Notions of purity and the cohomology of quiver moduli
type: Item
instance of: scholarly article
zbMATH Open document ID: 1248.14023
DOI: 10.1142/S0129167X12500978
author: Michel Brion
author: Roy Joshua
published in: International Journal of Mathematics
publication date: 2012-09-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1205.0629
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14L24
zbMATH DE Number: 6078202
zbMATH Keywords: weak purity
zbMATH Keywords: strong purity
zbMATH Keywords: quiver representations
zbMATH Keywords: GIT quotients
MaRDI profile type: Publication
arXiv ID: 1205.0629
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OpenAlex ID: W2962780001
title: Notions of purity and the cohomology of quiver moduli
review text: Let \(G\) be a smooth group scheme which acts on a scheme \(X\) of finite type over the finite field \(\mathbb{F}_{q},\) \(q\) a power of a prime \(p.\) The work under review is an analysis of some of the formulations of the term ``purity'' of schemes and related concepts, with particular attention to quiver representations. A finite-dimensional \(\mathbb{Q}_{l}\)-vector space \(V\) endowed with an endomorphism \(F\) is said to be strongly pure if each eigenvalue \(\alpha\in\mathbb{\bar{Q}}_{l}\) of \(F\) is of the form \(\alpha =q^{j}\) for some \(j\geq0\). On the other hand, \(\left( V,F\right) \) is called weakly pure pure if \(\alpha^{n}=q^{jn}\) for some \(n\).NEWLINENEWLINEFor \(l\neq p\) a prime one has the notion of a \(G\)-equivariant \(l\)-adic local system on \(X\). Let \(\mathcal{C}\) be a class of \(G\)-equivariant local systems on \(X\) such that (1) each local system is mixed and of weight bounded by a nonnegative integer \(w,\) (2) it is closed under extensions, and (3) it contains the local system \(\mathbb{Q}_{l}.\) It is shown that \(X\) is weakly (resp. strongly) pure with respect to \(\mathcal{C}\) if \(\left( H_{\text{et} }^{\ast}\left( \bar{X},\mathcal{\bar{L}}\right) ,F\right) \) is weakly (resp. strongly) pure for each \(\mathcal{L\in C},\) where \(F\) is the Frobenius. In the case where \(G\) acts trivially and \(\mathcal{C}\) is the class generated by the constant \(l\)-adic local system \(\mathbb{Q}_{l}\) then \(X\) is weakly (resp. strongly) pure with respect to \(\mathbb{Q}_{l}.\)NEWLINENEWLINELet \(\pi:X\rightarrow Y\) be a torsor under a linear algebraic group \(G\), and let \(\mathcal{C}_{X}\) (resp. \(\mathcal{C}_{Y}\)) be a class of \(l\)-adic local systems on \(X\) (resp. \(Y,\) where \(G\) acts trivially on \(Y\)). If \(\mathcal{C} _{X}\supseteq\pi^{\ast}\left( \mathcal{C}_{Y}\right) \) and \(X\) is weakly pure with respect to \(\mathcal{C}_{X},\) then \(Y\) is weakly pure with respect to \(\mathcal{C}_{Y}.\) If \(\mathcal{C}_{X}…
reviewed by: Alan Koch
description: scientific article; zbMATH DE number 6078202