## Journal of Mathematics of Kyoto University

### Homological codimension of modular rings of invariants and the Koszul complex

Larry Smith

#### Abstract

Let $\rho : G \hookrightarrow \mathrm{GL}(n, \mathbf{F})$ be a representation of a finite group over the field $\mathbf{F}$ of characteristic $p$, and $h_{1}, \ldots ,h_{m} \in \mathbf{F}[V]^{G}$ invariant polynomials that form a regular sequence in $\mathbf{F}[V]$. In this note we introduce a tool to study the problem of whether they form a regular sequence in $\mathbf{F}[V]^{G}$. Examples show they need not. We define the cohomology of $G$ with coefficients in the Koszul complex $(\mathscr{K},\partial )=(\mathbf{F}[V]\bigotimes E(s^{-1}h_{1}, \ldots ,s^{-1}h_{n}), \partial (s^{-1}h_{i}:i=1,\ldots ,n),$ which we denote by $H^{*}(G;(\mathscr{K}, \partial ))$, and use it to study the homological codimension of rings of invariants of permutation representations of the cyclic group of order $p$, for $p \neq 0$, and to answer the above question in this case.

#### Article information

Source
J. Math. Kyoto Univ., Volume 38, Number 4 (1998), 727-747.

Dates
First available in Project Euclid: 17 August 2009

https://projecteuclid.org/euclid.kjm/1250518006

Digital Object Identifier
doi:10.1215/kjm/1250518006

Mathematical Reviews number (MathSciNet)
MR1670003

Zentralblatt MATH identifier
0951.13002

Subjects
Primary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24]
Secondary: 13D25

#### Citation

Smith, Larry. Homological codimension of modular rings of invariants and the Koszul complex. J. Math. Kyoto Univ. 38 (1998), no. 4, 727--747. doi:10.1215/kjm/1250518006. https://projecteuclid.org/euclid.kjm/1250518006