Here are a few of the things that I am thinking about:
- For a regular local ring of equal characteristic 0, when is the injective dimension of a local cohomology modules equal to the dimension of its support?
- For a regular local ring of equal characteristic 0, are their non-trivial bounds on the projective dimension of local cohomology modules?
- Suppose $R=k[[x_1,...,x_n]]$ with $k$ a field of characteristic $0$. If $M$ is a holonomic $D_{R|k}$ module, then is $\textrm{dim}(\textrm{Supp}(M))-1\leq\textrm{inj.dim}_R(M)$?
- Understanding the structure of the $D_{R|k}$ when $R=\frac{k[x,y,u,v]}{(xy-uv)}$.
Please contact me at tmurray11@husker.unl.edu if you would like to discuss any of these projects with me.