Arithmetical ranks of ideals associated to symmetric and alternating matrices

Let X be an n×n matrix of indeterminates over an algebraically closed field K, and let K[X] be the corresponding polynomial ring. If X is symmetric, we denote by It(X) the ideal of K[X] generated by the t×t minors of X, for 1≤t≤n; if X is alternating, we denote by Pft(x) the ideal generated by the t×t Pfaffians of X, for all even t such that 2≤t≤n. In the paper a formula is given which expresses the arithmetical rank (ara) of It(X) and Pft(X) in terms of n and t. The arithmetical rank of an ideal is the minimal number of elements generating the ideal up to radical. We use the theory of algebras with straightening laws to reduce the problem to the computation of a certain étale cohomology module. Using a cohomological criterion we conclude that the arithmetical rank of Pft(X) coincides with its cohomological dimension. The same is true for It(X) when t is odd. For the symmetric matrices the case of characteristic 2 represents an exception: for even t the cohomological criteria fail and the arithmetical rank has to be computed directly, by combinatorial means.