The Frobenius morphism in invariant theory
type: Item
instance of: scholarly article
title: The Frobenius morphism in invariant theory
zbMATH Open document ID: 1430.14098
author: Theo Raedschelders
author: Špela Špenko
author: Michel Van den Bergh
published in: Advances in Mathematics
publication date: 2019-05-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/1705.01832
review text: Let \(k\) be an algebraically closed field of positive characteristic \(p=\text{char}(k)\). Let \(X\) be a smooth projective \(k\)-variety, and let \(\mathcal{T}\) denote the pushforward of the structure sheaf of \(X\) by \textit{the} Frobenius morphism \(\text{Fr}:X\to X\), \(f\mapsto f^p\). It is a natural question to ask when \(\mathcal{T}\) satisfies the following two properties: 1) \(\mathcal{T}\) is a generator for the bounded derived category of cohorent modules on \(X\), 2) \(\text{Ext}^i_X (\mathcal{T},\mathcal{T})=0\) for every \(i>0\). If these two conditions hold, then \(\mathcal{T}\) is said to be a \textit{tilting bundle} on \(X\). These are rather stringent conditions, so, in general, \(\mathcal{T}\) will not be a tilting bundle on every \(k\)-variety. Nevertheless, depending on the characteristic of the field, for several well-known families of varieties \(\mathcal{T}\) gives a tilting bundle. The examples include all partial flag varieties, quadrics, as well as some Fano toric varieties. Let \(\mathbb{G}\) denote the Grassmann variety of 2 dimensional subspaces of \(k^n\), where \(n\) is assumed to be \(\geq 4\). Let \(\mathcal{T}\) denote the bundle \(\text{Fr}_*(\mathcal{O}_{\mathbb{G}})\). In this article under review, the authors present several important calculations about \(\mathcal{T}\), and they formulate a series of general conjectures. As one of the main results of the article, in Theorem 16.5 and Corollary 16.6, all indecomposable summands of the bundle \(\mathcal{T}\) are described. We will not reproduce these summands in here since they form a rather lengthy list; we refer the reader to the article for details. As a corollary of this main decomposition result, the authors are able to conclude that \(\mathcal{T}\) is not a tilting bundle on \(X\) unless \(n=4\) and \(p>3\). This conclusion is recorded as Theorem 1.1 in the introduction and as Corollary 16.7 in the main body of the paper. The proof of Theorem 16.5 is obtained from a more …
reviewed by: Mahir Bilen Can
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 13A50
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 14G17
Mathematics Subject Classification ID: 32S45
Mathematics Subject Classification ID: 14A22
zbMATH DE Number: 7055759
zbMATH Keywords: invariant theory
zbMATH Keywords: Frobenius summand
zbMATH Keywords: FFRT
zbMATH Keywords: Grassmannian
zbMATH Keywords: tilting bundle
zbMATH Keywords: noncommutative resolution
MaRDI profile type: Publication
OpenAlex ID: W2963288673
arXiv ID: 1705.01832
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The Frobenius morphism in invariant theory (Wikidata QID)
Wikidata QID: Q128208389

The Frobenius morphism in invariant theory (DOI)
DOI: 10.1016/J.AIM.2019.03.013

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