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Spring 2007 Socle degrees of Frobenius powers
Andrew R. Kustin, Adela N. Vraciu
Illinois J. Math. 51(1): 185-208 (Spring 2007). DOI: 10.1215/ijm/1258735332


Let $k$ be a field of positive characteristic $p$, $R$ be a Gorenstein graded $k$-algebra, and $S=R/J$ be an artinian quotient of $R$ by a homogeneous ideal. We ask how the socle degrees of $S$ are related to the socle degrees of $F_R^e(S)=R/J^{[q]}$. If $S$ has finite projective dimension as an $R$-module, then the socles of $S$ and $F_R^e(S)$ have the same dimension and the socle degrees are related by the formula $D_i=qd_i-(q-1)a(R)$, where $d_1\le \dots\le d_{\ell}$ and $D_1\le \dots \le D_{\ell}$ are the socle degrees of $S$ and $F_R^e(S)$, respectively, and $a(R)$ is the $a$-invariant of the graded ring $R$, as introduced by Goto and Watanabe. We prove the converse when $R$ is a complete intersection.


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Andrew R. Kustin. Adela N. Vraciu. "Socle degrees of Frobenius powers." Illinois J. Math. 51 (1) 185 - 208, Spring 2007.


Published: Spring 2007
First available in Project Euclid: 20 November 2009

zbMATH: 1131.13003
MathSciNet: MR2346194
Digital Object Identifier: 10.1215/ijm/1258735332

Primary: 13A35
Secondary: 13D05 , 13H10

Rights: Copyright © 2007 University of Illinois at Urbana-Champaign


Vol.51 • No. 1 • Spring 2007
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