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On the de Rham homology and cohomology of a complete local ring in equicharacteristic zero

Published online by Cambridge University Press:  07 August 2017

Nicholas Switala*
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 322 SEO (M/C 249), 851 S. Morgan Street, Chicago, IL 60607, USA email nswitala@uic.edu
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Abstract

Let $A$ be a complete local ring with a coefficient field $k$ of characteristic zero, and let $Y$ be its spectrum. The de Rham homology and cohomology of $Y$ have been defined by R. Hartshorne using a choice of surjection $R\rightarrow A$ where $R$ is a complete regular local $k$-algebra: the resulting objects are independent of the chosen surjection. We prove that the Hodge–de Rham spectral sequences abutting to the de Rham homology and cohomology of $Y$, beginning with their $E_{2}$-terms, are independent of the chosen surjection (up to a degree shift in the homology case) and consist of finite-dimensional $k$-spaces. These $E_{2}$-terms therefore provide invariants of $A$ analogous to the Lyubeznik numbers. As part of our proofs we develop a theory of Matlis duality in relation to ${\mathcal{D}}$-modules that is of independent interest. Some of the highlights of this theory are that if $R$ is a complete regular local ring containing $k$ and ${\mathcal{D}}={\mathcal{D}}(R,k)$ is the ring of $k$-linear differential operators on $R$, then the Matlis dual $D(M)$ of any left ${\mathcal{D}}$-module $M$ can again be given a structure of left ${\mathcal{D}}$-module, and if $M$ is a holonomic ${\mathcal{D}}$-module, then the de Rham cohomology spaces of $D(M)$ are $k$-dual to those of $M$.

Type
Research Article
Copyright
© The Author 2017 

1 Introduction

In [Reference HartshorneHar75], Hartshorne constructs local and global algebraic de Rham homology and cohomology theories for schemes over a field $k$ of characteristic zero. The global theories, defined for any scheme $Y$ of finite type over $k$ , are defined using a choice of embedding $Y{\hookrightarrow}X$ into a smooth scheme over $k$ (or a local system of such embeddings if a global embedding does not exist): one computes the hypercohomology of certain complexes of sheaves on $X$ or on the formal completion of $Y$ in $X$ . Hartshorne’s primary interest [Reference HartshorneHar75, Remark, p. 70] in constructing the local theories is the case where $Y$ is the spectrum of a complete local ring. For technical reasons, he defines the local theories for a larger class of schemes (‘category ${\mathcal{C}}$ ’: see [Reference HartshorneHar75, p. 65]), and proves the corresponding finiteness and duality results by reducing to the global case and using resolution of singularities.

There is a sketch in [Reference HartshorneHar75, pp. 70–71] of a failed attempt to prove that local algebraic de Rham homology and cohomology are dual using Grothendieck’s local duality theorem. This sketch is the inspiration for the present paper, as its ideas can now be profitably pursued using Lyubeznik’s work on the ${\mathcal{D}}$ -module structure of local cohomology [Reference LyubeznikLyu93]. This structure allows us to speak of the de Rham complex of a local cohomology module, and knowledge of such complexes in turn enables us to better understand the early terms of the spectral sequences appearing in Hartshorne’s work. We are able to give a purely local proof for both the embedding-independence and the finiteness of local de Rham homology and cohomology, replacing the global methods of algebraic geometry (including resolution of singularities) with the theory of algebraic ${\mathcal{D}}$ -modules over a formal power series ring in characteristic zero. Along the way, we will define a new set of invariants for complete local rings, analogous to the Lyubeznik numbers. Our proof shows more than is contained in [Reference HartshorneHar75], namely that the entire Hodge–de Rham spectral sequences for homology and cohomology (with the exception of the first term) are embedding-independent (up to a degree shift in the homology case) and consist of finite-dimensional $k$ -spaces.

