[Paper Review] Strongly $n$-Gorenstein projective, injective and flat modules

Najib Mahdou, Mohammed Tamekkante|ArXiv.org|Apr 26, 2009

Algebraic structures and combinatorial models4 references3 citations

TL;DR

This paper introduces and studies strongly $n$-Gorenstein projective, injective, and flat modules—generalizations of strongly Gorenstein modules that provide simpler characterizations of Gorenstein projective, injective, and flat modules with finite Gorenstein dimension. The key contribution is a new characterization of $n$-Gorenstein flat modules via short exact sequences and vanishing Tor conditions, extending previous results on strongly Gorenstein modules.

ABSTRACT

This paper generalize the idea of the authors in \cite{Bennis and Mahdou1}. Namely, we define and study a particular case of modules with Gorenstein projective, injective, and flat dimension less or equal than $n\geq 0$, which we call, respectively, strongly n-Gorenstein projective, injective and flat modules. These three classes of modules give us a new characterization of the first modules, and they are a generalization of the notions of strongly Gorenstein projective, injective, and flat modules respectively.

Motivation & Objective

• To generalize the concept of strongly Gorenstein modules to higher finite Gorenstein dimensions.
• To provide simpler, more effective characterizations of modules with Gorenstein projective, injective, and flat dimensions at most $n$.
• To establish equivalences between strongly $n$-Gorenstein flat modules and modules with finite $n$-presentations over coherent rings.
• To clarify the relationship between Gorenstein flat dimension and flat dimension in the context of strongly $n$-Gorenstein modules.

Proposed method

• Define strongly $n$-Gorenstein projective, injective, and flat modules via exact sequences of modules with finite flat or projective dimension.
• Use short exact sequences of the form $0 \to M \to X \to M \to 0$ where $X$ has finite flat or projective dimension and certain Ext or Tor vanishing conditions.
• Apply homological algebra techniques, including long exact sequences of Tor and Ext functors, to derive dimension bounds.
• Leverage results from previous works on strongly Gorenstein modules and extend them to the $n$-Gorenstein setting.
• Use the notion of $n$-presentations and $n$-step resolutions to relate projective and flat dimension conditions.
• Prove equivalences between strongly $n$-Gorenstein flat and projective modules under finite presentation assumptions.

Experimental results

Research questions

• RQ1 What conditions characterize a module as strongly $n$-Gorenstein flat?
• RQ2 How do strongly $n$-Gorenstein modules relate to Gorenstein flat modules with finite Gorenstein flat dimension?
• RQ3 Under what conditions is a finitely presented module strongly $n$-Gorenstein flat if and only if it is strongly $n$-Gorenstein projective?
• RQ4 Can the Gorenstein flat dimension of a module be bounded using properties of its resolutions and Tor vanishing?
• RQ5 What is the role of $n$-presentations in characterizing strongly $n$-Gorenstein modules?

Key findings

• A module $M$ is strongly $n$-Gorenstein flat if and only if there exists a short exact sequence $0 \to M \to F \to M \to 0$ with $fd(F) \leq n$ and $Tor^i(M, I) = 0$ for all $i > n$ and all injective modules $I$.
• If $M$ is strongly $n$-Gorenstein flat and $R$ is coherent, then $M$ admits a finite $(n+1)$-presentation.
• For a finitely presented module $M$ over a coherent ring $R$, $M$ is strongly $n$-Gorenstein projective if and only if it is strongly $n$-Gorenstein flat.
• The Gorenstein flat dimension of a module $M$ satisfies $Gfd(M) \leq n$ if $M$ is strongly $n$-Gorenstein flat, but the converse does not hold in general.
• There exists a short exact sequence $0 \to M \to F \to M \to 0$ with $fd(F) \leq n$ if $M$ is strongly $n$-Gorenstein flat, and this sequence induces vanishing of higher Tor groups.
• The class of strongly $n$-Gorenstein flat modules is strictly larger than the class of $n$-Gorenstein flat modules, as shown by counterexamples in the paper.