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Differentials of a symmetric generic determinantal singularity. (English) Zbl 0887.13008

From the authors’ introduction: Let \(k\) be a field, \(X=(X_{ij})\) an \((n,n)\)-matrix of indeterminates over \(k\), and \(r\) an integer with \(1\leq r<\min(m,n)\). Let \(R\) be the factor ring of \(k[X_{11},\ldots,X_{mn}]\) with respect to the ideal \(I_{r+1}\) generated by all \((r+1)\)-rowed minors of \(X\), and let \(D\) be the module of Kähler differentials of \(R\) over \(k\). U. Vetter [Commun. Algebra 11, 1701-1724 (1983; Zbl 0513.13013)] showed that \(\text{depth}(D)=(m+n-r+1)(r-1)+2\).
In the paper under review the authors consider the case of a generic symmetric matrix \(X\). In the case treated by U. Vetter (loc. cit.) there is an obvious filtration which can be used to give a lower bound for the depth; it is the right-hand side of the equality above. In the case of a generic symmetric matrix the authors use a filtration which comes from a combinatorial structure of the module of differentials given by the Jacobians of \(I_{r+1}\) to get a lower bound for the depth. This lower bound cannot be expected to be sharp.

MSC:

13C40 Linkage, complete intersections and determinantal ideals
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
14M12 Determinantal varieties

Citations:

Zbl 0513.13013

Software:

Macaulay2
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Full Text: DOI

References:

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