Local cohomology with support in generic determinantal ideals
Let \(k\) be a field of characteristic \(0\). Let \(X\) be a generic \(m\times n\) matrix, i.e. the entries of \(X\) are distinct variables \(x_{1,1},\ldots,x_{m,n}\). For \(p\leq \min\{m,n\}\), the ideal \(I_p\subseteq k[x_{1,1},\ldots,x_{m,n}]\) of \(p\)-minors of \(X\) is generated by the determinants of \(p\times p\) submatrices of \(X\). The study of generic determinantal ideals, namely ideals of type \(I_p\) (where \(p\leq \min\{m,n\}\)), is a classical and beautiful subject lying in the intersection of commutative algebra and representation theory. The paper under review gives significant new information about such ideals, namely the authors compute the local cohomology with support in generic determinantal ideals. In fact, they compute the associated equivariant \(\text{Ext}\) modules of a large class of ideals invariant under the action of \(\text{GL}_m\times \text{GL}_n\). The local cohomology modules supported in such ideals are derived using the formula NEWLINE\[NEWLINE H^i_I(S)=\varinjlim_{d} \text{Ext}^i(S/I^d,S), NEWLINE\]NEWLINE for a proper ideal \(I\) over a noetherian local ring \(S\). The main theorems extend previous results due to \textit{C. Raicu} et al. [Adv. Math. 250, 596--610 (2014; Zbl 1295.13025)].NEWLINENEWLINEIn more detail, assume that \(m\geq n\). Let \(F,G\) be \(k\)-vector spaces of dimensions \(m,n\), respectively. Let \(S\) be the symmetric algebra \(\text{Sym}(F\otimes G)\). Then \(\text{GL}(F)\times \text{GL}(G)\cong \text{GL}_m\times \text{GL}_n\) acts on \(S\) and makes \(S\) into a \(\text{GL}(F)\times \text{GL}(G)\)-representation. By Cauchy formula, \(S\) is decomposed as a direct sum of irreducible \(\text{GL}(F)\times \text{GL}(G)\)-representations: NEWLINE\[NEWLINE S=\mathop{\bigoplus_{\underline x=(x_1,\ldots,x_n)}}_{x_1\geq x_2\geq \cdots \geq x_n\geq 0}S_{\underline x}F\otimes S_{\underline x}G, NEWLINE\]NEWLINE where \(S_{\underline x}\) denotes the Schur functor. A sequence \(\underline x=(x_1,\ldots,x_n)\) where \(x_1\geq x_2\geq \cdots \geq x_n\geq 0\) is called a partition of \(|\underline x|=x_1+\cdots+x_n\). Let \(I_{\underline x}\) be the ideal generated by \(S_{\underline x}F\otimes S_{\underline x}G\). The authors compute the \(\text{GL}_m\times \text{GL}_n\)-equivariant modules \(\text{Ext}^j_S(S/I_{\underline x},S)\), and the equivariant local cohomology modules \(H^j_{I_{\underline x}}(S)\). The computation of \(\text{Ext}^j_S(S/I_{\underline x},S)\) employs intricate geometric methods, which involve filtering \(S/I_{\underline x}\) by modules which can be realized as global sections of vector bundles with strong vanishing condition on sheaf cohomology.NEWLINENEWLINEAs corollary, a formula for regularity of \(I_{\underline x}\) is given, which implies that \(I_{\underline x}\) has linear resolution if and only if it is a power of the ideal of \(n\)-minors \(I_n\) (when \(x_1=\cdots=x_n\)), or a power of \(I_n\) times the maximal ideal (when \(x_1-1=x_2=\cdots=x_n\)). An interesting vanishing theorem for \(H^j_{I_p}(S)\) is provided (corollary of Theorem 6.1): \(H^j_{I_p}(S)\neq 0\) if and only if NEWLINE\[NEWLINE j = (n-p+1)^2 +(n-s)(m-n)+2k NEWLINE\]NEWLINE for \(0 \leq s \leq p-1, 0 \leq k \leq ( p-s-1)(n-p)\).NEWLINENEWLINEThis is in contrast with the case \(\text{char}~k>0\), where \(H^i_{I_p}(S)\neq 0\) only for \(i=(m-p+1)(n-p+1)\).
scientific article; zbMATH DE number 6348605