A question on curved algebras, papers by Positselski and E. Segal

Question

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.

Answer 1

The answer is that you shouldn't believe everything you read on the arxiv... The result I claim in that paper is wrong, at least in that level of generality. The problem is that I try to use the completion of the bar resolution to compute Hochschild homology, but this isn't a free resolution. Your example is a good one. What I assume is going on here is the following: there are two simple objects in this category, given by the rank one free module with differential either $x$ or $-x$. Presumably they generate the category. Then the category is equivalent to the derived category of your algebra $A$, and its HH is $A$. If we only took one simple object it would generate a subcategory equivalent to $D^b(\mathbb{C})$, whose HH is just $\mathbb{C}$. I believe my complex is computing this answer, but I don't have much justification for this! If you're seriously thinking about this I would love to talk to you, and hopefully resolve this problem.

Answer 2

From what you describe my guess is that you may have made a mistake in the argument with the spectral sequence. When you have a multiplicative spectral sequence, assuming that it converges, the term E_\infty is still not the limit, but rather, the associated graded algebra to a filtration on the limit. The algebra C[y]/(y^2) is the associated graded algebra to an increasing filtration on the algebra C[y]/(y^2-1).

I would not venture to discuss here the much more delicate issues raised in Ed Segal's answer, which require a careful consideration. I do have a point of view on these issues; those desiring to get a taste of this point of view can just read the beginning of the introduction to my paper, where the story of two kinds of derived functors and derived categories is told.