We now give more detail, recalling Hartshorne’s results and stating ours. Let $A$ be a complete local ring with coefficient field $k$ of characteristic zero (that is, $k$ is the residue field of $A$ , and $A$ contains a field isomorphic to $k$ ). We view $A$ as a $k$ -algebra via this coefficient field. By Cohen’s structure theorem, there exists a surjection of $k$ -algebras $\unicode[STIX]{x1D70B}:R\rightarrow A$ where $R$ is a complete regular local $k$ -algebra, which must take the form $R=k[[x_{1},\ldots ,x_{n}]]$ for some $n$ . Let $I\subset R$ be the kernel of this surjection. We have a corresponding closed immersion $Y{\hookrightarrow}X$ where $Y=\operatorname{Spec}(A)$ and $X=\operatorname{Spec}(R)$ . In [Reference HartshorneHar75], the de Rham homology of the local scheme $Y$ is defined as $H_{i}^{\text{dR}}(Y)=\mathbf{H}_{Y}^{2n-i}(X,\unicode[STIX]{x1D6FA}_{X}^{\bullet })$ , the hypercohomology (supported at $Y$ ) of the complex of continuous differential forms on $X$ . The differentials in this complex are merely $k$ -linear, so the $H_{i}^{\text{dR}}(Y)$ are $k$ -spaces.

Now let $\widehat{X}$ be the formal completion of $Y$ in $X$ [Reference HartshorneHar77, § II.9], that is, the topological space $Y$ equipped with the structure of a locally ringed space via the sheaf of rings $\varprojlim \,{\mathcal{O}}_{X}/{\mathcal{I}}^{n}$ where ${\mathcal{I}}\subset {\mathcal{O}}_{X}$ is the quasi-coherent ideal sheaf defining the chosen embedding $Y{\hookrightarrow}X$ (every ${\mathcal{O}}_{X}/{\mathcal{I}}^{n}$ is supported at $Y$ and thus can be viewed as a sheaf of rings on $Y$ ). The differentials in the complex $\unicode[STIX]{x1D6FA}_{X}^{\bullet }$ are ${\mathcal{I}}$ -adically continuous and thus pass to ${\mathcal{I}}$ -adic completions. We obtain in this way a complex $\widehat{\unicode[STIX]{x1D6FA}}_{X}^{\bullet }$ of sheaves on $\widehat{X}$ , the formal completion of $\unicode[STIX]{x1D6FA}_{X}^{\bullet }$ , whose differentials are again merely $k$ -linear. In [Reference HartshorneHar75], the (local) de Rham cohomology of the local scheme $Y$ is defined as $H_{P,\text{dR}}^{i}(Y)=\mathbf{H}_{P}^{i}(\widehat{X},\widehat{\unicode[STIX]{x1D6FA}}_{X}^{\bullet })$ , where $P$ is the closed point of $Y$ . After making these definitions, Hartshorne establishes the following properties.

Theorem 1.1 [Reference HartshorneHar75, Theorems III.1.1 and III.2.1].

Let $A$ be a complete local ring with coefficient field $k$ of characteristic zero and $Y=\operatorname{Spec}(A)$ .

  1. (a) The de Rham homology spaces $H_{i}^{\text{dR}}(Y)$ and cohomology spaces $H_{P,\text{dR}}^{i}(Y)$ as defined above are independent of the surjection of $k$ -algebras $R\rightarrow A$ used in their definitions.

  2. (b) For all $i$ , $H_{i}^{\text{dR}}(Y)$ and $H_{P,\text{dR}}^{i}(Y)$ are finite-dimensional $k$ -spaces.

  3. (c) For all $i$ , the $k$ -spaces $H_{i}^{\text{dR}}(Y)$ and $H_{P,\text{dR}}^{i}(Y)$ are $k$ -dual to each other.

The de Rham homology and cohomology of $Y=\operatorname{Spec}(A)$ are both defined using hypercohomology. As is well known, there are in general two spectral sequences converging to the hypercohomology of a complex ([Reference Grothendieck and DieudonnéGD61, 11.4.3]; also see § 2.2). If $K^{\bullet }$ is a complex of sheaves of Abelian groups on a topological space $Z$ , the first of these spectral sequences begins $E_{1}^{p,q}=H^{q}(Z,K^{p})$ and has abutment $\mathbf{H}^{p+q}(Z,K^{\bullet })$ . In our case, this takes the form of the Hodge–de Rham homology spectral sequence, which begins $E_{1}^{n-p,n-q}=H_{Y}^{n-q}(X,\unicode[STIX]{x1D6FA}_{X}^{n-p})$ and has abutment $H_{p+q}^{\text{dR}}(Y)$ , as well as the Hodge–de Rham cohomology spectral sequence, which begins $\tilde{E}_{1}^{p,q}=H_{P}^{q}(\widehat{X},\widehat{\unicode[STIX]{x1D6FA}}_{X}^{p})$ and has abutment $H_{P,\text{dR}}^{p+q}(Y)$ . A priori, these spectral sequences depend on the choice of surjection $R\rightarrow A$ from a complete regular local $k$ -algebra. We prove stronger versions of Hartshorne’s results for these spectral sequences. Our theorem for de Rham homology is the following.

