On the depth of the symmetric algebra

Abstract:

Let $(R,\mathfrak {m})$ be a local ring. Assume that $R = A/I$, where $(A,\mathfrak {n})$ is a regular local ring and $I \subseteq {\mathfrak {n}^2}$ is an ideal. The depth of the symmetric algebra $S(\mathfrak {m})$ of $\mathfrak {m}$ over $R$ is computed in terms of the depth of the associated graded module ${\text {gr}_\mathfrak {n}}(I)$ and the so-called "strong socle condition." Explicit results are obtained, for instance, if $I$ is generated by a super-regular sequence, if $I$ has a linear resolution or if $I$ has projective dimension one.