Gaps in Hochschild cohomology imply smoothness for commutative algebras

Hasse--Schmidt derivations, divided powers and differential smoothness

Let $k$ be a commutative ring, $A$ a commutative $k$-algebra and $D$ the filtered ring of $k$-linear differential operators of $A$. We prove that: (1) The graded ring $\gr D$ admits a canonical embedding $θ$ into the graded dual of the symmetric algebra of the module $Ω_{A/k}$ of differentials of $A$ over $k$, which has a canonical divided power structure. (2) There is a canonical morphism $\vartheta$ from the divided power algebra of the module of $k$-linear Hasse-Schmidt integrable derivations of $A$ to $\gr D$. (3) Morphisms $θ$ and $\vartheta$ fit into a canonical commutative diagram.

Hochschild cohomology commutes with adic completion

For a flat commutative $k$-algebra $A$ such that the enveloping algebra $A\otimes_k A$ is noetherian, given a finitely generated bimodule $M$, we show that the adic completion of the Hochschild cohomology module $HH^n(A/k,M)$ is naturally isomorphic to $HH^n(\widehat{A}/k,\widehat{M})$. To show this, we (1) make a detailed study of derived completion as a functor $D(A) \to D(\widehat{A})$ over a non-noetherian ring $A$; (2) prove a flat base change result for weakly proregular ideals; and (3) Prove that Hochschild cohomology and analytic Hochschild cohomology of complete noetherian local rings…

Smooth flat maps over commutative DG-rings

We study smooth maps that arise in derived algebraic geometry. Given a map $A \to B$ between non-positive commutative noetherian DG-rings which is of flat dimension $0$, we show that it is smooth in the sense of Toën-Vezzosi if and only if it is homologically smooth in the sense of Kontsevich. We then show that $B$, being a perfect DG-module over $B\otimes^{\mathrm{L}}_A B$ has, locally, an explicit semi-free resolution as a Koszul complex. As an application we show that a strong form of Van den Bergh duality between (derived) Hochschild homology and cohomology holds in this setting.

Completion and torsion over commutative DG rings

Let $\operatorname{CDG}_{cont}$ be the category whose objects are pairs $(A,\bar{\mathfrak{a}})$, where $A$ is a commutative DG-algebra and $\bar{\mathfrak{a}}\subseteq \mathrm{H}^0(A)$ is a finitely generated ideal, and whose morphisms $f:(A,\bar{\mathfrak{a}}) \to (B,\bar{\mathfrak{b}})$ are morphisms of DG-algebras $A \to B$, such that $(\mathrm{H}^0(f)(\bar{\mathfrak{a}})) \subseteq \bar{\mathfrak{b}}$. Letting $\mathrm{Ho}(\operatorname{CDG}_{cont})$ be its homotopy category, obtained by inverting adic quasi-isomorphisms, we construct a functor $\mathrm{L}Λ:\mathrm{Ho}(\operatorname{CDG}_…

A descent theorem for formal smoothness

Let u be a local homomorphism of noetherian local rings forming part of a commutative square vf=gu. We give some conditions on the square which imply that u is formally smooth. This result encapsulates a variety of (apparently unrelated) results in commutative algebra greatly improving some of them: Greco's theorem on descent of quasi-excellence property by finite surjective morphisms, Kunz's characterization of regular local rings in positive characteristic by means of the Frobenius homomorphism (and in fact the relative version obtained by Andre and Radu), etc. In the second part of the pape…

交换DG-环上的有限维数

Finitistic dimensions over commutative DG-rings

本文研究有限振幅交换诺特非正DG-环的有限维数，证明有限平坦维数DG-模的投射维数上界，构造指定投射维数的DG-模并给出大有限投射维数的上下界及可达性，还得到同调光滑代数导出Hochschild（上）同调的新零化结果。

The negative side of cohomology for Calabi-Yau categories

We study integer-graded cohomology rings defined over Calabi-Yau categories. We show that the cohomology in negative degree is a trivial extension of the cohomology ring in non-negative degree, provided the latter admits a regular sequence of central elements of length two. In particular, the product of elements of negative degrees are zero. As corollaries we apply this to Tate-Hochschild cohomology rings of symmetric algebras, and to Tate cohomology rings over group algebras. We also prove similar results for Tate cohomology rings over commutative local Gorenstein rings.