The Frobenius morphism in invariant theory
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 general decomposition result described by the authors. The main ingredients of this more general result can be summarized as follows: let \(G\) denote \(\text{SL}(V)\), where \(\dim_k V =2\). Let \(S\) denote the symmetric algebra \(\text{Sym}(V^{\oplus n})\) (\(n\geq 4\)), and let \(R\) denote the ring of \(G\)-invariants \(S^G\). One knows that, coincidentally, the homogeneous coordinate ring of \(\mathbb{G}\) is equal to \(R\). Roughly speaking, the main result of the paper is a theorem about the decomposition of \(R\) into \(R^p\)-modules, where \(R^p\) is the Frobenius image of \(R\). It turns out that this decomposition is equivalent to the decomposition of \(S^{G_1}\) as a \((G^{(1)}, S^p)\)-module, where \(S^p\) is the Frobenius image of \(S\), \(G_1\) is the kernel of the Frobenius map on \(G\), and \(G^{(1)} := G/G_1\). Once again, since the indexing sets that appear in this decomposition are rather lengthy, we refer the reader to the text, specifically, to Theorem 6.2. Note that Theorem 6.2 leads to the decomposition theorem for \(R\), which is recorded as Theorem 15.1. In turn, the decomposition theorem for \(R\) leads to the decomposition theorem for \(\mathcal{T}\), which is Theorem 16.5. Now, let \(Y\) be a singular variety, and set \(\mathcal{M}:= \text{Fr}_*(\mathcal{O}_Y)\). Then \(\mathcal{M}\) is said to provide a \textit{noncommutative resolution} (abbreviated to NCR) if \(\mathcal{E}nd_Y(\mathcal{M})\) is a sheaf of algebras on \(Y\) which is locally of finite global dimension. As an example for such a variety, the authors consider the cone \(Y\) over the Pücker embedding of \(\mathbb{G}\). The second main result of the present paper states that, for \(p \geq n-2\), the bundle \(\text{Fr}_*(\mathcal{O}_Y)\) provides an NCR for \(Y\). The proof of this result, which is essentially a corollary of the authors' decomposition theorems is in Section 17 of the article. Next, we will discuss some general conjectures that are presented in the article. Let \(R=k+R_1+\cdots \) be an integral finitely generated \(\mathbb{N}\)-graded algebra, where \(k\) is, as before, an algebraically closed field of positive characteristic \(p=\text{char}(k)\). A \textit{higher Frobenius summand} is an indecomposable summand of \(R^{1/p^r}\) for some \(r\geq 1\). \(R\) is said to have \textit{finite \(F\)-presentation type}, abbreviated to FFRT, if the number of isomorphism classes of higher Frobenius summands is finite. It is known for sometime that the invariant rings for reductive group actions have FFRT, see [\textit{K. E. Smith} and \textit{M. Van den Bergh}, Proc. Lond. Math. Soc. (3) 75, No. 1, 32--62 (1997; Zbl 0948.16019)]. The authors' definition of \textit{FFRT in characteristic zero} is as follows: Assume that \(k\) is algebraically closed and of characteristic zero, and let \(R=k+R_1+\cdots \) be an integral finitely generated \(\mathbb{N}\)-graded algebra. Then \(R\) is said to satisfy \textit{FFRT} if there exists a finitely generated \(\mathbb{Z}\)-algebra \(A\subset k\), a finitely generated graded flat \(A\)-algebra \(R_A\) such that \(k\otimes_A R_A = R\) and such that for every geometric point \(x : \text{Spec}\ l \to \text{Spec}\ A\), where \(\text{char}(l) >0\), \(x^*(R_A)\) satisfies FFRT. The authors conjecture that if \(W\) is a finite dimensional representation of a reductive group \(G\), then \(k[W]^G\) satisfies FFRT. This conjecture is called the \textit{FFRT conjecture in characteristic zero.} Another interesting conjecture that they propose is called the \textit{\(p\)-uniformity for invariant rings}: Assume that \(k\) is an algebraically closed field of characteristic zero. Let \(R=k+R_1+\cdots \) be an integral finitely generated \(\mathbb{N}\)-graded algebra. Then \(R\) is said to satisfy \textit{\(p\)-uniformity} if there exists a finitely generated \(\mathbb{Z}\)-algebra \(A\subset k\), a finitely generated graded flat \(A\)-algebra \(R_A\) such that \(k\otimes_A R_A = R\), a finite set \((M_{A,i})_{i=1}^m\) of finitely generated graded \(R_A\)-modules such that for every geometric point \(x : \text{Spec}\ l \to \text{Spec}\ A\), where \(\text{char}(l) >0\), the Frobenius summands of \(x^*(R_A)\) are given by the shifts of \((x^*(M_{A,i})^{\text{Fr}})_{i=1}^m\). Then the authors' \textit{\(p\)-uniformity conjecture for invariant rings (in characteristic zero)} states that if \(W\) is a finite dimensional representation of a reductive group \(G\), then \(k[W]^G\) satisfies \(p\)-uniformity. Finally, authors' \textit{resolution conjecture} goes as follows: Let \(k\) be an algebraically closed field of characteristic \(p>0\), and let \(G\) be an algebraic \(k\)-group. Assume that \(G\) acts on a commutative \(k\)-algebra \(S\). Let \(M\) be a finitely generated \((G,S)\)-module with a good filtration. Then there exists a tilting module \(T\) for \(G\) and a surjective \((G,S)\)-morphism \(T\otimes_k S \to M\) whose kernel has a good filtration. There is also a graded version of this conjecture, where \(T\) is assumed to be a graded \(G\)-module. In Section 9 of the article, the authors show that the graded resolution conjecture holds for \(\text{SL}_2\).
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