I am trying to understand something about curved dg algebras as studied by Positselski, E. Segal. These come up in mirror symmetry and when one wants to study Kozsul duality for algebras that are more general then those that fit into the classical framework.
Let C denote the complex numbers. Suppose we take a cdg-algebra B that is C[x] with a curving x^2, where x is now a variable in odd degree and so does not fit into the matrix factorization framework. As I understand the situation, then the Koszul dual of this should be an algebra A which is C[y]/(y^2-1), where y is in even degree.In the framework of Positselski, as I understand it one can compute A as Ext(C,C) and pass back to B by taking the Cobar construction. The curving arises from the fact that the coalgebra dual to A is no longer co-augmented. I think that in this simple example one equivalently can also take a more down to earth approach, as was taken by Dyckerhoff in his paper on matrix factorizations, using explicit Koszul resolutions.
My question is about a confusion I have about the Hochschild cohomology of this dg-category. As I understand it, the Hochschild cohomology of D(B) should be isomorphic to the center of A which is A. Yet if one uses the complex defined in Segal's paper, I believe one gets C[y]/y^2, with y an even variable. The complex he claims should compute HH*(D(B)) is given by the usual Hochschild complex with differential on the algebra B + Gerstenhaber bracket with x^2. To compute, I first computed ordinary HH*(C[x]) and the Gerstenhaber bracket on HH* and then concluded the answer using a spectral sequence.
Segal's justification seems to be that this is what you get when you regard B as a curved A-infinity algebra, but as I understand Positselski, there seems to be some subtelties with this and that one can often end up with no objects in the category. So I was made a little bit nervous by the justification. Most likely everything is ok and I am missing something stupid, but it would be great if someone would be so kind as to point out where my mistake is.