Theorem A. Let $A$ be a complete local ring with coefficient field $k$ of characteristic zero. Viewing $A$ as a $k$ -algebra via this coefficient field, let $R\rightarrow A$ be a choice of $k$ -algebra surjection from a complete regular local $k$ -algebra. Associated with this surjection we have a Hodge–de Rham spectral sequence for homology as above.

  1. (a) Beginning with the $E_{2}$ -term, the isomorphism class of the homology spectral sequence with its abutment is independent of the choice of regular $k$ -algebra and surjection $R\rightarrow A$ , up to a degree shift.

  2. (b) The $k$ -spaces $E_{2}^{p,q}$ appearing in the $E_{2}$ -term of the homology spectral sequence are finite-dimensional.

The meaning of ‘up to a degree shift’ in the statement of part (a) is the following: given two surjections $R\rightarrow A$ and $R^{\prime }\rightarrow A$ from complete regular local $k$ -algebras, where $\dim (R)=n$ and $\dim (R^{\prime })=n^{\prime }$ , we obtain two Hodge–de Rham spectral sequences $E_{\bullet ,R}^{\bullet ,\bullet }$ and $\mathbf{E}_{\bullet ,R^{\prime }}^{\bullet ,\bullet }$ . Part (a) asserts that there is a morphism $E_{\bullet ,R}^{\bullet ,\bullet }\rightarrow \mathbf{E}_{\bullet ,R^{\prime }}^{\bullet ,\bullet }$ of bidegree $(n^{\prime }-n,n^{\prime }-n)$ between these spectral sequences which is an isomorphism on the objects of the $E_{2}$ - (and later) terms (see § 2.2 for the precise definitions of the terms used here).

We also have the analogue (without a degree shift) for de Rham cohomology, as follows.

Theorem B. Let $A$ and $R$ be as in Theorem A. Associated with this surjection we also have a local Hodge–de Rham spectral sequence for cohomology.

  1. (a) Beginning with the $E_{2}$ -term, the isomorphism class of the cohomology spectral sequence with its abutment is independent of the choice of regular $k$ -algebra and surjection $R\rightarrow A$ .

  2. (b) The $k$ -spaces $\tilde{E}_{2}^{pq}$ appearing in the $E_{2}$ -term of the cohomology spectral sequence are finite-dimensional.

Remark 1.2. In § 2.2, the notion of an isomorphism of spectral sequences is defined. As described in this subsection, the ingredients of a spectral sequence are the objects and differentials in the $E_{r}$ -term for all $r$ , the abutment objects, and the filtrations on the abutment objects, together with the isomorphisms relating the terms with their successors and with the abutment. The assertion of Theorems A and B is that all of these ingredients (except for the $E_{1}$ -terms) are independent of the chosen regular $k$ -algebra and surjection. Therefore parts (a) and (b) of Theorem 1.1 are subsumed by Theorems A and B, which provide more information: the isomorphism classes of $H_{p+q}^{\text{dR}}(Y)$ and $H_{P,\text{dR}}^{p+q}(Y)$ as filtered objects are independent of the surjection. We obtain many numerical invariants of $(A,k)$ from the spectral sequence: the (finite) dimensions of the kernels and cokernels of the differentials $d_{r}$ for all $r\geqslant 2$ , together with the dimensions of the filtered pieces of the abutment.

