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- Title
The endomorphism ring of the trivial module in a localized category.
- Authors
Carlson, Jon F.
- Abstract
Suppose that G is a finite group and k is a field of characteristic p>0$p >0$. Let M${\mathcal {M}}$ be the thick tensor ideal of finitely generated modules, whose support variety is in a fixed subvariety V of the projectivized prime ideal spectrum ProjH∗(G,k)$\operatorname{Proj}\nolimits \operatorname{H}\nolimits ^*(G,k)$. Let C${\mathcal {C}}$ denote the Verdier localization of the stable module category stmod(kG)$\operatorname{{\bf stmod}}\nolimits (kG)$ at M${\mathcal {M}}$. We show that if V is a finite collection of closed points and if the p‐rank of every maximal elementary abelian p‐subgroups of G is at least 3, then the endomorphism ring of the trivial module in C${\mathcal {C}}$ is a local ring, whose unique maximal ideal is infinitely generated and nilpotent. In addition, we show an example where the endomorphism ring in C${\mathcal {C}}$ of a compact object is not finitely presented as a module over the endomorphism ring of the trivial module.
- Subjects
ENDOMORPHISM rings; ENDOMORPHISMS; NOETHERIAN rings; PRIME ideals; LOCAL rings (Algebra); FINITE groups
- Publication
Mathematische Nachrichten, 2023, Vol 296, Issue 9, p4264
- ISSN
0025-584X
- Publication type
Article
- DOI
10.1002/mana.202200160