A Quadratic-Linear-Constant (QLC) algebra $U$ is an algebra which can be written as $T(V)/P$ where $T(V) = \bigoplus_{n \in \mathbb{N}} V^{\otimes n}$ is the tensor algebra and $P \subseteq k \oplus V \oplus V^{\otimes 2}$ is a subspace with $P \cap (k \oplus V) = \{ 0 \}$.

A curved differential graded algebra (cdga) $(B,d,c)$ is a graded algebra $B$ with a $\deg 1$ morphism $d$ and a $\deg 2$ curvature element $c$ such that $d(c) = 0$ and $d^2(b) = [c,b]$.

Given a QLC algebra $U$, there is a simple receipt to construct its Koszul dual $(A^!,c,d)$ a cdga which involves writing $P$ as a graph of its $\deg 2$ projection $R$ to $k \oplus V$ and dualizing all the data.

The Koszul duality is supposed to be an equivalence of (stable $\infty$/triangulated) categories between certain categories of modules : $U - \operatorname{Mod}$ and $(A^!,d,c) - cdg\operatorname{Mod}$. And the functors are just tensor - Hom-out adjunction given by the $U - A^!$-bimodule $U\otimes_k A$ with right cdg $A^!$ module structure.

When there is no constant term, the statement is that the derived category of chain complexes of $U$-modules are equivalent to what they call the coderived category of $(A^!,d)$-dg modules. (See FLO YSTAD). Also, when $U$ is the Clifford algebra $Cl(V,Q)$ , a case when there's no linear term, the correspondence is the equivalence between the bounded derived category of Clifford modules of $Cl(V,Q)$ and matrix factorization $MF(Q)$. (See Bertin)

My questions are the following. The reason that coderived category showed up is that there are different ways to complete the category. If I just want to work with small categories to avoid convergence issues and small algebras to avoid coalgebras, what can be a easy description of the category dual to the (small) category of derived $U$-modules? Because of the existence of curvature, we cannot take cohomology so what should be the correct modification? Also, how can one see the matrix factorization in this kind of framework?


I realize I need to ask some more basic questions first. Consider the simplest case $k[x]$. The goal is to have some Morita theory so we consider $\operatorname{Hom}_{k[x]}(k,k) = k[\epsilon]$ with $\deg \epsilon = 1$. The hope is that this should tell us that $k[x] - \operatorname{Mod} = k[\epsilon] - \operatorname{Mod} $ but this is not true. One reason is $\operatorname{Hom}_{k[x]}(k,k[x]/(x-a)) = k \xrightarrow{a} k = 0$ for any $a\in k^\times$. The latter is a simple $k[x]$ module which means $k$ is not a generator in the derived category.

One way to fix this is to throw away those modules and restricting to graded modules. This is the approach in for example Beilinson-Ginzburg-Soergel and what they got is $D^\uparrow(\text{graded}\ k[x]) \cong D^\downarrow(\text{graded}\ k[\epsilon])$. The other approach is modified the $\operatorname{Hom}$s so that an acyclic chain complexes can be non-zero in the target category. That's the approach in FL\O YSTAD or more general Positselski and the corresponding category is the coderived category (which can fix the convergence issue at the same time for some reason).

So conceptual why those methods work? I know graded things have better spectral sequences but what I'm doing to the category when I consider the grading? The second method seems straightforward but I have no idea why we should consider the coderived category? What I mean is whenever we introduce the derived category, we can just say we that formally invert quasi-isomorphisms as long as we ignore the details. So how about the coderived category? And then is the question for Clifford algebras and general QSL algebras. We have curved dg modules as the objects before we derive anything. Are they compatible with the above constructions of categories?

  • $\begingroup$ If I understood what your question is, I could try to answer it, but at present I cannot see a well-formed question. Perhaps just some time is needed for you to come to grips with the material. $\endgroup$ – Leonid Positselski Oct 10 '18 at 16:45
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    $\begingroup$ One thing I can perhaps comment on is the following. The derived category construction is not applicable to curved dg-modules (because curved dg-modules have no cohomology modules, so one cannot speak about their quasi-isomorphisms in the conventional sense). But the coderived (as well as the contraderived) category construction applies to curved dg-modules very well (because one can consider short exact sequences of cdg-modules, the totalizations of such short exact sequences, etc). $\endgroup$ – Leonid Positselski Oct 10 '18 at 16:51

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