On the Ext Analog of the Euler Characteristic

TL;DR

We study the Ext-analogue of Serre's Euler characteristic by defining the Euler form $\xi^R(M,N)$ and partial forms $\xi_j^R(M,N)$, linking their vanishing to Ext behavior and contrasting with the Tor-focused $\chi_j^R(M,N)$. The authors prove a main result that $\xi_j^R(M,N)\ge 0$ with equivalences relating $\mathrm{grade}_R(M,N)$ and $\chi_{\mathrm{pd}_R M - j + 1}^R(M,N)$, and they apply this to Jorgensen's question and to Chan's graded formula, plus a graded setting translation. They further relate the higher Herbrand difference to Ext-vanishing through a zero-criterion that reduces to a complete intersection and yields vanishing beyond depth differences, enriching the toolkit for asymptotic Ext behavior in complete intersections. The work connects partial Euler forms, graded invariants, and vanishing phenomena, with concrete consequences in complete intersections and graded rings.

Abstract

The Euler form is an Ext analog of the Euler characteristic, and in this paper we study the Euler form and give some applications. The first being a question of Jorgensen, which bounds the projective dimension of a module over a complete intersection by using the vanishing of self extensions. Our second application uses the Euler form to yield a new result involving the vanishing of the higher Herbrand difference. Along the way we translate some of our results to the graded setting.

Theorems & Definitions (37)

• Theorem 1.1: Hochster and Lichtenbaum
• Theorem A: Lemma \ref{['signsofpartialeulerforms']} and Theorem \ref{['mainresult']}
• Proposition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof