# Do almost commutative flat degenerations induce equality in K-theory? (Or: Is the characteristic variety actually a support of a class in $K$-theory?)

I intentionally phrased the title to match a different question which is almost identical to the one i'm asking. However similar, the answer there, which uses commutative algebraic geometry, is not directly applicable here as the setting is non-commutative.

Let $A$ be an almost commutative algebra over a field $k$. Meaning $A$ is a ($\mathbb{N}$-)filtered associative algebra whose associated graded $grA$ is commutative. There's a flat degeneration from $A$ to $grA$ which goes by the name of "Rees algebra". This is the graded algebra $R=\oplus_{n \ge0} t^n F_nA$. We may view this geometrically as follows:

Consider "$SpecR$" as a flat family over $\mathbb{A}^1_k$. The fiber over $0$ is $Spec(grA)$ while the rest of the fibers are isomorphic to "$SpecA$".

The only example I have in mind is the algebra of differential operators on an smooth affine scheme $X$ in char $0$. In that case $A=\mathcal{D}_X$ and $Spec(grA)= T^*X$. The notion of good filtration has a natural interpretation in this picture:

A good filtration on a coherent $\mathcal{D}_X$-module $\mathcal{M}$ is the same as the data of an $R$-module $\mathcal{N}$ with an isomorphism $\mathcal{N} \otimes_R R[t^{-1}] \cong \mathcal{M} \otimes_{\mathcal{D}_X} R[t^{-1}].$. That is a flat degeneration over the Rees family from $\mathcal{M}$ to some module over $grA$.

Now if it were the case that "flat degenerations" still respect (coherent sheaf) $K$-theory in this setting one could conclude that any $\mathcal{D}_X$ module which has a good filtration gives $[\mathcal{M}]$ a well defined class in $K(T^*X)$. So my question is (slightly generalized form of the above):

Question: Let $A$ be a flat associative noetherian $k[t]$-algebra with a regular commutative fiber over $0$. Let $M$ and $N$ be two finite type $A$-modules whose generic fibers are isomorphic (as $A[t^{-1}]$-modules). Do $M$ and $N$ define the same class in the (coherent) $K$ theory of the $0$ fiber $K(A \otimes_{k[t]} k)$?

Then the obvious generalization in the context of $D$-modules will be:

Does the above give rise to a globally well defined canonical map between $K$-theory of coherent $\mathcal{D}$-modules and $\mathbb{C}^\times$-equivariant $K$-theory of the cotangent bundle? $$K(\mathcal{D}_X-mod) \to K_{\mathbb{C}^\times}(T^*X)$$ when $X$ is a smooth complex algebraic variety?

• Have you looked at Ginsburg's paper "Characteristic varieties and vanishing cycles"? Specifically, sections 1.1-1.3. – Avi Steiner Jun 24 '17 at 18:56