nLab Euler bilinear form

Expression Euler form may mean either a differential form (“Chern-Euler form”) representing the Euler characteristic class, but also a $\mathbf{Z}$-bilinear form on a Grothendieck group of a $k$-linear abelian or triangulated category, related to an alternative sum of dimensions of Ext-groups. This entry is about the latter. The terminology is allegedly due Claus Michael Ringel.

Definition

If $C$ a $k$-linear abelian category such that $dim(Ext^i(M,N))\lt \infty$ for all $i$ and all objects $M,N$ in $C$, then the Euler form is a $\mathbf{Z}$-bilinear form $\langle -,-\rangle : K_0(C)\times K_0(C)\to\mathbf{Z}$ defined by

$$\langle [M],[N]\rangle = \sum_{i\geq 0} (-1)^i dim_k(Ext^i(M,N)).$$

Similarly, for suitable $k$-linear triangulated category $D$ one sets

$$\langle [M],[N]\rangle = \sum_{i\geq 0} (-1)^i dim_k(Hom_D(M,\Sigma^i N))$$

Literature

• C. Geiß, Derived tame algebras and Euler-forms, Math Z. 239 (2002) 829–862 doi
• Helmut Lenzing, On the K-theory of weighted projective curves, arXiv:1702.03445
• Helmut Lenzing, A K-theoretic study of canonical algebras, in: Representation theory of algebras (Cocoyoc, 1994), CMS Conf. Proc, 1996 RG
• Sefi Ladkani, Refined Coxeter polynomials, arXiv:2110.15329