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

TL;DR

This paper extends classical results on injective modules and flat modules over commutative noetherian rings to the setting of commutative noetherian DG-rings, showing that certain derived functors preserve homological dimensions.

Contribution

It generalizes the invariance of injective and flat dimensions under local cohomology and completion functors to the context of commutative noetherian DG-rings.

Findings

Local cohomology functor does not increase injective dimension.

Derived completion functor does not increase flat dimension.

Results extend classical module theory to DG-ring setting.

Abstract

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 $\Gamma_{\mathfrak{a}}(I)$ 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 $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}$ associated to $\bar{\mathfrak{a}}$ does not increase injective dimension. Dually, the derived $\bar{\mathfrak{a}}$-adic completion functor $\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}$ does not increase flat dimension.

Equations

$$\mathrm{R}{\Gamma }_{\mathfrak{a}}:D(A)\to D(A)$$

$${\mathrm{injdim}}_A(\mathrm{R}{\Gamma }_{\mathfrak{a}}(M))\le {\mathrm{injdim}}_A(M).$$

$$\bar{A}:={\mathrm{H}}^0(A).$$

$$\mathrm{R}{\Gamma }_{\bar{\mathfrak{a}}}:D(A)\to D(A)$$

$${\mathrm{injdim}}_A\left(\mathrm{R}{\Gamma }_{\bar{\mathfrak{a}}}(M)\right)\le {\mathrm{injdim}}_A(M).$$

$$\mathrm{L}{\Lambda }_{\mathfrak{a}}:D(A)\to D(A).$$

$${\mathrm{fldim}}_A\left(\mathrm{L}{\Lambda }_{\mathfrak{a}}(M)\right)\le {\mathrm{fldim}}_A(M).$$

$$\mathrm{L}{\Lambda }_{\bar{\mathfrak{a}}}:D(A)\to D(A)$$

$${\mathrm{fldim}}_A\left(\mathrm{L}{\Lambda }_{\bar{\mathfrak{a}}}(M)\right)\le {\mathrm{fldim}}_A(M).$$

$$A=\underset{n=-\infty }{\overset{\infty }{⨁}}{A}^n$$

$$0\to \underset{i=0}{\overset{\infty }{⨁}}A\stackrel{d}{\to }\underset{i=0}{\overset{\infty }{⨁}}A\to 0$$

$$d({\delta }_i)=\{\begin{array}{ll}{\delta }_0& \text{if i=0}\\ {\delta }_{i-1}-a\cdot {\delta }_i& \text{if i≥1}\end{array}$$

$$\mathrm{Tel}(A;\mathbf{a}):=\mathrm{Tel}(A;a_1){\otimes }_A\mathrm{Tel}(A;a_2){\otimes }_A\cdots {\otimes }_A\mathrm{Tel}(A;a_n).$$

$$\mathrm{R}{\Gamma }_{\bar{\mathfrak{a}}}(-)\cong \mathrm{Tel}(A^0;\mathbf{a}){\otimes }_{A^0}-$$

$$\mathrm{L}{\Lambda }_{\bar{\mathfrak{a}}}(-)\cong {\mathrm{Hom}}_{A^0}(\mathrm{Tel}(A^0;\mathbf{a}),-)$$

$$\inf (M):=\inf \{i\in \mathbb{Z}\mid {\mathrm{H}}^i(M)\ne 0\},\sup (M):=\sup \{i\in \mathbb{Z}\mid {\mathrm{H}}^i(M)\ne 0\}$$

$$\inf \{n\in \mathbb{Z}\mid {\mathrm{Ext}}_A^i(N,M)=0\text{ for any bounded DG-module }N\text{ and any }i>n-\inf N\}$$

$$\inf \{n\in \mathbb{Z}\mid {\mathrm{Tor}}_A^i(N,M)=0\text{ for any bounded DG-module }N\text{ and any }i>n-\inf N\}$$

$$M,N,K\in D(A),$$

$${\eta }_{M,N,K}:\mathrm{R}{\mathrm{Hom}}_A(M,N){\otimes }_A^{\mathrm{L}}K\to \mathrm{R}{\mathrm{Hom}}_A(M,N{\otimes }_A^{\mathrm{L}}K)$$

$$\mathrm{R}{\mathrm{Hom}}_A(M,N){\otimes }_A^{\mathrm{L}}K\cong {\mathrm{Hom}}_A(P,N){\otimes }_{A}F\stackrel{\varphi }{\to }$$

$${\mathrm{Hom}}_A(P,N{\otimes }_{A}F)\cong \mathrm{R}{\mathrm{Hom}}_A(M,N{\otimes }_A^{\mathrm{L}}K)$$

$${\eta }_{M,N,K}:\mathrm{R}{\mathrm{Hom}}_A(M,N){\otimes }_A^{\mathrm{L}}K\to \mathrm{R}{\mathrm{Hom}}_A(M,N{\otimes }_A^{\mathrm{L}}K)$$

$${\zeta }_M:\mathrm{R}{\mathrm{Hom}}_A(M,N){\otimes }_A^{\mathrm{L}}K\to \mathrm{R}{\mathrm{Hom}}_A(M,N{\otimes }_A^{\mathrm{L}}K)$$