The proof of Theorem B requires the development of a theory of Matlis duality for ${\mathcal{D}}$ -modules. This theory is worked out in §§ 34, and 5, which form a unit of independent interest. If $R$ is a complete local ring with coefficient field $k$ of characteristic zero, we can consider the ring ${\mathcal{D}}(R,k)$ of $k$ -linear differential operators on $R$ . Given any left module over this ring, we can define its de Rham complex and speak of its de Rham cohomology spaces. We prove that the Matlis dual of a left ${\mathcal{D}}(R,k)$ -module has a natural structure of right ${\mathcal{D}}(R,k)$ -module. Specializing to the case of a complete regular local ring, we are able to obtain information about the de Rham cohomology of a Matlis dual. In this case, the dual of a left ${\mathcal{D}}(R,k)$ -module can again be viewed as a left ${\mathcal{D}}(R,k)$ -module. The following is the main result of our theory of Matlis duality for ${\mathcal{D}}$ -modules (its two assertions are proved separately below as Proposition 4.17 and Theorem 5.1).

Theorem C. Let $k$ be a field of characteristic zero, let $R=k[[x_{1},\ldots x_{n}]]$ be a formal power series ring over $k$ , and let ${\mathcal{D}}={\mathcal{D}}(R,k)$ be the ring of $k$ -linear differential operators on $R$ . If $M$ is a left ${\mathcal{D}}$ -module, the Matlis dual $D(M)$ of $M$ with respect to $R$ can also be given a natural structure of left ${\mathcal{D}}$ -module. We write $H_{\text{dR}}^{\ast }(M)$ for the de Rham cohomology of a left ${\mathcal{D}}$ -module. If $M$ is a holonomic left ${\mathcal{D}}$ -module, then for every $i$ , we have an isomorphism of $k$ -spaces

$$\begin{eqnarray}(H_{\text{dR}}^{i}(M))^{\vee }\simeq H_{\text{dR}}^{n-i}(D(M))\end{eqnarray}$$

where $\vee$ denotes $k$ -linear dual.

Remark 1.3. If $M$ is holonomic, its de Rham cohomology spaces are known to be finite-dimensional (see Theorem 2.2), and so it follows from Theorem C that $D(M)$ also has finite-dimensional de Rham cohomology. Since $D(M)$ is not, in general, a holonomic ${\mathcal{D}}$ -module (see Remark 1.4 below), this is not clear a priori.

When applied to the $E_{1}$ - and $E_{2}$ -terms of the homology and cohomology spectral sequences, Theorem C has the following consequence.

Theorem D. Let $A$ , $k$ , and $R$ be as in the statement of Theorem A. The objects in the $E_{2}$ -terms of the homology and cohomology spectral sequences are $k$ -dual to each other: for all $p$ and $q$ , $E_{2}^{n-p,n-q}\simeq (\tilde{E}_{2}^{pq})^{\vee }$ .

We conjecture that the entire spectral sequences, beginning with $E_{2}$ , should be $k$ -dual to each other (see § 8), but at present we are able only to prove the preceding statement.

As was already clear to Hartshorne [Reference HartshorneHar75, p. 71], the Matlis duals of the objects appearing in the $E_{1}$ -term of the homology spectral sequence are exactly the corresponding $E_{1}$ -objects for de Rham cohomology. What is much less clear is the relationship between the differentials. The $E_{1}$ -differentials for de Rham homology are merely $k$ -linear, so the usual definition of Matlis duality over $R$ cannot be applied to them. A large part of this paper is devoted to the problem of applying Matlis duality to these $k$ -linear maps and establishing that, in our setting, Matlis duality at the $E_{1}$ -term gives rise to $k$ -duality at the $E_{2}$ -term.

In more detail, the outline of the paper is as follows. In § 2, after reviewing some preliminary material on differential operators, de Rham complexes, and spectral sequences, we prove Theorem A using local algebra. In the course of this proof, we define a new set of invariants for complete local rings with coefficient fields of characteristic zero. In § 3, Matlis duality for local rings containing a field $k$ is interpreted in terms of $k$ -linear maps, after SGA2 [Reference Grothendieck and RaynaudGR05]. This allows us to dualize continuous maps between finitely generated modules over such rings. We pass to direct limits in order to dualize $k$ -linear maps between arbitrary modules that satisfy certain finiteness and continuity conditions. In § 4, we consider the case of ${\mathcal{D}}$ -modules for complete local rings containing $k$ ; we describe a natural right ${\mathcal{D}}$ -module structure on the Matlis dual $D(M)$ of a left ${\mathcal{D}}$ -module $M$ . In the special case of a formal power series ring, we can regard $D(M)$ as a left ${\mathcal{D}}$ -module as well using a simple ‘transpose’ operation, and thus define its de Rham complex. We determine the cohomology of this complex in § 5 in the case of holonomic $M$ , completing the proof of Theorem C. The specific case of local cohomology is considered next. In § 6, we work out precisely what happens to the action of derivations on a local cohomology module. Finally, in § 7, we give a self-contained proof of Theorem B(a) and combine the results of §§ 25, and 6 to prove Theorems B(b) and D.

