Depth Zero Ideals in the Homogenized Weyl Algebra

• Let $mathcal{D}$ be the $n$th Weyl algebra $mathcal{D} :=k[x_1,...,x_n,partial_1,...,partial_n]$, where $partial_ix_i-x_ipartial_i=1$.


• Let $widetilde{mathcal{D}}$ be its Rees algebra, which is $mathcal{D} :=k[t, x_1,...,x_n,partial_1,...,partial_n]$, where $partial_ix_i-x_ipartial_i=t$, and $t$ is central.


• Let $mathcal{O}_X$ denote the polynomial algebra $k[x_1,...,x_n]$, which is a left $widetilde{mathcal{D}}$-module, where $t$ and all the $partial_i$ act by zero. NOTE: this is different than the homogenization of the standard $mathcal{D}$-module structure on $mathcal{O}_X$.

The question I am interested in is, how many generators does a left ideal $M$ in $widetilde{mathcal{D}}$ need before $Hom_{widetilde{mathcal{D}}}(mathcal{O}_X,widetilde{mathcal{D}}/M)$ can be non-zero? My conjecture is that $M$ needs at least $n+1$ generators. NOTE: Savvy Weyl algebra veterans will know every left ideal in $mathcal{D}$ can be generated by two elements; however, this is not true in $widetilde{mathcal{D}}$. There can be ideals generated by $n+1$ elements and no fewer.

The functor $Hom_{widetilde{mathcal{D}}}(mathcal{O}_X,-)$ acts as a relative analog of the more familiar functor $Hom_R(k,-)$ (where $R=k[t,y_1,...y_n]$). Therefore, the above question is analogous to asking "how many generators must an ideal $Isubseteq R$ have before $R/I$ can have depth zero?" The answer here is $n+1$, which follows from Thm 13.4, pg 98 of Matsumura (essentially a souped up version of the Hauptidealsatz).

In the noncommutative case, if you try to make this work with the $widetilde{mathcal{D}}$-module $k$ (where $t$, $x_i$ and $partial_i$ all act by zero), it doesn't work. The natural conjecture would be that $Hom_{widetilde{D}}(k,widetilde{mathcal{D}}/M)neq 0$ implies $M$ had at least $2n+1$ generators, except this fails even for the first Weyl algebra and $M=widetilde{mathcal{D}}x_1+widetilde{mathcal{D}}partial_1$ (since $widetilde{mathcal{D}}/M=k$).

However, it seems that things might work right for the relative module $O_X$, based on a fair amount of experimentation. It is easily true in the first Weyl algebra. Oh, and equivalent condition is to ask when $Hom_{overline{mathcal{D}}}(mathcal{O}_X,overline{M})neq 0$, where $overline{mathcal{D}}=widetilde{mathcal{D}}/t$, and $overline{M}$ is $M/Mt$.

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