On the Ext Analog of the Euler Characteristic

2025/04/08 by Benjamin Katz, Katz, Benjamin, Andrew J. Soto Levins +1
Mathematics · #13D07 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2504.06450

openalex publication_date 2025/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• A Rigidity Theorem for Ext
• The Grade Conjecture and Asymptotic Intersection Multiplicity
• Asymptotic Behavior of Ext functors for modules of finite complete intersection dimension
• Asymptotic behavior of Tor over complete intersections and applications
• Some observations on local and projective hypersurfaces
• A counterexample to the rigidity conjecture for rings

Related

• An Euler characteristic for modules of finite G-dimension
• The quadratic Euler characteristic of a smooth projective same-degree complete intersection
• A roadmap to the (vanishing of the) Euler characteristic
• The signed Euler characteristic of very affine varieties
• On the Euler characteristic of a relative hypersurface