$${\iota }_K:\mathrm{R}{\mathrm{Hom}}_A(M,N){\otimes }_A^{\mathrm{L}}K\to \mathrm{R}{\mathrm{Hom}}_A(M,N{\otimes }_A^{\mathrm{L}}K).$$

$${\mathrm{injdim}}_B\left(\mathrm{R}{\mathrm{Hom}}_A(B,M)\right)\le {\mathrm{injdim}}_A(M).$$

$$\mathrm{R}{\mathrm{Hom}}_B(-,\mathrm{R}{\mathrm{Hom}}_A(B,M))\simeq \mathrm{R}{\mathrm{Hom}}_A(-,M)$$

$${\mathrm{injdim}}_A(M)={\mathrm{injdim}}_{{\mathrm{H}}^0(A)}\left(\mathrm{R}{\mathrm{Hom}}_A({\mathrm{H}}^0(A),M)\right)$$

$${\mathrm{injdim}}_A(M)\ge {\mathrm{injdim}}_{\bar{A}}\left(\mathrm{R}{\mathrm{Hom}}_A(\bar{A},M)\right).$$

$$N\simeq {\mathrm{H}}^m(N)[-m]$$

$$\mathrm{R}{\mathrm{Hom}}_A(N,M)\simeq \mathrm{R}{\mathrm{Hom}}_A({\mathrm{H}}^m(N)[-m],M)$$

Full text

Mathematics Subject Classification 2010: 13D05, 13D45, 13B35, 16E45

Keywords: Local cohomology, Derived completion, Homological dimension, Commutative DG-rings

0. Introduction

0.1. Torsion of injective modules

Let $A$ be a commutative noetherian ring, and $\mathfrak{a}\subseteq A$ an ideal. The $\mathfrak{a}$-torsion functor is the functor $\Gamma_{\mathfrak{a}}(M):=\varinjlim\operatorname{Hom}_{A}(A/\mathfrak{a}^{n},M)$ which maps an $A$-module $M$ to its submodule consisting $\mathfrak{a}$-torsion elements. An important consequence of Matlis’ structure theory of injective modules is the following result:

Theorem A**.**

Let $A$ be a commutative noetherian ring, let $\mathfrak{a}\subseteq A$ be an ideal, and let $J$ be an injective $A$-module. Then $\Gamma_{\mathfrak{a}}(J)$ is also an injective $A$-module.

This basic result depends on $A$ being noetherian. It is false in general if $A$ is not noetherian, even if $\mathfrak{a}$ is finitely generated (see Remark 2.8 below).

We denote by $\operatorname{\mathsf{D}}(A)$ the unbounded derived category of $A$-modules, and by $\operatorname{\mathsf{D}}^{\mathrm{b}}(A)$ its full triangulated subcategory of complexes with bounded cohomology. For $M\in\operatorname{\mathsf{D}}(A)$, its injective dimension, denoted by $\operatorname{inj\,dim}_{A}(M)$ was defined in [1, Definition 2.1.I]. The functor $\Gamma_{\mathfrak{a}}$ has a right derived functor

[TABLE]

It is calculated using K-injective resolutions. An immediate corollary of Theorem A is the following result:

Corollary 0.1**.**

Let $A$ be a commutative noetherian ring, and let $\mathfrak{a}\subseteq A$ be an ideal. Then for any $M\in\operatorname{\mathsf{D}}^{\mathrm{b}}(A)$, we have the following inequality:

[TABLE]

We now switch our attention to commutative non-positive noetherian DG-rings. The definition of a commutative DG-ring is recalled in Section 1.1 below. Just like commutative rings represent affine schemes, commutative DG-rings represent affine derived schemes. Given a commutative DG-ring $A$, the fact that $A^{i}=0$ for $i>0$ implies that $\mathrm{H}^{0}(A)$ is a commutative ring. Following [11], we set

[TABLE]

A commutative DG-ring $A$ is called noetherian if the commutative ring $\bar{A}$ is noetherian, and $\mathrm{H}^{i}(A)$ is a finitely generated $\bar{A}$-module for every $i<0$. We will denote by $\operatorname{\mathsf{D}}(A)$ the unbounded derived category of DG-modules over $A$, and by $\operatorname{\mathsf{D}}^{\mathrm{b}}(A)$ its full triangulated subcategory of DG-modules with bounded cohomology.

Commutative noetherian DG-rings arise naturally in algebraic geometry: If $\mathbb{K}\hskip 0.50003pt$ is a commutative noetherian ring, and $A,B$ are finite type $\mathbb{K}\hskip 0.50003pt$-algebras then $A\otimes^{\mathrm{L}}_{\mathbb{K}\hskip 0.35002pt}B$ is a commutative noetherian DG-ring, and if $\mathbb{K}\hskip 0.50003pt$ is not a field, it is often cannot be represented using ordinary commutative rings.

Let $A$ be a commutative DG-ring, and let $\bar{\mathfrak{a}}\subseteq\bar{A}$ be a finitely generated ideal. In our recent paper [9] we introduced a triangulated functor

[TABLE]

called the right derived $\bar{\mathfrak{a}}$-torsion or local cohomology at $\bar{\mathfrak{a}}$ functor. When $A$ is a commutative noetherian ring, this functor coincides with the usual local cohomology functor. The definition of $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}$ as well as an explicit construction of it using the telescope complex is recalled in Section 1.2 below. To any $M\in\operatorname{\mathsf{D}}(A)$ we can associate its injective dimension, denoted by $\operatorname{inj\,dim}_{A}(M)$. The definition is recalled in Section 1.3.

