Differentials of a symmetric generic determinantal singularity
Titel:
Differentials of a symmetric generic determinantal singularity
Auteur:
Vetter, Udo Warneke, Klaus
Verschenen in:
Communications in algebra
Paginering:
Jaargang 25 (1997) nr. 7 pagina's 2193-2209
Jaar:
1997
Inhoud:
Let k be a field, X a symmetric matrix of indeterminates over k, and R the factor ring of the polynomial ring k[X]with respect to the ideal generated by all (r[d]1)-minors of X where r is a fixed positive integer. We give lower bounds for the depth of the module of Kahler differentials of R over k and of its R-dual. Let k be a field, X [d] (X$sub:ij$esub:)an m x n matrix of indeterminates over k, and r an integer, 1 [d] r < min(m,n). We denote by R [d] R$sub:r$esub:[d]1 the factor ring of k[X11,…,Xmn] with respect to the ideal I$sub:r[d]1$esub:generated by all (r[d]1)-minors of X, and by D [d] D$sub:k$esub:(R) the module of Kahler differentials of R over k. It is well known (see [V1]) that (*)depth D [d] (m[d]n[d]r[d]l)(r[d]1) [d] 2. (Here depth means depth with respect to the irrelevant maximal ideal.) If sider the analogous setup in case X is a symmetric n x n matrix of indeterminates, what can be said about depth D ? The proof of (*) runs as follows: there is an obvious filtration of the first R-syzygy of D (i.e. the module given by the Jacobian of I$sub:r$esub:[d]1), the quotients of
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