Tate Homology Research Articles

Mini-Workshop: Flavors of Rabinowitz Floer and Tate Homology

Rabinowitz Floer homology originated 15 years ago in symplectic geometry. Recent developments have related it to algebraic topology via string toplogy and Tate homology, and to mirror symmetry via Fukaya categories. This mini-workshop brought together researchers from these different communities, in order to foster exchange and collaborations across research fields.

Tate homology and powered flybys

Tate homology and powered flybys

A Study of Tate Homology via the Approximation Theory with Applications to the Depth Formula

A Study of Tate Homology via the Approximation Theory with Applications to the Depth Formula

Homological independence of infinite syzygies

We consider modules that appear as syzygies of acyclic complexes of flat modules and examine a certain condition on pairs of such modules, that generalizes the vanishing of Tate homology. If this condition is satisfied for two modules over a commutative ring , then (a) the tensor product of the two modules is also a syzygy of an acyclic complex of flat modules and (b) the tensor product of the corresponding acyclic complexes of flat modules is an acyclic complex of flat modules as well, whose syzygies can be expressed in terms of the syzygies of the factor complexes. We also examine the analogous (dual) case of homomorphism groups of infinite syzygies.

Tate (Co)homology of invariant group chains

Let [Formula: see text] be a finite group acting on a group [Formula: see text] as a group automorphisms, [Formula: see text] the bar complex, [Formula: see text] the homology of invariant group chains and [Formula: see text] the cohomology invariant, both defined in Knudson’s paper “The homology of invariant group chains”. In this paper, we define the Tate homology of invariants [Formula: see text] and the Tate cohomology of invariants [Formula: see text]. When the coefficient [Formula: see text] is the abelian group of the integers, we proved that these groups are isomorphics, [Formula: see text]. Further, we prove that the homology and cohomology of invariant group chains are duals, [Formula: see text], [Formula: see text].

Remarks on Balance for Tate and Generalized Tate (Co)homology

Based on the analysis of Tate and generalized Tate cohomology in abelian categories, in this note, we consider the balance for Tate and generalized Tate cohomology. Some existing results are generalized and some applications in the category of modules are given. We also consider the balance for Tate and generalized Tate homology in broader sense.

Injective stabilization of additive functors, I. Preliminaries

This paper is the first one in a series of three dealing with the concept of injective stabilization of the tensor product and its applications. Its primary goal is to collect known facts and establish a basic operational calculus that will be used in the subsequent parts. This is done in greater generality than is necessary for the stated goal. Several results of independent interest are also established. They include, among other things, connections with satellites, an explicit construction of the stabilization of a finitely presented functor, various exactness properties of the injectively stable functors, a construction, from a functor and a short exact sequence, of a doubly-infinite exact sequence by splicing the injective stabilization of the functor and its derived functors. When specialized to the tensor product with a finitely presented module, the injective stabilization with coefficients in the ring is isomorphic to the 1-torsion functor. The Auslander-Reiten formula is extended to a more general formula, which holds for arbitrary (i.e., not necessarily finite) modules over arbitrary associative rings with identity. Weakening of the assumptions in the theorems of Eilenberg and Watts leads to characterizations of the requisite zeroth derived functors.The subsequent papers, provide applications of the developed techniques. Part II deals with new notions of torsion module and cotorsion module of a module. This is done for arbitrary modules over arbitrary rings. Part III introduces a new concept, called the asymptotic stabilization of the tensor product. The result is closely related to different variants of stable homology (these are generalizations of Tate homology to arbitrary rings). A comparison transformation from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be epic.

Tate Homology of Modules Based on Tate $$\mathcal {F}_C$$ F C -Resolutions

Let R be a commutative Noetherian ring and C a semidualizing module. Based on a kind of Tate \(\mathcal {F}_C\)-resolutions of modules constructed by Hu et al., in this paper, we introduce and investigate Tate homology for R-modules. For modules admitting a Tate \(\mathcal {F}_C\)-resolution, we first connect the Tate homology and corresponding relative homologies via a long exact sequence. Then we show that the vanishing of such Tate homology characterizes the finiteness of \(\mathcal {F}_C\)-projective dimension of modules. Finally, if in addition R is a Cohen–Macaulay ring with a dualizing module, we establish a balance result for such Tate homology.

Tate homology with respect to cotorsion pairs

ABSTRACTGiven two complete hereditary cotorsion pairs (𝒬,ℛ) and in the category of modules which satisfy the conditions 𝒬′⊆𝒬, , and ℛ′ is enveloping, a kind of Tate homology of modules is introduced and investigated in this article based on complete -resolutions. It is shown that a module admits complete -resolutions precisely when it has finite 𝒬-projective dimension. In particular, an Avramov–Martsinkovsky type exact sequence, which is natural in both variables, is constructed to connect such Tate homology functors and relative homology functors. Applications given for the subcategory 𝒢ℱ of Gorenstein flat modules go in three directions. The first is to improve the Avramov–Martsinkovsky type exact sequence appeared in Liang’s work [Tate homology of modules of finite Gorenstein flat dimension. Algebr. Represent. Theory 16:1541–1560 (2013)] by showing that it is indeed natural in both variables. The second is to characterize the finiteness of Gorenstein flat (resp., flat) dimension by using vanishing of Tate homology functors . Finally, by virtue of a conclusion provided by Enochs et al., some balance results are established for .

