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Register a new userLocal Homology, Koszul Homology and Serre Classes
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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.
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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 $R$-algebra automorphism of $K$ induces the identity map on $H(K)$. In such cases, it is possible to define an action of the automorphism group of $R$ on $H(K)$. On the other hand, numerous rings are described for which $K$ has automorphisms that do not induce the identity on $H(K)$. For any $R$, it is shown that the group of automorphisms of $H(K)$ induced by automorphisms of $K$ is abelian.
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 multiplicative structure of $operatorname{H}^R$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincare series. As an application, we show that the Poincare series of all finitely generated modules over a stretched Cohen-Macaulay local ring are rational, sharing a common denominator.
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 commutative and Lie types in computations of the cohomology of the outer automorphism group of a free group. Another consequence is an explicit computation of dualizing sheaves on spaces of metric graphs, thus characterizing to which extent these spaces are different from oriented orbifolds. We also provide a relation between the cohomology of the space of metric ribbon graphs, known to be homotopy equivalent to the moduli space of Riemann surfaces, and the cohomology of a certain sheaf on the space of usual metric graphs.
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 $R$-module $Tor^R_i(R/fa,M)$ is finitely generated for every $0leq ileq n$. This provides a hands-on and computable finitely-many-steps criterion to examine $mathfrak{a}$-confiniteness. Our approach relies heavily on the theory of local homology which demonstrates the effectiveness and indispensability of this tool.
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