We study a question raised by Eisenbud, Mustaţǎ, and Stillman regarding the injectivity of natural maps from Ext modules to local cohomology modules. We obtain some positive answers to this question which extend earlier results of Lyubeznik. In the process, we also prove a vanishing theorem for local cohomology modules which connects theorems previously known in the case of positive characteris...

The category of representations of a finite quiver in the category of sheaves of modules on a ringed space is abelian. We show that this category has enough injectives by constructing an explicit injective resolution. From this resolution we deduce a long exact sequence relating the Ext groups in these two categories. We also show that under some hypotheses, the Ext groups are isomorphic to cer...

This works concerns cohomological support varieties of modules over commutative local rings. The main result is that the a derived tensor product pair differential graded Koszul complex join supports modules. generalizes, and gives another proof of, Dao third author dealing with Tor-independent complete intersection for complexes has broader applicability, including to exterior algebras

The goal of this paper is to provide homological descriptions of monomial ideals. The key concepts in these descriptions are the minimal free resolution, the Koszul homology and the multigraded Betti numbers. These three objects are strongly related, being Tor modules a simple way to describe this relation. We introduce a new tool, Mayer-Vietoris trees which provides a good way to compute the h...

We prove that for modules M and N over a local ring R, the depth formula: depthR M + depthR N − depthR = depthR Tor R s (M,N) − s, where s = sup{i | Tor i (M,N) 6= 0}, holds under certain conditions. This adds to the list cases where the depth formula, which extends the classical Auslander-Buchsbaum equality, is satisfied.

If A and B are Abelian groups, then Hom(A,B) is also an Abelian group under pointwise addition of functions. In this section we will see how Hom gives rise to classes of functors. Let A denote the category of Abelian groups. A covariant functor T : A → A associates to every Abelian group A an Abelian group T (A), and for every homomorphism f : A → B a homomorphism T (f) : T (A)→ T (B), such tha...