Local Cohomology Modules and Motivic Chern Class Computations

Abstract

We give a new proof of a result by Puthenpurakal on Lyubeznik's conjecture regarding the associated primes of local cohomology modules. The conjecture states that if $R$ is a regular ring and $I subset R$ is an ideal, then, for all $i$, the local cohomology module $H_I^i(R)$ has finitely many associated primes. The result of interest is for $H_I^{n-1}(R)$ where $R$ is of dimension $n$, contains $mathbb{Q}$, and satisfies a certain condition on the singular loci of its reduced quotient rings. We use the theory of $D$-modules over formal power series rings to show this, and also give a result on the codimension of the support of $H_I^i(R)$ when $R$ is further catenary, using the same methods.

Next, we compute the equivariant motivic Chern class for the nilpotent cone in $M_n$, the space of $n times n$ matrices, and for the affine cone over a smooth hypersurface. In the nilpotent cone case, we consider the action of $text{GL}_n times mathbb{C}^star$ acting on $M_n$ by conjugation in $text{GL}_n$ and by scaling in $mathbb{C}^star$. With this action, the orbits of the nilpotent cone are the nilpotent orbits, indexed by partitions of $n$. Following the techniques of Feher, Rimanyi, and Weber, we compute the motivic Chern class of these nilpotent orbits. In the affine cone case, we consider the action of $mathbb{C}^star$ on $mathbb{A}^{n+1}$ by scaling and compute the motivic Chern class of the affine cone $C(D)$, in $mathbb{A}^{n+1}$, where $Dsubset mathbb{P}^n$ is a smooth hypersurface.