Gorenstein injective filtrations over rings with dualizing complexes

TL;DR

This work extends finite filtrations of Gorenstein injective modules from Gorenstein rings to commutative noetherian rings admitting a dualizing complex $D$. It develops two parallel filtrations: (i) a derived-category approach via Auslander categories, yielding a finite filtration $0=G_{d+1}\subset\cdots\subset G_0=G$ with $G_k/G_{k+1}\cong\bigoplus_{\frak p\in X_k}\mathrm{Tor}_k^R(E(R/\frak p), \mathbf{RHom}_R(D,G))$, where $X_k=\{\frak p\,|\, \ht\frak p-\sup D_{\frak p}=k\}$; (ii) a constructive filtration using section functors for $\dim R=d$, with $G_k=\sum_{\frak p\in Y_k}\Gamma_{\frak p}(G)$ and $G_k/G_{k+1}=\sum_{\frak p\in Y_k}\Gamma_{\frak p}(G/G_{k+1})$, where $Y_k=\{\frak p\,|\, \ht\frak p=k\}$. Both filtrations are unique and functorial in $G$, and the two constructions agree; in the special cases where $R$ is Gorenstein or Cohen–Macaulay with a dualizing module, the filtrations specialize to known results. The paper thus connects dualizing complexes, Auslander categories, and local-cohomology techniques to provide a robust, structured description of Gorenstein injective modules over a broad class of rings.

Abstract

Let $R$ be a commutative noetherian ring. Enochs and Huang [EH] proved that over a Gorenstein ring of Krull dimension $d$, every Gorenstein injective module admits a finite filtration of Gorenstein injective submodules. In this paper, we extend this result to rings admitting a dualizing complex and we provide such filtrations using Auslander categories and section functors.

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1
• Definition 2
• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 1