Complete homology over associative rings

We compare two generalizations of Tate homology to the realm of associative rings: stable homology and the J-completion of Tor, also known as complete homology. For finitely generated modules, we show that the two theories agree over Artin algebras and over commutative noetherian rings that are Gorenstein, or local and complete.

Avramov–Martsinkovsky type exact sequences with tor functors

For two classes of right R-modules W, X such that P ⊂ W ⊂ X, where P is the class of projective right R-modules, we show that there is an Avramov–Martsinkovsky type exact sequence with generalized Tate homology functor \({\widehat {Tor}^{X.W}}\), relative homology functors Tor W and Tor X . Many results in Iacob [Comm. Algebra, 35, 1589–1606 (2007)] and Liang [Algebr. Represent. Theory, 16, 1541–1560 (2013)] are generalized and improved.

Stable homology over associative rings

We analyze stable homology over associative rings and obtain results over Artin algebras and commutative noetherian rings. Our study develops similarly for these classes; for simplicity we only discuss the latter here. Stable homology is a broad generalization of Tate homology. Vanishing of stable homology detects classes of rings---among them Gorenstein rings, the original domain of Tate homology. Closely related to gorensteinness of rings is Auslander's G-dimension for modules. We show that vanishing of stable homology detects modules of finite G-dimension. This is the first characterization of such modules in terms of vanishing of (co)homology alone. Stable homology, like absolute homology, Tor, is a theory in two variables. It can be computed from a flat resolution of one module together with an injective resolution of the other. This betrays that stable homology is not balanced in the way Tor is balanced. In fact, we prove that a ring is Gorenstein if and only if stable homology is balanced.

Complete flat resolutions, Tate homology and the depth formula

We introduce Tate homology of complexes of finite Gorenstein flat dimension based on complete flat resolutions and give a new method of computing Tate homology in Christensen and Jorgensen's sense. We also investigate the relationship between Tate homology and Tate cohomology. As an application, a more brief proof of the main result on derived depth formula of [Vanishing of Tate homology and depth formula over local rings, J. Pure Appl. Algebra 219 (2015) 464-481] is given.

Vanishing of relative homology and depth of tensor products

For finitely generated modules M and N over a Gorenstein local ring R, one has depthM+depthN=depth(M⊗RN)+depthR, i.e., the depth formula holds, if M and N are Tor-independent and Tate homology Torˆi(M,N) vanishes for all i∈Z. We establish the same conclusion under weaker hypotheses: if M and N are G-relative Tor-independent, then the vanishing of Torˆi(M,N) for all i⩽0 is enough for the depth formula to hold. We also analyze the depth of tensor products of modules and obtain a necessary condition for the depth formula in terms of G-relative homology.

Vanishing of Tate homology — An application of stable homology for complexes

It has been proved that the vanishing of Tate homology is a sufficient condition for the derived depth formula to hold in [J. Pure Appl. Algebra, 219, 464–481 (2015)]. In this paper, we investigate when Tate homology vanishes by studying the stable homology theory for complexes. Properties such as the balancedness and vanishing of stable homology for complexes are studied. Our results show that the vanishing of this homology can detect finiteness of homological dimensions of complexes and regularness of rings.

Gorenstein FI-Flat Dimension and Tate Homology

In this paper, we are concerned with Tate homology $\widehat {\text {tor}}$ of modules of finite Gorenstein FI-flat dimension. We determine connections between Tate, relative and absolute homology. Furthermore, we discuss Tate homology $\widehat {\text {tor}}$ of a FI-cotorsion modules of finite Gorenstein FI-flat dimension over right coherent rings.

Symplectic Tate homology

For a Liouville domain $W$ satisfying $c_1(W)=0$, we propose in this note two versions of symplectic Tate homology $\underrightarrow{H}\underleftarrow{T}(W)$ and $\underleftarrow{H}\underrightarrow{T}(W)$ which are related by a canonical map $\kappa \colon \underrightarrow{H}\underleftarrow{T}(W) \to \underleftarrow{H}\underrightarrow{T}(W)$. Our geometric approach to Tate homology uses the moduli space of finite energy gradient flow lines of the Rabinowitz action functional for a circle in the complex plane as a classifying space for $S^1$-equivariant Tate homology. For rational coefficients the symplectic Tate homology $\underrightarrow{H}\underleftarrow{T}(W)$ has the fixed point property and is therefore isomorphic to $H(W;\mathbb{Q}) \otimes \mathrm{Q}[u,u^{-1}]$, where $\mathbb{Q}[u,u^{-1}]$ is the ring of Laurent polynomials over the rationals. Using a deep theorem of Goodwillie, we construct examples of Liouville domains where the canonical map $\kappa$ is not surjective and examples where it is not injective.

Gorenstein $$n$$ n -flat dimension and Tate homology

In this paper, we investigate Tate homology \(\widehat{tor}\) of modules of finite Gorenstein \(n\)-flat dimension. In particular, we give a tight connection between Tate homology, relative homology and absolutely homology. We also study the Tate homology \(\widehat{tor}\) of a \(n\)-cotorsion module of finite Gorenstein \(n\)-flat dimension.

Gorenstein X-Flat dimension and Tate homology

In this paper, we study a Gorenstein X-flat dimension and investigate Tate homology Xtor of modules of finite Gorenstein X-flat dimension. We prove that if R is a right GXF-closed ring and M a R-module admitting a Tate X-flat resolution, and if N is a left R-module admitting a Tate X-flat resolution TN, then for any i ∈ ℤ, XtoriR (M, N) ≅ Hi(M ⊗R TN).

Relative and Tate homology 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].