Let X be a projective scheme; let M and N be two coherent OX modules. Given an integer m, we present an algorithm for computing the global extension module Ext(X;M,N ). In particular, this allows one to calculate the sheaf cohomology H(X,N ) and to construct the sheaf corresponding to an element of the module Ext(X;M,N ). This algorithm can be implemented using only the computation of Gröbner b...

Let S = K[X 1 ,. .. , X n ] be the polynomial ring over a field K. For bounded below Z n-graded S-modules M and N we show that if Tor S p (M, N) = 0, then for 0 ≤ i ≤ p, the dimension of the K-vector space Tor S i (M, N) is at least p i. In particular, we get lower bounds for the total Betti numbers. These results are related to a conjecture of Buchsbaum and Eisenbud.

Let R be a commutative Noetherian ring, a an ideal of R, M and N be two finitely generated R-modules. Let t be a positive integer. We prove that if R is local with maximal ideal m and M ⊗R N is of finite length then H t m (M, N) is of finite length for all t ≥ 0 and lR(H t m (M, N)) ≤ ∑t i=0 lR(Ext i R (M, H m (N))). This yields, lR(H t m (M, N)) = lR(Ext t R(M, N)). Additionally, we show that ...

A commutative Noetherian ring R is said to be Tor-persistent if, for any finitely generated R-module M, the vanishing of ToriR(M,M) i?0 implies M has finite projective dimension. An open question Avramov, et al. asks whether such Tor-persistent. In this work, we exploit properties exterior powers modules and complexes provide several partial answers question; in particular, show that every loca...

Let $(R,underline{m})$ be a commutative Noetherian local ring and $M$ be a non-zero finitely generated $R$-module. We show that if $R$ is almost Cohen-Macaulay and $M$ is perfect with finite projective dimension, then $M$ is an almost Cohen-Macaulay module. Also, we give some necessary and sufficient condition on $M$ to be an almost Cohen-Macaulay module, by using $Ext$ functors.

Let $R=\Bbbk[x_1,\dots,x_n]$ be a polynomial ring over field $\Bbbk$ and let $I\subset R$ monomial ideal preserved by the natural action of symmetric group $\mathfrak S_n$ on $R$. We give combinatorial method to determine S_n$-module structure $\mathrm{Tor}_i(I,\Bbbk)$. Our formula shows that $\mathrm{Tor}_i(I,\Bbbk)$ is built from induced representations tensor products Specht modules associat...

Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed space. We show that the category of such representations is an abelian category with enough injectives by constructing an explicit injective resolution. Usin...