The first main result of this paper generalizes Corollary 0.1 to commutative noetherian DG-rings:

Theorem 0.2**.**

Let $A$ be a commutative noetherian DG-ring, and let $\bar{\mathfrak{a}}\subseteq\mathrm{H}^{0}(A)$ be an ideal. Then for any $M\in\operatorname{\mathsf{D}}^{\mathrm{b}}(A)$, there is an inequality

[TABLE]

We will prove this result in Theorem 2.7 below.

0.2. Completion of flat modules

Given a commutative ring $A$ and a finitely generated ideal $\mathfrak{a}\subseteq A$, the $\mathfrak{a}$-adic completion functor is the functor $\Lambda_{\mathfrak{a}}(M):=\varprojlim A/{\mathfrak{a}}^{n}\otimes_{A}M$. This construction is dual to the $\mathfrak{a}$-torsion functor. This is best demonstrated by the Greenlees-May duality([5]), which states that if $A$ is noetherian then the derived functors of $\mathfrak{a}$-torsion and $\mathfrak{a}$-adic completion are adjoint to each other. This intimate connection between $\mathfrak{a}$-torsion and $\mathfrak{a}$-adic completion raises the question: is there a dual result to Theorem A for the $\mathfrak{a}$-adic completion functor? We have the following result:

Theorem B**.**

Let $A$ be a commutative noetherian ring, let $\mathfrak{a}\subseteq A$ be an ideal, and let $F$ be a flat $A$-module. Then $\Lambda_{\mathfrak{a}}(F)$ is also a flat $A$-module.

Theorem A is a classical result which has been known for many decades. In contrast, Theorem B was first proved by de Jong in 2013 (see [10, Tag 0AGW]). If $F$ is a finitely generated $A$-module then this result is trivial. Prior to de Jong’s proof, the best known result was by Enochs, who showed in [3] that this holds under the additional assumption that $A$ has finite Krull dimension. After de Jong’s proof, other proofs were given by Gabber and Ramero, and by Yekutieli ([13, Theorem 0.1]).

For a complex of $A$-modules $M$, its flat dimension, denoted by $\operatorname{fl\,dim}_{A}(M)$ was defined in [1, Definition 2.1.F]. The functor $\Lambda_{\mathfrak{a}}$ has a left derived functor

[TABLE]

It is calculated using K-flat resolutions. The following is am immediate corollary of Theorem B:

Corollary 0.3**.**

Let $A$ be a commutative noetherian ring, and let $\mathfrak{a}\subseteq A$ be an ideal. Then for any $M\in\operatorname{\mathsf{D}}^{\mathrm{b}}(A)$, we have the following inequality:

[TABLE]

Now let $A$ be a commutative DG-ring. Given a finitely generated ideal $\bar{\mathfrak{a}}\subseteq\bar{A}$, there is a triangulated functor

[TABLE]

called the derived $\bar{\mathfrak{a}}$-adic completion functor. Its definition is recalled in Section 1.2 below. If $A$ is a commutative noetherian ring, it coincides with the usual left derived functor of adic completion. To any $M\in\operatorname{\mathsf{D}}(A)$ we can associate its flat dimension, denoted by $\operatorname{fl\,dim}_{A}(M)$. The definition is recalled in Section 1.3.

The second main result of this paper generalizes Corollary 0.3 to commutative noetherian DG-rings:

Theorem 0.4**.**

Let $A$ be a commutative noetherian DG-ring, and let $\bar{\mathfrak{a}}\subseteq\mathrm{H}^{0}(A)$ be an ideal. Then for any $M\in\operatorname{\mathsf{D}}^{\mathrm{b}}(A)$, there is an inequality

[TABLE]

This will be proved in Theorem 3.3 below.

1. Preliminaries

In this section we recall some basic facts concerning commutative DG-rings.

1.1. Commutative DG-rings

A DG-ring $A$ is a $\mathbb{Z}$-graded ring

[TABLE]

together with an additive differential $d:A\to A$ of degree $+1$, such that $d\circ d=0$, and such that the Leibniz rule holds: $d(a\cdot b)=d(a)\cdot b+(-1)^{i}\cdot a\cdot d(b)$ for all $a\in A^{i}$ and $b\in A^{j}$. A DG-ring $A$ is called non-positive if $A^{i}=0$ for all $i>0$. We say that $A$ is commutative if $b\cdot a=(-1)^{i\cdot j}\cdot a\cdot b$ for all $a\in A^{i}$ and $b\in A^{j}$, and moreover $a\cdot a=0$ if $i$ is odd.

In this paper, all DG-rings are assumed to be commutative and non-positive. Given a commutative DG-ring $A$, a DG-module over it is graded $A$-module $M$ with a differential $d:M\to M$ of degree $+1$ satisfying a graded Leibniz rule. The category of all DG-modules is denoted by $\operatorname{DGMod}(A)$. Inverting quasi-isomorphisms in it, we obtain the derived category of DG-modules over $A$, denoted by $\operatorname{\mathsf{D}}(A)$.

