Some Homological Conjectures Over Idealization Rings

TL;DR

This paper investigates homological conjectures over idealization rings, providing positive results and characterizations that extend classical conjectures to these rings, especially for modules with infinite projective dimension.

Contribution

It offers new criteria and proofs for long-standing homological conjectures over idealization rings, including the Auslander-Reiten, Jorgensen-Leuschke, and Buchsbaum-Eisenbud-Horrocks conjectures.

Findings

Vanishing of certain Ext groups implies module freeness over idealization rings.

Betti number characterizations confirm the Jorgensen-Leuschke conjecture for idealization rings.

Classical conjectures hold over idealization rings for modules with infinite projective dimension.

Abstract

Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module. The main focus of this paper is to give positive answers for some long-standing homological conjectures over the idealization ring $R\ltimes M$. First, if $N$ is a $R\ltimes k$-module, we show that the vanishing of $\operatorname{Ext}_{R\ltimes k}^{i}(N,N\oplus (R\ltimes k))$ for $i=1,2,3$ gives that $N$ is free, and this provides a sharpened version of the Auslander-Reiten conjecture over $R\ltimes k$. Also, we give a characterization of the Betti numbers of an $R$-module over the idealization ring $R\ltimes M$ and, as a biproduct, we derive that the Jorgensen-Leuschke conjecture holds true for $R\ltimes M$. Further, we show that the true of Buchsbaum-Eisenbud-Horrocks and Total Rank conjectures over $R$ implies the true over $R\ltimes M$. This establishes particular answers for both conjectures for modules with infinite projective dimension, especially when $R$ is regular or a complete intersection ring. As applications of the idealization ring theory, we show that the Zariski-Lipman conjecture holds for any ring $R$ provided the Betti numbers of the $R$-derivation module $\operatorname{Der}_k(R)$, seen as $R\ltimes k$-module, satisfy the inequality $\beta_{n}^{R\ltimes k}(\operatorname{Der}_k(R))\leq\beta_{n-1}^{R\ltimes k}(\operatorname{Der}_k(R))$ for some $n>0$. Some implications regarding the Herzog-Vasconcelos conjecture are also provided.