Arithmetical ranks of ideals associated to symmetric and alternating matrices
The author computes the arithmetical rank of two classes of determinantal varieties over a field \(K\): those given by the \(t\)-minors of a symmetric \(n \times n\) matrix \(X\) of indeterminates, and those given by the \(t\)-Pfaffians, \(t\) even, of an alternating \(n \times n\) matrix \(X\) of indeterminates. If \(K\) is algebraically closed and \(I\) is an ideal in a polynomial ring over \(K\), then \(\text{ara} I\) is the number of equations defining the affine variety \(V(I)\) set-theoretically. -- The arithmetical rank is given by \(n (n - 1)/2 - t(t - 1)/2 + 1\) in the alternative case, independently of \(K\), and by \(n (n + 1)/2 - t (t + 1)/2 + 1\) in the symmetric case, provided \(\text{char} K \neq 2\) or \(t\) is odd; if \(\text{char} K = 2\) and \(t\) is even, then it is \(n(n - 1)/2 - t(t - 1)/2 + 1\). This seems to be the first class of ideals defined over \(\mathbb{Z}\) for which, after specialization to a field \(K\), the arithmetical rank depends on \(K\). The paper uses the methods developed by \textit{W. Bruns} and \textit{R. Schwänzl} [Bull. Lond. Math. Soc. 22, No. 5, 439-445 (1990; Zbl 0725.14039)], but several extra arguments are necessary, especially in the symmetric case. That the arithmetical rank is bounded above by the number given is shown by a constructive method that uses the structure of \(K[X]\) as an ASL [on a so-called doset in the symmetric case; see \textit{C. De Concini}, \textit{D. Eisenbud} and \textit{C. Procesi}, ``Hodge algebras'', Astérisque 91 (1982; Zbl 0509.13026)]. That it is not smaller is shown by methods of étale and group cohomology: in all the cases, the `critical' étale cohomology group (with suitable coefficients) does not vanish. In characteristic 0, one can also compute the cohomological dimension (with respect to the Zariski topology) of the given ideals (or varieties), except for even \(t\) in the symmetric case: it equals the arithmetical rank. To this end one replaces étale cohomology by De Rham cohomology (or singular cohomology) and uses Grothendieck's comparison theorem. (In characteristic \(p > 0\) the cohomological dimension of a perfect ideal \(I\) is not very interesting: it equals \(\text{codim} I\) by a theorem of Peskine and Szpiro).
scientific article; zbMATH DE number 802369