If $A$ is a commutative DG-ring, recall from the introduction that we denote by $\bar{A}$ the commutative ring $\mathrm{H}^{0}(A)$. The DG-ring $A$ is called noetherian if $\bar{A}$ is noetherian and $\mathrm{H}^{i}(A)$ is a finitely generated $\bar{A}$-module for all $i<0$.

1.2. Local (co)homology over commutative DG-rings and the telescope complex

Let $A$ be a commutative DG-ring, and let $\bar{\mathfrak{a}}\subseteq\bar{A}$ be a finitely generated ideal. The category of derived $\bar{\mathfrak{a}}$-torsion DG-modules, denoted by $\operatorname{\mathsf{D}}_{\bar{\mathfrak{a}}-\operatorname{tor}}(A)$, is the full triangulated subcategory of $\operatorname{\mathsf{D}}(A)$, consisting of DG-modules $M$, such that for all $n\in\mathbb{Z}$, the $\bar{A}$-module $\mathrm{H}^{n}(M)$ is $\bar{\mathfrak{a}}$-torsion.

According to [9, Theorem 2.13(1)], the inclusion functor $i_{\bar{\mathfrak{a}}}:\operatorname{\mathsf{D}}_{\bar{\mathfrak{a}}-\operatorname{tor}}(A)\hookrightarrow\operatorname{\mathsf{D}}(A)$ has a right adjoint $F_{\bar{\mathfrak{a}}}:\operatorname{\mathsf{D}}(A)\to\operatorname{\mathsf{D}}_{\bar{\mathfrak{a}}-\operatorname{tor}}(A)$. The composition $i_{\bar{\mathfrak{a}}}\circ F_{\bar{\mathfrak{a}}}:\operatorname{\mathsf{D}}(A)\to\operatorname{\mathsf{D}}(A)$ is denoted by $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}:\operatorname{\mathsf{D}}(A)\to\operatorname{\mathsf{D}}(A)$ and called the local cohomology functor of $A$ with respect to $\bar{\mathfrak{a}}$.

By [9, Theorem 2.13(2)], the functor $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}$ has a left adjoint $\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}:\operatorname{\mathsf{D}}(A)\to\operatorname{\mathsf{D}}(A)$ called the derived completion (or local homology) functor of $A$ with respect to $\bar{\mathfrak{a}}$.

Below, we will give explicit formulas for the functors $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}},\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}$.

Remark 1.1**.**

If $A$ is an ordinary commutative noetherian ring, by [6, Theorem 7.12], these constructions of local cohomology and derived completion with respect to $\bar{\mathfrak{a}}$ coincide with the usual right derived functor of the functor $\Gamma_{\bar{\mathfrak{a}}}$ and left derived functor of $\Lambda_{\bar{\mathfrak{a}}}$.

To give an explicit formula for the local cohomology $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}$ and derived completion $\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}$, we recall the construction of the telescope complex from [6, Section 5]: given a commutative ring $A$ and some $a\in A$, the telescope complex $\operatorname{Tel}(A;a)$ is the cochain complex

[TABLE]

with non-zero components in degrees $0,1$. Letting $\{\delta_{i}\mid i\geq 0\}$ be the basis of the countably generated free A-module $\bigoplus_{i=0}^{\infty}A$, the differential $d$ is defined by

[TABLE]

Given a finite sequence $\mathbf{a}=(a_{1},\dots,a_{n})$ of elements of $A$, the telescope complex associated to $\mathbf{a}$ is the complex

[TABLE]

This is a bounded complex of free $A$-modules. The telescope complex has the following base change property: if $f:A\to B$ is a homomorphism between commutative rings, and $\mathbf{b}=(f(a_{1}),f(a_{2}),\dots,f(a_{n}))$, there is an isomorphism of complexes of $B$-modules $\operatorname{Tel}(A;\mathbf{a})\otimes_{A}B\cong\operatorname{Tel}(B;\mathbf{b})$.

Let $A$ be a commutative DG-ring, and let $\bar{\mathfrak{a}}\subseteq\bar{A}$ be a finitely generated ideal. Let $\bar{\mathbf{a}}=(\bar{a}_{1},\dots,\bar{a}_{n})$ be a finite sequence of elements in $\bar{A}$ that generates $\bar{\mathfrak{a}}$, and using the surjection $A^{0}\to\bar{A}$, choose some lifts $\mathbf{a}=(a_{1},\dots,a_{n})$ of $\bar{\mathbf{a}}$ to the commutative ring $A^{0}$. By [9, Proposition 2.4] and the base change property of the telescope complex, there are isomorphisms

[TABLE]

and

[TABLE]

of functors $\operatorname{\mathsf{D}}(A)\to\operatorname{\mathsf{D}}(A)$.

1.3. Homological dimensions over commutative DG-rings

Let $A$ be a commutative DG-ring. Given a DG-module $M$ over $A$, we set

[TABLE]

and $\operatorname{amp}(M):=\sup(M)-\inf(M)$.