Remark 1.4. After we identify the objects in the $E_{2}$ -term of the homology spectral sequence with de Rham cohomology spaces of local cohomology modules in § 2, part (b) of Theorem A follows from Lyubeznik’s result ([Reference LyubeznikLyu93, 2.2(d)]; cf. [Reference MebkhoutMeb77]) that local cohomology modules are holonomic ${\mathcal{D}}$ -modules, as it is known that holonomic ${\mathcal{D}}$ -modules have finite-dimensional de Rham cohomology. However, the objects in the $E_{2}$ -term of the cohomology spectral sequence are not de Rham cohomology spaces of holonomic ${\mathcal{D}}$ -modules: as shown in § 6, they are de Rham cohomology spaces of Matlis duals of local cohomology modules. Hellus has shown [Reference HellusHel07, Corollary 2.6] that Matlis duals of local cohomology modules have, in general, infinitely many associated primes, implying that they need not be holonomic ${\mathcal{D}}$ -modules (which always have finitely many associated primes [Reference LyubeznikLyu93, Corollary 3.6(c)]). It is therefore surprising that the cohomology $E_{2}$ -objects, which are, in general, de Rham cohomology spaces of non-holonomic ${\mathcal{D}}$ -modules, are still finite-dimensional. For this reason, the proof of Theorem B(b) is significantly more difficult than the proofs of the other parts of our main theorems.

2 The de Rham homology of a complete local ring

In this section, after reviewing some preliminary material on modules over rings of differential operators and spectral sequences, we recall Hartshorne’s definition of algebraic de Rham homology [Reference HartshorneHar75, ch. III] in the case of the spectrum of a complete local ring in equicharacteristic zero, and examine the associated Hodge–de Rham spectral sequence. We give proofs that this de Rham homology is intrinsically defined and finite-dimensional which are purely local; in fact, we prove more, namely that up to a degree shift, the entire Hodge–de Rham spectral sequence (with the exception of the first term) is intrinsically defined and has finite-dimensional objects. As a byproduct of our proof, we obtain a new set of invariants for complete local rings analogous to the Lyubeznik numbers [Reference LyubeznikLyu93, Theorem–Definition 4.1].

2.1 Preliminaries on ${\mathcal{D}}$ -modules

Let $R$ be a commutative ring and $k\subset R$ a commutative subring. The ring ${\mathcal{D}}(R,k)$ of $k$ -linear differential operators on $R$ , a subring of $\operatorname{End}_{k}(R)$ , is defined recursively as follows [Reference Grothendieck and DieudonnéGD67, § 16]. A differential operator $R\rightarrow R$ of order zero is multiplication by an element of $R$ . Supposing that differential operators of order less than or equal to $j-1$ have been defined, $d\in \operatorname{End}_{k}(R)$ is said to be a differential operator of order less than or equal to $j$ if, for all $r\in R$ , the commutator $[d,r]\in \operatorname{End}_{k}(R)$ is a differential operator of order less than or equal to $j-1$ , where $[d,r]=dr-rd$ (the products being taken in $\operatorname{End}_{k}(R)$ ). We write ${\mathcal{D}}^{j}(R)$ for the set of differential operators on $R$ of order less than or equal to $j$ and set ${\mathcal{D}}(R,k)=\bigcup _{j}{\mathcal{D}}^{j}(R)$ . Every ${\mathcal{D}}^{j}(R)$ is naturally a left $R$ -module. If $d\in {\mathcal{D}}^{j}(R)$ and $d^{\prime }\in {\mathcal{D}}^{l}(R)$ , it is easy to prove by induction on $j+l$ that $d^{\prime }\circ d\in {\mathcal{D}}^{j+l}(R)$ , so ${\mathcal{D}}(R,k)$ is a ring.

