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Given a Serre class $mathcal{S}$ of modules, we compare the containment of the Koszul homology, Ext modules, Tor modules, local homology, and local cohomology in $mathcal{S}$ up to a given bound $s geq 0$. As some applications, we give a full characterization of noetherian local homology modules. Further, we establish a comprehensive vanishing result which readily leads to the formerly known descriptions of the numerical invariants width and depth in terms of Koszul homology, local homology, and local cohomology. Also, we immediately recover a few renowned vanishing criteria scattered about the literature.
This work concerns the Koszul complex $K$ of a commutative noetherian local ring $R$, with its natural structure as differential graded $R$-algebra. It is proved that under diverse conditions, involving the multiplicative structure of $H(K)$, any dg
We investigate various module-theoretic properties of Koszul homology under mild conditions. These include their depth, $S_2$-property and their Bass numbers
Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $operatorname{H}^R$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicativ
We show that Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This in particular gives a conceptual explanation of the appearance of graph cohomology of both the
Let $frak a$ be an ideal of a commutative noetherian ring $R$ with unity and $M$ an $R$-module supported at $V(fa)$. Let $n$ be the supermum of the integers $i$ for which $H^{fa}_i(M) eq 0$. We show that $M$ is $fa$-cofinite if and only if the $R$-mo