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### Localized homology

Afra Zomorodian, Gunnar Carlsson
2008 Computational geometry
In this paper, we provide the theoretical foundation and an effective algorithm for localizing topological attributes such as tunnels and voids.  ...  Unlike previous work that focused on 2-manifolds with restricted geometry, our theory is general and localizes arbitrary-dimensional attributes in arbitrary spaces.  ...  The U -localized homology of X is the image of ι * . We say that a homology class in the image is U -local.  ...

### Formal local homology [article]

Tran Tuan Nam
2016 arXiv   pre-print
The dual theorem and the non-vanishing theorem of formal local homology modules will be shown. We also give some conditions for formal local homology modules being finitely generated or artinian.  ...  We introduce a concept of formal local homology modules which is in some sense dual to P. Schenzel's concept of formal local cohomology modules.  ...  The last section is devoted to study the finiteness of formal local homology modules.  ...

### Formal local homology

Tran Tuan Nam, Do Ngoc Yen, Nguyen Minh Tri
2020 Colloquium Mathematicum
We introduce a concept of formal local homology modules which is in some sense dual to P. Schenzel's concept of formal local cohomology modules.  ...  The duality theorem and the non-vanishing theorem for formal local homology modules are shown.  ...  The finiteness of formal local homology modules.  ...

Kanishka Perera
1998 Abstract and Applied Analysis
We generalize the notion of local linking to include certain cases where the functional does not have a local splitting near the origin. Applications to second-order Hamiltonian systems are given.  ...  An Example As an example of homological local linking, we prove the following proposition: Proposition 2.1 Assume that 0 is an isolated critical point of F and F (u) = P (u) + o( u s ) at u = 0 where P  ...  Then F has a local (q, β)-linking near the origin.  ...

### Local Homology and Local Cohomology [article]

Zhongming Tang
2004 arXiv   pre-print
Let (R, m) be a local ring, I a proper ideal of R and M a finitely generated R-module of dimension d. We discuss the local homology modules of H^d_I(M).  ...  Local Homology of Top Local Cohomology Modules Let (A, n) be a commutative quasi-local ring and X an Artinian A-module.  ...  . , x r ) in n and an integer i, the i-th local homology module H i−1 M ).  ...

### Local Homology, Koszul Homology and Serre Classes [article]

Kamran Divaani-Aazar, Hossein Faridian, Massoud Tousi
2018 arXiv   pre-print
Given a Serre class S of modules, we compare the containment of the Koszul homology, Ext modules, Tor modules, local homology, and local cohomology in S up to a given bound s ≥ 0.  ...  As some applications, we give a full characterization of noetherian local homology modules.  ...  However, the theory of local homology has not been addressed as it merits and numerous counterparts of the results in local cohomology are yet to be explored and unearthed in local homology.  ...

### Extended Local Cohomology and Local Homology [article]

Sean Sather-Wagstaff, Richard Wicklein
2016 arXiv   pre-print
We present an in-depth exploration of the module structures of local (co)homology modules (moreover, for complexes) over the completion R^ a of a commutative noetherian ring R with respect to a proper  ...  We apply this to the study of two recent versions of homological finiteness for complexes, and to certain isomorphisms, with a view toward further applications.  ...  More generally, given an R-complex X, the derived local cohomology complex RΓ a (X) and the derived local homology complex LΛ a (X) are naturally complexes over R a .  ...

### Homology over local homomorphisms [article]

Luchezar L. Avramov, Srikanth Iyengar, Claudia Miller
2004 arXiv   pre-print
The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism f: R --> S.  ...  Endomorphisms Let φ be a local endomorphism, that is, a local homomorphism φ : (R, m, k) → (R, m, k) and let N be a homologically finite complex of R-modules. 12.1. Contractions.  ...  Namely, for each prime ideal q of S the complex N q is homologically finite over the induced local homomorphism ϕ q : R p → S q , where p = q ∩ R.  ...

### Homology over local homomorphisms

L. L. (Luchezar L.) Avramov, Srikanth Iyengar, Claudia Miller
2006 American Journal of Mathematics
The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism ϕ : R → S.  ...  Let Q be a local ring such that N has a structure of homologically finite complex of Q-modules.  ...  Endomorphisms Let φ be a local endomorphism, that is, a local homomorphism φ : (R, m, k) → (R, m, k) and let N be a homologically finite complex of R-modules. 12.1. Contractions.  ...

### Local-global intersection homology [article]

Jonathan Fine
1997 arXiv   pre-print
This paper defines new intersection homology groups. The basic idea is this. Ordinary homology is locally trivial. Intersection homology is not. It may have significant local cycles.  ...  This leads to the topological definition of local-global intersection homology, and a formula for the expected Betti numbers of toric varieties. Various related questions are also discussed.  ...  Ordinary homology is locally trivial. Intersection homology is not. It may have significant local cycles.  ...

### Localization in Khovanov homology [article]

Matthew Stoffregen, Melissa Zhang
2018 arXiv   pre-print
By identifying the fixed-point sets, we obtain rank inequalities for odd and even Khovanov homologies, and their annular filtrations, for prime-periodic links in S^3.  ...  We can also describe the maps F Ann (φ op v,u ) in local pictures.  ...  That is, locally the 0-resolution of (respectively, ) is (respectively, ) and the 1-resolution is (respectively, ).  ...

### Localization in Hochschild homology [article]

Markus J. Pflaum
2022 arXiv   pre-print
Localization methods are ubiquitous in cyclic homology theory, but vary in detail and are used in different scenarios.  ...  We will then indicate how this method can be used to determine the Hochschild homology theory of more complicated algebras out of simpler ones.  ...  Localization in cyclic homology.  ...

### Topological Cyclic Homology of Local Fields [article]

Ruochuan Liu, Guozhen Wang
2022 arXiv   pre-print
We introduce a new approach to determining the structure of topological cyclic homology by means of a descent spectral sequence.  ...  We carry out the computation for a p-adic local field with Fp-coefficients, including the case p=2 which was only covered by motivic methods except in the totally unramified case.  ...  Especially, he proposed to us using the descent along the base change map (1.1) in topological Hochschild homology.  ...

### Local contact homology and applications [article]

Umberto Hryniewicz, Leonardo Macarini
2013 arXiv   pre-print
We introduce a local version of contact homology for an isolated periodic orbit of the Reeb flow and prove that its rank is uniformly bounded for isolated iterations.  ...  Invariance of local contact homology. Lemma 4.12.  ...  The local contact homology of HC(H, γ) is defined as the homology of the complex (C * (H , K, γ), d).  ...

### Localization for coarse homology theories [article]

Ulrich Bunke, Luigi Caputi
2019 arXiv   pre-print
We show that such a Bredon-style equivariant coarse homology theory satisfies localization theorems and that a general equivariant coarse homology theory can be approximated by a Bredon-style version.  ...  We discuss the special case of algebraic and topological equivariant coarse K-homology and obtain the coarse analog of Segal's localization theorem.  ...  The localization theorems for the equivariant coarse algebraic K-homology theory KMX G implies a localization theorem for the equivariant homology theory KM G := KMX •O ∞ hlg which is similar to Corollary  ...
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