Foxby equivalence over associative rings

On the invariance of certain types of generalized Cohen-Macaulay modules under Foxby equivalence

Let R be a local ring and C a semidualizing module of R. We investigate the behavior of certain classes of generalized Cohen-Macaulay R-modules under the Foxby equivalence between the Auslander and Bass classes with respect to C. In particular, we show that generalized Cohen-Macaulay R-modules are invariant under this equivalence and if M is a finitely generated R-module in the Auslander class with respect to C such that C\otimes_RM is surjective Buchsbaum, then M is also surjective Buchsbaum.

关于 $C$-$fp_n$-内射与 $C$-$fp_{n}$-平坦模的 Foxby 等价

Foxby equivalence relative to $C$-$fp_n$-injective and $C$-$fp_{n}$-flat modules

本文引入 $C$-$fp_n$-内射与 $C$-$fp_n$-平坦模类，研究其维数、Foxby 等价、预包络与覆盖的存在性，并在几乎优扩张下探讨其交换性质。

Foxby equivalence, local duality and Gorenstein homological dimensions

Let $(R,\fm)$ be a local ring and $(-)^{\vee}$ denote the Matlis duality functor. We investigate the relationship between Foxby equivalence and local duality through generalized local cohomology modules. Assume that $R$ possesses a normalized dualizing complex $D$ and $X$ and $Y$ are two homologically bounded complexes of $R$-modules with finitely generated homology modules. We present several duality results for $\fm$-section complex ${\bf R}Γ_{\fm}({\bf R}\Hom_R(X,Y))$. In particular, if G-dimension of $X$ and injective dimension of $Y$ are finite, then we show that $${\bf R}Γ_{\fm}({\bf R}\…

Adic Foxby Classes

We continue our work on adic semidualizing complexes over a commutative noetherian ring $R$ by investigating the associated Auslander and Bass classes (collectively known as Foxby classes), following Foxby and Christensen. Fundamental properties of these classes include Foxby Equivalence, which provides an equivalence between the Auslander and Bass classes associated to a given adic semidualizing complex. We prove a variety of stability results for these classes, for instance, with respect to $F\otimes^{\mathbf{L}}_R-$ where $F$ is an $R$-complex finite flat dimension, including special conver…

Automorphic equivalence problem for free associative algebras of rank two

Let K<x,y> be the free associative algebra of rank 2 over an algebraically closed constructive field of any characteristic. We present an algorithm which decides whether or not two elements in K<x,y> are equivalent under an automorphism of K<x,y>. A modification of our algorithm solves the problem whether or not an element in K<x,y> is a semiinvariant of a nontrivial automorphism. In particular, it determines whether or not the element has a nontrivial stabilizer in Aut K<x,y>. An algorithm for equivalence of polynomials under automorphisms of C[x,y] was presented by Wightwick. Another, much s…

相对 Cohen-Macaulay 模范畴的对偶与等价

Dualities and equivalences of the category of relative Cohen-Macaulay modules

本文发展相对 Cohen-Macaulay 模理论，引入 a-相对对偶化模与大 Cohen-Macaulay 模，建立局部代数对偶的整体版本，并考察其在 Foxby 等价下的范畴行为。