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Liran Shaul

Publications and source records attributed to Liran Shaul.

29 records · Page 2Linked to original sources
arXiv2017

Homological dimensions of local (co)homology over commutative DG-rings

Let $A$ be a commutative noetherian ring, let $\mathfrak{a}\subseteq A$ be an ideal, and let $I$ be an injective $A$-module. A basic result in the structure theory of injective modules states that the $A$-module $Γ_{\mathfrak{a}}(I)$ consisting of $\mathfrak{a}$-torsion elements is also an injective $A$-module. Recently, de Jong proved a dual result: If $F$ is a flat $A$-module, then the $\mathfrak{a}$-adic completion of $F$ is also a flat $A$-module. In this paper we generalize these facts to commutative noetherian DG-rings: let $A$ be a commutative non-positive DG-ring such that $\mathrm{H}^0(A)$ is a noetherian ring, and for each $i<0$, the $\mathrm{H}^0(A)$-module $\mathrm{H}^i(A)$ is finitely generated. Given an ideal $\bar{\mathfrak{a}} \subseteq \mathrm{H}^0(A)$, we show that the local cohomology functor $\mathrm{R}Γ_{\bar{\mathfrak{a}}}$ associated to $\bar{\mathfrak{a}}$ does not increase injective dimension. Dually, the derived $\bar{\mathfrak{a}}$-adic completion functor $\mathrm{L}Λ_{\bar{\mathfrak{a}}}$ does not increase flat dimension.

math.AC↗
arXiv2017

Adic reduction to the diagonal and a relation between cofiniteness and derived completion

We prove two results about the derived functor of $a$-adic completion: (1) Let $K$ be a commutative noetherian ring, let $A$ be a flat noetherian $K$-algebra which is $a$-adically complete with respect to some ideal $a\subseteq A$, such that $A/a$ is essentially of finite type over $K$, and let $M,N$ be finitely generated $A$-modules. Then adic reduction to the diagonal holds: $A\otimes^{L}_{ A\hat{\otimes}_{K} A } ( M\hat{\otimes}^{L}_{K} N ) \cong M \otimes^{L}_A N$. A similar result is given in the case where $M,N$ are not necessarily finitely generated. (2) Let $A$ be a commutative ring, let $a\subseteq A$ be a weakly proregular ideal, let $M$ be an $A$-module, and assume that the $a$-adic completion of $A$ is noetherian (if $A$ is noetherian, all these conditions are always satisfied). Then $\mbox{Ext}^i_A(A/a,M)$ is finitely generated for all $i\ge 0$ if and only if the derived $a$-adic completion $Ł\hatΛ_{a}(M)$ has finitely generated cohomologies over $\hat{A}$. The first result is a far reaching generalization of a result of Serre, who proved this in case $K$ is a field or a discrete valuation ring and $A = K[[x_1,\dots,x_n]]$.

math.AC↗
arXiv2016

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 are isomorphic, answering a question of Buchweitz and Flenner. Our results makes it possible for the first time to compute the Hochschild cohomology of $k[[t_1,\dots,t_n]]$ over any noetherian ring $k$, and open the door for a theory of Hochschild cohomology over formal schemes.

math.AC↗
arXiv2016

Tensor product of dualizing complexes over a field

Let $k$ be a field, and let $X,Y$ be two locally noetherian $k$-schemes (respectively $k$-formal schemes) with dualizing complexes $R_X$ and $R_Y$ respectively. We show that $R_X \boxtimes_{k} R_Y$ (respectively its derived completion) is a dualizing complex over $X\times_{k} Y$ if and only if $X\times_{k} Y$ is locally noetherian of finite Krull dimension.

math.AC↗
arXiv2015

Relations between derived Hochschild functors via twisting

Let $k$ be a regular ring, and let $A,B$ be essentially finite type $k$-algebras. For any functor $F:{D}(A)\times\dots\times{D}(A)\to{D}(B)$ between their derived categories, we define its twist $F^{!}:{D}(A)\times\dots\times{D}(A)\to{D}(B)$ with respect to dualizing complexes, generalizing Grothendieck's construction of $f^{!}$. We show that relations between functors are preserved between their twists, and deduce that various relations hold between derived Hochschild (co)-homology and the $f^{!}$ functor. We also deduce that the set of isomorphism classes of dualizing complexes over a ring (or a scheme) form a group with respect to derived Hochschild cohomology, and that the twisted inverse image functor is a group homomorphism.

math.AG↗
arXiv2014

Reduction of Hochschild cohomology over algebras finite over their center

We borrow ideas from Grothendieck duality theory to noncommutative algebra, and use them to prove a reduction result for Hochschild cohomology for noncommutative algebras which are finite over their center. This generalizes a result over commutative algebras by Avramov, Iyengar, Lipman and Nayak.

math.RA↗
arXiv2013

Derived Hochschild functors over commutative adic algebras

Let $\k$ be a commutative ring, and let $(A,\mfrak{a})$ be an adic ring which is a $\k$-algebra. We study complete and torsion versions of the derived Hochschild homology and cohomology functors of $A$ over $\k$. To do this, we first establish weak proregularity of certain ideals in flat base changes of noetherian rings. Next, we develop a theory of DG-affine formal schemes, extending the Greenlees-May duality and the MGM equivalence to this setting. Finally, we define complete and torsion derived Hochschild homology and cohomology functors in this setting, and show that if $\k$ is noetherian and $(A,\mfrak{a})$ is essentially of finite type (in the adic sense) over $\k$, then there are formulas to compute them that stay inside the noetherian category. In the classical case, where $\k$ is a field, we deduce that topological Hochschild cohomology and discrete Hochschild cohomology are isomorphic.

math.AC↗
arXiv2013

Cohomologically Cofinite Complexes

Let A be a commutative noetherian ring, and \a an ideal in it. In this paper we continue the study, begun in [PSY1], of the derived \a-adic completion and the derived \a-torsion functors. Here are our results: (1) a structural characterization of bounded above cohomologically complete complexes; (2) the Cohomologically Complete Nakayama Theorem; and (3) a characterization of cohomologically cofinite complexes.

math.AC↗
arXiv2013

Completion by Derived Double Centralizer

Let A be a commutative ring, and let \a be a weakly proregular ideal in A. (If A is noetherian then any ideal in it is weakly proregular.) Suppose M is a compact generator of the category of cohomologically \a-torsion complexes. We prove that the derived double centralizer of M is isomorphic to the \a-adic completion of A. The proof relies on the MGM equivalence from [PSY] and on derived Morita equivalence. Our result extends earlier work of Dwyer-Greenlees-Iyengar [DGI] and Efimov [Ef].

math.AC↗
arXiv2012

On the Homology of Completion and Torsion

Let A be a commutative ring, and \a a weakly proregular ideal in A. This includes the noetherian case: if A is noetherian then any ideal in it is weakly proregular; but there are other interesting examples. In this paper we prove the MGM equivalence, which is an equivalence between the category of cohomologically \a-adically complete complexes and the category of cohomologically \a-torsion complexes. These are triangulated subcategories of the derived category of A-modules. Our work extends earlier work by Alonso- Jeremias-Lipman, Schenzel and Dwyer-Greenlees.

math.AC↗