of complexes

Journal of Algebra and Its Applications ◽

10.1142/s021949882050005x ◽

2019 ◽

Vol 19 (01) ◽

pp. 2050005

Author(s):

Zhenxing Di ◽

Bo Lu ◽

Junxiu Zhao

Keyword(s):

Projective Dimension ◽

Cohomology Groups ◽

Injective Dimension ◽

Projective Generator ◽

Arbitrary Ring ◽

Relative Cohomology ◽

Injective Modules ◽

Injective Cogenerator ◽

Semidualizing Modules

Let [Formula: see text] be an arbitrary ring. We use a strict [Formula: see text]-resolution [Formula: see text] of a complex [Formula: see text] with finite [Formula: see text]-projective dimension, where [Formula: see text] denotes a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits an injective cogenerator [Formula: see text], to define the [Formula: see text]th relative cohomology functor [Formula: see text] as [Formula: see text]. If a complex [Formula: see text] has finite [Formula: see text]-injective dimension, then one can use a dual argument to define a notion of a relative cohomology functor [Formula: see text], where [Formula: see text] is a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits a projective generator. Under several orthogonal conditions, we show that there exists an isomorphism [Formula: see text] of relative cohomology groups for each [Formula: see text]. This result simultaneously encompasses a balance result of Holm on Gorenstein projective and injective modules, a balance result of Sather-Wagstaff, Sharif and White on Gorenstein projective and injective modules with respect to semidualizing modules, and a balance result of Liu on Gorenstein projective and injective complexes. In particular, as an application of this result, we extend the above balance result of Sather-Wagstaff, Sharif and White to the setting of complexes.

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Coarse cohomology with twisted coefficients

Mathematica Slovaca ◽

10.1515/ms-2017-0440 ◽

2020 ◽

Vol 70 (6) ◽

pp. 1413-1444

Author(s):

Elisa Hartmann

Keyword(s):

Sheaf Cohomology ◽

Grothendieck Topology ◽

Coarse Structure ◽

Relative Cohomology ◽

Number Of Ends ◽

Constant Coefficients ◽

Coarse Space ◽

Grothendieck Topologies

AbstractTo a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

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WITTEN DEFORMATION FOR NONCOMPACT MANIFOLDS WITH BOUNDED GEOMETRY

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748021000232 ◽

2021 ◽

pp. 1-38

Author(s):

Xianzhe Dai ◽

Junrong Yan

Keyword(s):

Essential Role ◽

Morse Function ◽

Noncompact Manifold ◽

Morse Inequalities ◽

Bounded Geometry ◽

Noncompact Manifolds ◽

Witten Laplacian ◽

Relative Cohomology ◽

Witten Deformation ◽

Small Eigenvalues

Abstract Motivated by the Landau–Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function f near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of the Thom–Smale complex of f as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large T. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten’s instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application, we obtain the strong Morse inequalities in this setting.

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