Let $(A,\mathfrak{m})$ be a Gorenstein local ring and let $M$, $N$ two Cohen-Macaulay $A$-modules with $M$ linked to via ideal $\mathfrak{q}$. $L$ another finitely generated $A$-module. We show that $\mathrm{Ext}^i_A(L,M) = 0 $ for all $i \gg 0$ if only $\mathrm{Tor}^A_i(L,N) 0$. If $D$ is module then we $\mathrm{Ext}^i_A(M, D) $\mathrm{Ext}^i_A(D^\dagger , N) 0$, where $D^\dagger \mathrm{Ext}^...

Let R be a local complete intersection and M,N are R-modules such that l(Tor i (M,N)) < ∞ for i ≫ 0. Imitating an approach by Avramov and Buchweitz, we investigate the asymptotic behavior of l(Tor i (M,N)) using Eisenbud operators and show that they have well-behaved growth. We define and study a function η(M,N) which generalizes Serre’s intersection multiplicity χ(M,N) over regular local rings...

In this paper, we introduce a new homological invariant called quasi-projective dimension, which is generalization of projective dimension. We discuss various properties Among other things, prove the following. (1) Over quotient regular local ring by sequence, every finitely generated module has finite (2) The Auslander--Buchsbaum formula and depth for modules dimension remain valid (3) Several...

We develop some aspects of the homological algebra persistence modules, in both one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. consider module sheaf tensor product Hom bifunctors well their derived functors, Tor Ext, give explicit computations for interval a classification injective, projective, flat state Kunneth theo...

1. Basic homological algebra 1 1.1. Chain complexes 2 1.2. Maps and homotopies of maps of chain complexes 2 1.3. Tensor products of chain complexes 3 1.4. Short and long exact sequences 3 1.5. Dual cochain complexes and Hom complexes 4 1.6. Relations between ⊗ and Hom 4 2. The universal coefficient and Künneth theorems 5 2.1. Universal coefficients in homology 5 2.2. The Künneth theorem 6 2.3. ...

Let $R$ be a commutative Noetherian ring with non-zero identity, $fa$ an ideal of $R$, and $X$ an $R$--module. Here, for fixed integers $s, t$ and a finite $fa$--torsion $R$--module $N$, we first study the membership of $Ext^{s+t}_{R}(N, X)$ and $Ext^{s}_{R}(N, H^{t}_{fa}(X))$ in the Serre subcategories of the category of $R$--modules. Then, we present some conditions which ensure the exi...

This is a course in algebraic topology for anyone who has seen the fundamental group and homology. Traditionally (20 years ago), the syllabus included cohomology, the universal coefficient theorem, products, Tor and Ext and duality. We will cover all those topics, while also building a modern framework of derived categories and derived functors using the Cartan–Eilenberg method. We will apply t...