Let $M$ be a DG-module over $A$. Following [1, Section 2.I], we let the injective dimension of $M$, denoted by $\operatorname{inj\,dim}_{A}(M)$ to be the number:

[TABLE]

where $\operatorname{Ext}^{i}_{A}(N,M):=\mathrm{H}^{i}(\mathrm{R}\operatorname{Hom}_{A}(N,M))$. By [1, Theorem 2.4.I]111Unlike [1], in this paper we use a cohomological notation, hence the difference between the formulas. this coincides with the usual definition in case $A$ is a commutative ring. Similarly, we let the flat dimension of $M$, denoted by $\operatorname{fl\,dim}_{A}(M)$ to be the number:

[TABLE]

where $\operatorname{Tor}^{i}_{A}(N,M):=\mathrm{H}^{-i}(N\otimes^{\mathrm{L}}_{A}M)$. The projective dimension of $M$, denoted by $\operatorname{proj\,dim}_{A}(M)$ is defined similarly.

1.4. Subcategories of $\operatorname{\mathsf{D}}(A)$

Given a commutative DG-ring $A$, we denote by $\operatorname{\mathsf{D}}^{-}(A)$ (respectively by $\operatorname{\mathsf{D}}^{+}(A)$) the full triangulated subcategory of $A$ consisting of DG-modules with bounded above (resp. bounded below) cohomology. The full triangulated subcategory of DG-modules with bounded cohomology is $\operatorname{\mathsf{D}}^{\mathrm{b}}(A)=\operatorname{\mathsf{D}}^{-}(A)\cap\operatorname{\mathsf{D}}^{+}(A)$. Assume further that $A$ is noetherian. In particular, $\bar{A}$ is a noetherian ring. We say that $M\in\operatorname{\mathsf{D}}(A)$ has finitely generated cohomology if for all $i\in\mathbb{Z}$, $\mathrm{H}^{i}(M)$ is a finitely generated $\bar{A}$-module. The full triangulated subcategory of $\operatorname{\mathsf{D}}(A)$ consisting of DG-modules with finitely generated cohomologies is denoted by $\mathrm{D}_{\mathrm{f}}(A)$. We let $\operatorname{\mathsf{D}}^{+}_{\mathrm{f}}(A)=\operatorname{\mathsf{D}}^{+}(A)\cap\operatorname{\mathsf{D}}_{\mathrm{f}}(A)$, $\operatorname{\mathsf{D}}^{-}_{\mathrm{f}}(A)=\operatorname{\mathsf{D}}^{-}(A)\cap\operatorname{\mathsf{D}}_{\mathrm{f}}(A)$ and $\operatorname{\mathsf{D}}^{\mathrm{b}}_{\mathrm{f}}(A)=\operatorname{\mathsf{D}}^{b}(A)\cap\operatorname{\mathsf{D}}_{\mathrm{f}}(A)$.

1.5. The tensor-evaluation morphism

Let $A$ be a commutative DG-ring. Given

[TABLE]

there is a natural morphism

[TABLE]

in $\operatorname{\mathsf{D}}(A)$, defined as follows: Let $P\cong M$ be a K-projective resolution, and let $F\cong K$ be a K-flat resolution. Then, $\eta_{M,N,K}$ is the composition

[TABLE]

where the map $\phi$ is the usual tensor-evaluation morphism (see [11, Equation (5.6)] for its formula in the DG-case). The next result generalizes [8, Proposition 6.7].

Proposition 1.4**.**

Let $A$ be a commutative noetherian DG-ring, and let $M,N,K\in\operatorname{\mathsf{D}}(A)$. Assume one of the following holds:

1. (1)

$M\in\operatorname{\mathsf{D}}^{-}_{\mathrm{f}}(A)$, $N\in\operatorname{\mathsf{D}}^{\mathrm{b}}(A)$ and $\operatorname{fl\,dim}_{A}(K)<\infty$. 2. (2)

$\operatorname{proj\,dim}_{A}(M)<\infty$, $N\in\operatorname{\mathsf{D}}^{\mathrm{b}}(A)$ and $K\in\mathrm{D}^{-}_{\mathrm{f}}(A)$.

Then the morphism

[TABLE]

is an isomorphism in $\operatorname{\mathsf{D}}(A)$.

Proof.

1. (1)

Fixing such $N,K$, we have a natural morphism

[TABLE]

These assumptions on $N,K$ ensure that the functors $\mathrm{R}\operatorname{Hom}_{A}(-,N)\otimes^{\mathrm{L}}_{A}K$ and $\mathrm{R}\operatorname{Hom}_{A}(-,N\otimes^{\mathrm{L}}_{A}K)$ are both contravariant way-out right functors. Clearly, $\zeta_{A}$ is an isomorphism. Hence, by a DG-version of the lemma on way-out functors (for instance, [11, Theorem 2.11]), we deduce that $\zeta_{M}$ is an isomorphism for any $M\in\operatorname{\mathsf{D}}^{-}_{\mathrm{f}}(A)$. 2. (2)

Fixing such $M,N$, we have a natural morphism

[TABLE]

These assumptions on $M,N$ ensure that the functors $\mathrm{R}\operatorname{Hom}_{A}(M,N)\otimes^{\mathrm{L}}_{A}-$ and $\mathrm{R}\operatorname{Hom}_{A}(M,N\otimes^{\mathrm{L}}_{A}-)$ are both covariant way-out left functors, and it is clear that $\iota_{A}$ is an isomorphism. Hence, by the lemma on way-out functors, $\iota_{K}$ is an isomorphism for any $K\in\mathrm{D}^{-}_{\mathrm{f}}(A)$.

