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Hyperoctahedral homology is the homology theory associated to the hyperoctahedral crossed simplicial group. It is defined for involutive algebras over a commutative ring using functor homology and the hyperoctahedral bar construction of Fiedorowicz. The main result of the paper proves that hyper- octahedral homology is related to equivariant stable homotopy theory: for a discrete group of odd order, the hyperoctahedral homology of the group algebra is isomorphic to the homology of the fixed points under the involution of an equivariant infinite loop space built from the classifying space of the group. [ABSTRACT FROM AUTHOR]

Let ℓ be a commutative ring with unit. Garkusha constructed a functor from the category of ℓ -algebras into a triangulated category D , that is a universal excisive and homotopy invariant homology theory. Later on, he provided different descriptions of D , as an application of his motivic homotopy theory of algebras. Using these, it can be shown that D is triangulated equivalent to a category, denote it by K , whose objects are pairs (A , m) with A an ℓ -algebra and m an integer, and whose Hom-sets can be described in terms of homotopy classes of morphisms. All these computations, however, require a heavy machinery of homotopy theory. In this paper, we give a more explicit construction of the triangulated category K and prove its universal property, avoiding the homotopy-theoretic methods and using instead the ones developed by Cortiñas-Thom for defining kk -theory. Moreover, we give a new description of the composition law in K , mimicking the one in the suspension-stable homotopy category of bornological algebras defined by Cuntz-Meyer-Rosenberg. We also prove that the triangulated structure in K can be defined using either extension or mapping path triangles. [ABSTRACT FROM AUTHOR]

In this paper we develop an axiomatic setup for algorithmic homological algebra of Abelian categories. This is done by exhibiting all existential quantifiers entering the definition of an Abelian category, which for the sake of computability need to be turned into constructive ones. We do this explicitly for the often-studied example Abelian category of finitely presented modules over a so-called computable ring R, i.e. a ring with an explicit algorithm to solve one-sided (in)homogeneous linear systems over R. For a finitely generated maximal ideal 픪 in a commutative ring R, we show how solving (in)homogeneous linear systems over R픪 can be reduced to solving associated systems over R. Hence, the computability of R implies that of R픪. As a corollary, we obtain the computability of the category of finitely presented R픪-modules as an Abelian category, without the need of a Mora-like algorithm. The reduction also yields, as a byproduct, a complexity estimation for the ideal membership problem over local polynomial rings. Finally, in the case of localized polynomial rings, we demonstrate the computational advantage of our homologically motivated alternative approach in comparison to an existing implementation of Mora's algorithm. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring, 픞 be an ideal of R and M be an arbitrary R-module. In this paper, among other things, we show that if, for a non-negative integer t, the R-module $\textrm{Ext}^t_R(R/{\mathfrak a}, M)$ is weakly Laskerian and $H^i_{\mathfrak a}(M)$ is 픞-weakly cofinite for all i < t, then, for any weakly Laskerian submodule U of $H^t_{\mathfrak a}(M)$, the R-module $\textrm{Hom}_R(R/{\mathfrak a}, H^t_\mathfrak{a}(M)/U)$ is weakly Laskerian. As a consequence the set of associated primes of $H^t_\mathfrak{a}(M)/U$ is finite. [ABSTRACT FROM AUTHOR]

Studies the structure of generalized co-Cohen-Macaulay and co-Buchsbaum modules over commutative rings. Use of q-weak sequence and co-standard system of parameters; Proof theory; Localization method; Local homology modules..

It is well-known that the ith local cohomology of a finitely generated R-module M over a positively graded commutative Noetherian ring R, associated to the irrelevant ideal R+ is graded. Furthermore, for every integer n, the nth component HR+i(M)n of this local cohomology module HR+i(M) is finitely generated over R0 and vanishes for n≫0.In this paper, we want to understand the behavior of HR+i(M)n for n≪0 in the case where R is a Cohen–Macaulay ring and M is a Cohen–Macaulay R-module. When dim R0=1, we will show that AssR0(HR+i(M)n) becomes constant when n becomes negatively large. When R0 is local, and dim R0=2, we will show that there exists an integer N such that either HR+i(M)n=(0) for all n or, HR+i(M)n≠(0) for all n. [Copyright &y& Elsevier]

In this article, we compute the Hochschild homology group ofA = KΓ/(f(Xs)), whereKΓ is the path algebra of the cyclic quiver Γ withsvertices andsarrows over a commutative ringK,f(x) is a monic polynomial overK, andXis the sum of all arrows inKΓ. Moreover, we compute the cyclic homology group ofAin the casef(x) = (x − a)m, wherea ∈ K, so that we can determine the cyclic homology ofAin general whenKis an algebraically closed field. [ABSTRACT FROM PUBLISHER]

We generalize the homological characterization of sequentially Cohen–Macaulay modules over a graded Gorenstein algebra to sequentially reflexive modules over Noetherian, not necessarily commutative rings, with a N-partial cotilting bimodule playing the role of the graded Gorenstein algebra. In such a way we get a complete version of the“Cotilting Theorem”. Finally, conditions are found to insure that the“N-partial cotilting notion” pass through a finite ring extension. [ABSTRACT FROM AUTHOR]