Differential Hochschild Cohomology, general tools?

Question

Background: in deformation quantization one wants to construct formal associative deformations (the star products $\star$) of the algebra of smooth complex-valued functions on a Poisson manifold $M$ in direction of the Poisson bracket. The cochains in order $\hbar^r$, where $\hbar$ is the deformation parameter, are not just any $2$-cochains of $C^\infty(M)$ but typically one requires them to be either continuous with respect to the usual $C^\infty$-topology, or even bi-differential. Beyond the deformation of $C^\infty(M)$ one also is interested in deforming all kinds of modules over $C^\infty(M)$ once a star product $\star$ is fixed in such a way that one still obtains modules. Also here the "classical" modules usually come from geometry (sections of bundles etc) and hence allow for the notion of differential cochains.

The deformation problem is mainly governed by the Hochschild cohomology of $C^\infty(M)$ with values in the corresponding endomorphisms of the "classical" module one wants to deform (or the algebra itself). However, the additional requirement of being differential brings us to a sub-complex. Thus the computation of its cohomology is a new problem which in the cases I know has to be done more or less by hand...

For a general commutative algebra one still can define mutlidifferential operators (with values in modules) a la Grothendieck in a completely algebraic way (reproducing the above differential operators in the case of $C^\infty(M)$). This gives a sub-complex of the Hochschild complex which I would like to understand: for the algebraic situation and also for continuous cochains (under some conditions) one has the usual tools of homological algebra to identify the Hochschild cohomology as certain Ext groups etc.

However, the additional requirement of being differential does not seem to fit into this nice algebraic abstract nonsense theory. I guess it would be very nice to have these sort of tools also available in the differential setting, which, after all, is entirely algebraic in its nature.

So my question is: are there any possibilities to transfer the usual notions of Ext etc to the differential case?

Answer 1

If I understand the question correctly Stefan is asking for an Ext interpretation of the polydifferential Hochschild cochain complex. Elements of this are not just continuous linear maps $C^\infty(M)^{\otimes n} \to C^\infty(M)$, but they have to be polydifferential operators. (This version of Hochschild cohomology is used in Kontsevich's formality theorem).

Anyway, one can understand the polydifferential condition as follows. Consider the jet bundle $J$ on $M$; this is an infinite-rank vector bundle whose fibre at a point $p \in M$ is the algebra of formal power series at $p$. If we choose coordinates $x_1,\dots, x_n$ at $p$, then we can identify the fibre $J_p$ as $\mathbb{R}[[x_1,\dots,x_n]]$.

It's standard that $J$ is a left $D$-module. Further, the obvious product on the fibre of $J$ makes $J$ into a commutative algebra in the symmetric monoidal category of left $D$-modules.

Then, one can take Hochschild cochains of $J$ in the symmetric monoidal category of left $D$-modules.

This is the same as the complex of poly-differential Hochschild cochains. The key point is that $D$-module maps $J^{\otimes n} \to J$ are the same as polydifferential operators.

Of course, this means that you can apply any of the standard interpretations of Hochschild cohomology in this context (e.g. $\operatorname{Ext}_{J \otimes J}(J,J)$).

One needs a little care with these definitions, because $J$ is a topological $D$-module. However, if you take continuous $D$-module maps and appropriately completed tensor products you get the right answer.

Answer 2

Let me try an answer.

1. It seems to me that the appropriate language to use is the one of ringed spaces. For a given ring space $(X,\mathcal{O}_X)$ one can consider the category of sheaves of right $\mathcal{O}_X$-modules (see e.g. Section 7 of http://alpha.uhasselt.be/Research/Algebra/Publications/hochschild_ab.pdf, where you will find some evidences for such an approach).

2. Another approach, more in the spirit of David Ben-Zvi answer is to use the notion of Lie algebroid. In http://arxiv.org/abs/0908.2630 we interpret differential Hochschild cohomology as Ext's of $J_L$-modules, where $J_L$ is the jet algebra sheaf associated to the Lie algebroid $L=(\mathcal{O},Der(\mathcal{O}))$ (i.e. the $\mathcal{O}$-dual of its universal enveloping algebra).

3. Maybe yet another point of view could help. $J_L$ is the algebra of functions on the formal groupoid integrating the Lie algebroid $L$. Now for a groupoid $G=(G_1,G_0)$ we can consider $Ext_{G_1}(e_*(-),e_*(-))$, where $e:G_0\to G_1$ is the unit inclusion. Now it appears that this only depends on the inclusion of $G_0$ into its formal neighbourhood in $G_1$. In other words, it only depends on $J_L$, where $L$ is the Lie algebroid of $G$. SO (up to details) 2 and 3 are equivalent.

In David's answer $G=(X,X\times X)$, and its associated Lie algebroid is $(\mathcal{O}_X,Der(\mathcal{O}_X))$. In this case the relation between 3 and 1 is again explained in Section 7 of http://alpha.uhasselt.be/Research/Algebra/Publications/hochschild_ab.pdf for schemes. For complex analytic and differentiable manifolds the proof is basically the same, but you will probably have to use http://arxiv.org/abs/0908.2630

I hope this can help.

Answer 3

Given any bimodule over a commutative ring (or over a scheme, i.e. coherent sheaf on the product) we can consider its differential part, namely the part supported set-theoretically on the diagonal. Applying this to End(R) we find Grothendieck's definition of differential operators. Can you not repeat the Hochschild story working strictly in the world of differential bimodules? In this case you get a quasiisomorphic complex to the Hochschild complex -- ie in the algebraic setting self-Exts of the identity functor are already supported on the diagonal. (Put another way, if everything in sight is coherent, differential bimodules -- such as the diagonal -- are I think a full subcategory of bimodules, so it doesn't hurt to restrict to them). Is this the kind of picture you're looking for?

Another trick that may be useful is the setting of induced D-modules: there's an equivalence of categories between the category with objects quasicoherent sheaves and morphisms given by differential operators, and the full subcategory of D-modules given by induced D-modules (for right D-modules, these are things of the form $M\otimes_{O_X} D_X$). So working with induced D-modules might be a convenient place to think of differential Hochschild theory..