∎

2. Injective dimension of local cohomology

In this section we will prove Theorem 0.2. We begin by proving some basic results about injective dimension over commutative DG-rings.

Proposition 2.1**.**

Let $A\to B$ be a homomorphism between commutative DG-rings, and let $M$ be a DG-module over $A$. Then

[TABLE]

Proof.

This follows from the adjunction

[TABLE]

∎

Over a commutative ring $A$, it is well known that one can detect the injective dimension of a complex $M$ by checking the vanishing of $\operatorname{Ext}^{i}_{A}(N,M)$ for all $A$-modules $N$ (that is, complexes with zero amplitude). The proof of the next result is based on the same idea over a DG-ring $A$, together with the observation that a DG-module whose amplitude is zero is isomorphic in the derived category to the shift of an $\bar{A}$-module.

Theorem 2.2**.**

Let $A$ be a commutative DG-ring, and let $M$ be a DG-module over $A$. Then there is an equality

[TABLE]

Proof.

Applying Proposition 2.1 to the map $A\to\bar{A}$, we have that

[TABLE]

To prove the converse, assume $\operatorname{inj\,dim}_{\bar{A}}\left(\mathrm{R}\operatorname{Hom}_{A}(\bar{A},M)\right)=n<\infty$. Let $N$ be a bounded DG-module. We must show that $\operatorname{Ext}^{i}_{A}(N,M)=0$ for all $i>n-\inf N$. We will prove this by induction on $\operatorname{amp}N$. If $\operatorname{amp}N=0$, there is some $m\in\mathbb{Z}$ such that

[TABLE]

It follows by adjunction that

[TABLE]

so the fact that $\operatorname{inj\,dim}_{\bar{A}}\left(\mathrm{R}\operatorname{Hom}_{A}(\bar{A},M)\right)=n$ implies that $\operatorname{Ext}^{i}_{A}(N,M)=0$ for all $i>n-\inf N$.

Given $l>0$, assume now that for any bounded DG-module $N$ with $\operatorname{amp}N<l$ we have that $\operatorname{Ext}^{i}_{A}(N,M)=0$ for all $i>n-\inf N$, and let $N$ be a bounded DG-module with $\operatorname{amp}N=l$.

According to [4, Page 299], using truncations, the DG-module $N$ fits into a distinguished triangle

[TABLE]

in $\operatorname{\mathsf{D}}(A)$, such that the $\operatorname{amp}N^{\prime}<l$, $\operatorname{amp}N^{\prime\prime}<l$, and moreover $\inf N^{\prime}\geq\inf N$ and $\inf N^{\prime\prime}\geq\inf N$. Applying the contravariant triangulated functor $\mathrm{R}\operatorname{Hom}_{A}(-,M)$ to the triangle (2.3), we obtain a distinguished triangle

[TABLE]

The result now follows from the induction hypothesis and the long exact sequence in cohomology associated to the distinguished triangle (2.4).

∎

Before stating the next lemma, we shall need the following terminology:

Remark 2.5**.**

If $A$ is a commutative DG-ring, and $\bar{\mathfrak{a}}\subseteq\bar{A}$ is a finitely generated ideal, we can form the local cohomology functor of $A$ with respect to $\bar{\mathfrak{a}}$ and the local cohomology functor of $\bar{A}$ with respect to $\bar{\mathfrak{a}}$. The former is a functor $\operatorname{\mathsf{D}}(A)\to\operatorname{\mathsf{D}}(A)$, while the latter is a functor $\operatorname{\mathsf{D}}(\bar{A})\to\operatorname{\mathsf{D}}(\bar{A})$.

According to our notations, both should be denoted by $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}$. In cases where there will be such ambiguity, we solve it by using the notation

[TABLE]

for the local cohomology functor of $\bar{A}$ with respect to $\bar{\mathfrak{a}}$, and the notation

[TABLE]

for the local cohomology functor of $A$ with respect to $\bar{\mathfrak{a}}$. Similarly, we will write

[TABLE]

for the derived completion functor of $\bar{A}$ with respect to $\bar{\mathfrak{a}}$, and

[TABLE]

for the derived completion functor of $A$ with respect to $\bar{\mathfrak{a}}$.

Lemma 2.6**.**

Let $A$ be a commutative noetherian DG-ring, and let $\bar{\mathfrak{a}}\subseteq\mathrm{H}^{0}(A)$ be an ideal. Then for any DG-module $M$ over $A$ with bounded cohomology, there is a natural isomorphism

[TABLE]

in $\operatorname{\mathsf{D}}(\bar{A})$.

Proof.

