Presentations of rings with non-trivial semidualizing modules

Presentations of rings with a chain of semidualizing modules

Inspired by Jorgensen et. al., it is proved that if a Cohen--Macaulay local ring $R$ with dualizing module admits a suitable chain of semidualizing $R$--modules of length $n$, then $R\cong Q/(I_1+\cdots+I_n)$ for some Gorenstein ring $Q$ and ideals $I_1,\cdots, I_n$ of $Q$; and, for each $Λ\subseteq [n]$, the ring $Q/(Σ_{l\in Λ} I_l)$ has some interesting cohomological properties . This extends the result of Jorgensen et. al., and also of Foxby and Reiten.

Local rings of embedding codepth at most 3 have only trivial semidualizing complexes

We prove that a local ring $R$ of embedding codepth at most 3 has at most two semidualizing complexes up to shift-isomorphism, namely, $R$ itself and a dualizing $R$-complex if one exists.

On semidualizing modules of ladder determinantal rings

We identify all semidualizing modules over certain classes of ladder determinantal rings over a field ${\mathsf k}$. Specifically, given a ladder of variables $Y$, we show that the ring ${\mathsf k}[Y]/I_t(Y)$ has only trivial semidualizing modules up to isomorphism in the following cases: (1) $Y$ is a one-sided ladder, and (2) $Y$ is a two-sided ladder with $t=2$ and no coincidental inside corners.

Relative Tor functors with respect to a semidualizing module

We consider relative Tor functors built from resolutions described by a semidualizing module C over a commutative noetherian ring R. We show that the bifunctors Tor^{F_CM}_i (-,-) and Tor^{P_CM}_i (-,-), defined using flat-like and projective-like resolutions, are isomorphic. We show how the vanishing of these functors characterizes the finiteness of the homological dimension F_C-pd, and we use this to give a relation between the F_C-pd of a given module and that of a pure submodule. On the other hand, we show that other relations that one may expect to hold similarly, fail in general. In fact…

Vanishing of stable homology with respect to a semidualizing module

We investigate stable homology of modules over a commutative noetherian ring $R$ with respect to a semidualzing module $C$, and give some vanishing results that improve/extend the known results. As a consequence, we show that the balance of the theory forces $C$ to be trivial and $R$ to be Gorenstein.

数值半群环上的半对偶化模

Semidualizing Modules over Numerical Semigroup Rings

本文研究数值半群环上的半对偶化模，完全分类了重数至多为9的此类环具有非平凡半对偶化模的条件，并据此构造了任意重数至少为9的实例。