Let $C$ be a pre-triangulated dg-category over a field $k$ whose Hochschild cohomology groups $\operatorname{HH}^*(C)$ are concentrated in non-negative degree (cohomologically). Is $C$ Morita equivalent to a dg-category which is linear over $\operatorname{HH}^0(C)$? It's a folklore statement that this is equivalent to constructing an $E_2$-homomorphism $\operatorname{HH}^0(C) \to \underline{\operatorname{HH}}^*(C)$, where $\underline{\operatorname{HH}}^*(C)$ denotes Hochschild cochains. This seems amenable to "standard obstruction theory" arguments, but I wasn't able to find this claim in the literature (or maybe it's false).
$\begingroup$
$\endgroup$
4
asked Apr 22, 2020 at 17:38
-
$\begingroup$ This is certainly true, no obstruction theory needed. As you point out, specifying an $R$-linear structure is equivalent to constructing an $E_2$ homomorphism $R\to CH^*(C)$. Now for any coconnective $E_n$ algebra $A$, the map $H^0(A)\to A$ is a map of $E_n$ algebras (indeed, uniquely so). There are many ways to prove this fact, which essentially amounts to showing that the natural transformation $H^0(X)\to X$ is a symmetric monoidal transformation between symmetric monoidal endofunctors of the category of coconnective complexes. $\endgroup$– Dmitry VaintrobCommented Apr 22, 2020 at 22:54
-
$\begingroup$ @DmitryVaintrob: This looks promising, thanks! Sorry for being slow, but how are you defining your map H^0(A) \to A? Just by choosing lifts of the cohomology classes? $\endgroup$– Daniel PomerleanoCommented Apr 23, 2020 at 1:54
-
$\begingroup$ Ah I see your question. You can use that the category of coconnective complexes is equivalent (as a symmetric monoidal infinity category) to the full subcategory of the category of unbounded complexes with cohomology in nonnegative degrees. This means an E_n object in one is equivalent to an E_n object in the other. $\endgroup$– Dmitry VaintrobCommented Apr 23, 2020 at 16:12
-
$\begingroup$ After your prodding, I dug around and found Proposition 7.1.3.13 of Lurie's Higher Algebra which proves the existence of connective covers for E_n algebras. My case is just a special case of this because an E_n structure on a discrete object is just a commutative ring. The proof is along the lines you described (demonstrating the compatibility of the t-structure with the monoidal structure). Thanks! $\endgroup$– Daniel PomerleanoCommented Apr 23, 2020 at 18:43
Add a comment
|