Let $\mathbf{a}$ be a finite sequence of elements of the ring $A^{0}$ whose image in $\bar{A}$ generates the ideal $\bar{\mathfrak{a}}$, and let $\bar{\mathbf{a}}$ be its image in $\bar{A}$. The latter is a finite sequence of elements of the ring $\bar{A}$. By (1.2), there is a natural isomorphism

[TABLE]

By the base change property of the telescope complex, there is an isomorphism of complexes of $\bar{A}$-modules:

[TABLE]

This implies that

[TABLE]

Set $T:=A\otimes_{A^{0}}\operatorname{Tel}(A^{0};\mathbf{a})$. Since $\operatorname{Tel}(A^{0};\mathbf{a})$ is a K-flat complex of finite flat dimension over $A^{0}$, it follows that the DG-module $T$ is K-flat over $A$ and that $\operatorname{fl\,dim}_{A}(T)<\infty$. Hence, it holds that

[TABLE]

Let $M\cong I$ be a K-injective resolution over $A$, let $I\otimes_{A}T\cong J$ be a K-injective resolution over $A$, and let $A\to B\xrightarrow{\simeq}\bar{A}$ be a semi-free DG-algebra resolution of $\bar{A}$ over $A$. In particular, $B$ is K-projective over $A$. These resolutions and the naturality of the tensor evaluation morphism induce a commutative diagram in $\operatorname{DGMod}(A)$:

[TABLE]

Since $I$ is K-injective over $A$ and $T$ is K-projective over $A$, it follows that $\alpha$ is a quasi-isomorphism. Similarly, since $B$ is K-projective over $A$, it follows that $\chi$ is a quasi-isomorphism. K-injectivity of $J$ over $A$ implies that $\psi$ is a quasi-isomorphism. Finally, since $A$ is noetherian and $T$ has finite flat dimension over $A$, it follows by Proposition 1.4(1) that $\gamma$ is a quasi-isomorphism. Hence, by the 2-out-of-3 property, the $\bar{A}$-linear map $\varphi\circ\beta$ is also a quasi-isomorphism. Hence, there are natural isomorphisms

[TABLE]

in $\operatorname{\mathsf{D}}(\bar{A})$. Finally, note that

[TABLE]

Combining all the above natural isomorphisms gives the required result. ∎

We now prove the first main result of this paper.

Theorem 2.7**.**

Let $A$ be a commutative noetherian DG-ring, and let $\bar{\mathfrak{a}}\subseteq\mathrm{H}^{0}(A)$ be an ideal. Then for any $M\in\operatorname{\mathsf{D}}^{\mathrm{b}}(A)$, there is an inequality

[TABLE]

Proof.

According to Theorem 2.2, we have that

[TABLE]

Using Lemma 2.6, we have that

[TABLE]

and by Corollary 0.1 we obtain that

[TABLE]

where the last equality follows from Theorem 2.2. Combining all of the above, we obtain that

[TABLE]

as claimed.

∎

Remark 2.8**.**

Given a commutative ring $A$, and a finitely generated ideal $\mathfrak{a}\subseteq{A}$, unlike Remark 1.1, if $A$ is non-noetherian in general the right derived functor of the $\mathfrak{a}$-torsion functor $\Gamma_{\mathfrak{a}}$ might be different from the local cohomology functor with respect to $\mathfrak{a}$ from Section 1.2. However, if $\mathfrak{a}$ satisfies a technical condition called weak proregularity (see [6, Definition 4.21]), these functors coincide. In [7, Proposition 3.1], there is an example of a commutative ring $A$, a finitely generated (in fact, principal) weakly proregular ideal $\mathfrak{a}\subseteq A$, and an injective $A$-module $I$ such that $\Gamma_{\mathfrak{a}}(I)$ is not an injective $A$-module.

3. Flat dimension of derived completion

The aim of this section is to prove Theorem 0.4. The next result is dual to Theorem 2.2.

Theorem 3.1**.**

Let $A$ be a commutative DG-ring, and let $M$ be a DG-module over $A$. Then there is an equality

[TABLE]

Proof.

For any $N\in\operatorname{\mathsf{D}}(\bar{A})$, the isomorphism

[TABLE]

shows that

[TABLE]

To prove the converse, assume that $\operatorname{fl\,dim}_{\bar{A}}\left(\bar{A}\otimes^{\mathrm{L}}_{A}M\right)=n<\infty$, and let $N$ be a bounded DG-module. If $\operatorname{amp}(N)=0$, there is an isomorphism $N\cong\mathrm{H}^{m}(N)[-m]$ for some $m\in\mathbb{Z}$, which implies that

[TABLE]

Hence, the fact that

[TABLE]

implies that in this case $\operatorname{Tor}^{i}_{A}(N,M)=0$ for all $i>n-\inf N$. Now, proceeding by induction on $\operatorname{amp}(N)$, exactly as in the proof of Theorem 2.2, we obtain the general case for an arbitrary bounded DG-module $N$. ∎

We will now use again the terminology introduced in Remark 2.5.

Lemma 3.2**.**

Let $A$ be a commutative noetherian DG-ring, and let $\bar{\mathfrak{a}}\subseteq\mathrm{H}^{0}(A)$ be an ideal. Then for any DG-module $M$ over $A$ with bounded cohomology, there is a natural isomorphism

[TABLE]

in $\operatorname{\mathsf{D}}(\bar{A})$.

Proof.

Let $\mathbf{a}$ be a finite sequence of elements of the ring $A^{0}$ whose image in $\bar{A}$ generates the ideal $\bar{\mathfrak{a}}$, and let $\bar{\mathbf{a}}$ be its image in $\bar{A}$. The latter is a finite sequence of elements of the ring $\bar{A}$. By (1.3), there is a natural isomorphism

[TABLE]

As in the proof of Lemma 2.6, setting $T:=\operatorname{Tel}(A^{0};\mathbf{a})\otimes_{A^{0}}A$, observing that $T$ is K-projective over $A$, and using adjunctions, this is naturally isomorphic to

[TABLE]

Let $P\cong M$ be a K-flat resolution over $A$, let $F\cong\operatorname{Hom}_{A}(T,P)$ be a K-flat resolution over $A$, and let $A\to B\cong\bar{A}$ be a semi-free DG-algebra resolution of $\bar{A}$ over $A$. We obtain a commutative diagram in $\operatorname{DGMod}(A)$:

[TABLE]

The fact that $F$ is K-flat over $A$ implies that $\alpha$ is a quasi-isomorphism, while K-flatness of $B$ over $A$ implies that $\gamma$ is a quasi-isomorphism. The fact that $P$ is K-flat over $A$ and $T$ is K-projective over $A$ implies that $\psi$ is a quasi-isomorphism. Finally, since $A$ is noetherian and $\operatorname{proj\,dim}_{A}(T)<\infty$, by Proposition 1.4(2), the map $\chi$ is a quasi-isomorphism. It follows by the 2-out-of-3 property that the $\bar{A}$-linear map $\varphi\circ\beta$ is a quasi-isomorphism. Hence, there are natural isomorphisms

[TABLE]

in $\operatorname{\mathsf{D}}(\bar{A})$. The result now follows by combining all these isomorphisms and using the fact that there is a natural isomorphism

[TABLE]

in $\operatorname{\mathsf{D}}(A)$. ∎

Here is the second main result of this paper.

Theorem 3.3**.**

Let $A$ be a commutative noetherian DG-ring, and let $\bar{\mathfrak{a}}\subseteq\mathrm{H}^{0}(A)$ be an ideal. Then for any $M\in\operatorname{\mathsf{D}}^{\mathrm{b}}(A)$, there is an inequality

[TABLE]

Proof.

By Theorem 3.1,

[TABLE]

Using Lemma 3.2, we have that

[TABLE]

and by Corollary 0.3 we obtain that

[TABLE]

where the last equality follows from Theorem 3.1. Combining all of the above, we obtain that

[TABLE]

as claimed.

∎

Remark 3.4**.**

As in Remark 2.8, this result is false in general if the (DG-)ring is not assumed to be noetherian (even if the ideal is finitely generated by a regular sequence). See [13, Theorem 6.2] for an example.

Remark 3.5**.**

Let $A$ be a commutative noetherian ring such that the Krull dimension of $A$ is $\geq 1$, and let $B=A[x]$. Then $B$ is a projective $B$-module, and it follows from [2, Theorem 1] that the $B$-module $\Lambda_{(x)}(B)$ is not projective.

We finish the paper with an important application of Theorem 3.3. A basic result in commutative algebra states that if $A$ is a commutative noetherian ring, and $\mathfrak{a}\subseteq A$ is an ideal, then the canonical map $A\to\widehat{A}$ from $A$ to its $\mathfrak{a}$-adic completion is flat. Here is the analogue of this result in derived commutative algebra:

Let $A$ be a commutative noetherian DG-ring, and let $\bar{\mathfrak{a}}\subseteq\bar{A}$ be an ideal. The derived $\bar{\mathfrak{a}}$-adic completion of $A$ is a commutative DG-ring, denoted by $\mathrm{L}\Lambda(A,\bar{\mathfrak{a}})$. It was defined in [9, Theorem 0.2]. It is a derived analogue of the $\mathfrak{a}$-adic completion $\widehat{A}$. There is a natural map $A\to\mathrm{L}\Lambda(A,\bar{\mathfrak{a}})$, but this is not a map of DG-rings. Instead, it only exists in a suitable homotopy category (see [9] for details). Concretely, one can realize it as follows: there is a commutative DG-ring $P$, a quasi-isomorphism $P\to A$, and a natural map of DG-rings $P\to\mathrm{L}\Lambda(A,\bar{\mathfrak{a}})$. Since $P$ and $A$ are quasi-isomorphic, the triangulated categories $\operatorname{\mathsf{D}}(A)$ and $\operatorname{\mathsf{D}}(P)$ are isomorphic. Using this isomorphism and the map $P\to\mathrm{L}\Lambda(A,\bar{\mathfrak{a}})$, we can view $\mathrm{L}\Lambda(A,\bar{\mathfrak{a}})$ as an object of $\operatorname{\mathsf{D}}(A)$. According to [9, Proposition 3.58], this object is isomorphic to $\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}(A)$.

The above paragraph explains that the analogue of the number $\operatorname{fl\,dim}_{A}(\widehat{A})$ in derived commutative algebra is the number $\operatorname{fl\,dim}_{A}(\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}(A))$. This explains the importance of our final result:

Corollary 3.6**.**

Let $A$ be a commutative noetherian DG-ring, and assume that $A$ has bounded cohomology. Let $\bar{\mathfrak{a}}\subseteq\mathrm{H}^{0}(A)$ be an ideal. Then

[TABLE]

Proof.

Since $\operatorname{fl\,dim}_{A}(A)=0$, it follows from Theorem 3.3 that

[TABLE]

On the other hand, by [9, Proposition 6.1], we have that

[TABLE]

Hence, $\sup(\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}(A))=0$, and we deduce that

[TABLE]

which implies that

[TABLE]

∎

Acknowledgments. The author is grateful to the anonymous referee for comments and suggestions that helped improving this manuscript.