We consider now the special case in which $k$ is a field of characteristic zero and $R=k[[x_{1},\ldots ,x_{n}]]$ is a formal power series ring over $k$ . A standard reference for facts about the ring ${\mathcal{D}}={\mathcal{D}}(R,k)$ and left modules over ${\mathcal{D}}$ in this case is [Reference BjörkBjö79, ch. 3]; we summarize some of these facts now. The ring ${\mathcal{D}}$ , viewed as a left $R$ -module, is freely generated by monomials in the partial differentiation operators $\unicode[STIX]{x2202}_{1}=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{1},\ldots ,\unicode[STIX]{x2202}_{n}=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{n}$ ([Reference Grothendieck and DieudonnéGD67, Theorem 16.11.2]: here the characteristic-zero assumption is necessary). This ring has an increasing filtration $\{{\mathcal{D}}(\unicode[STIX]{x1D708})\}$ , called the order filtration, where ${\mathcal{D}}(\unicode[STIX]{x1D708})$ consists of those differential operators of order less than or equal to $\unicode[STIX]{x1D708}$ (the order of an element of ${\mathcal{D}}$ is the maximum of the orders of its summands, and the order of a single summand $\unicode[STIX]{x1D70C}\unicode[STIX]{x2202}_{1}^{a_{1}}\cdots \unicode[STIX]{x2202}_{n}^{a_{n}}$ with $\unicode[STIX]{x1D70C}\in R$ is $\sum a_{i}$ : this notion of order coincides with the one defined in the previous paragraph). The associated graded object $\operatorname{gr}({\mathcal{D}})=\oplus {\mathcal{D}}(\unicode[STIX]{x1D708})/{\mathcal{D}}(\unicode[STIX]{x1D708}-1)$ with respect to this filtration is isomorphic to $R[\unicode[STIX]{x1D701}_{1},\ldots ,\unicode[STIX]{x1D701}_{n}]$ (a commutative ring), where $\unicode[STIX]{x1D701}_{i}$ is the image of $\unicode[STIX]{x2202}_{i}$ in ${\mathcal{D}}(1)/{\mathcal{D}}(0)\subset \operatorname{gr}({\mathcal{D}})$ .

If $M$ is a finitely generated left ${\mathcal{D}}$ -module, there exists a good filtration  $\{M(\unicode[STIX]{x1D708})\}$ on $M$ , meaning that $M$ becomes a filtered left ${\mathcal{D}}$ -module with respect to the order filtration on ${\mathcal{D}}$ and  $\operatorname{gr}(M)=\oplus M(\unicode[STIX]{x1D708})/M(\unicode[STIX]{x1D708}-1)$ is a finitely generated $\operatorname{gr}({\mathcal{D}})$ -module. We let $J$ be the radical of $\operatorname{Ann}_{\operatorname{gr}({\mathcal{D}})}\operatorname{gr}(M)\subset \operatorname{gr}({\mathcal{D}})$ and set $d(M)=\dim \operatorname{gr}({\mathcal{D}})/J$ (Krull dimension). The ideal $J$ , and hence the number $d(M)$ , is independent of the choice of good filtration on $M$ . By Bernstein’s theorem, if $M\neq 0$ is a finitely generated left ${\mathcal{D}}$ -module, we have $n\leqslant d(M)\leqslant 2n$ . In the case $d(M)=n$ we say that $M$ is holonomic. It is known that submodules and quotients of holonomic ${\mathcal{D}}$ -modules are holonomic, an extension of a holonomic ${\mathcal{D}}$ -module by another holonomic ${\mathcal{D}}$ -module is holonomic, holonomic ${\mathcal{D}}$ -modules are of finite length over ${\mathcal{D}}$ , and holonomic ${\mathcal{D}}$ -modules are cyclic (generated over ${\mathcal{D}}$ by a single element).

Given any left ${\mathcal{D}}(R,k)$ -module $M$ , we can define its de Rham complex. This is a complex of length $n$ , denoted $M\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet }$ (or simply $\unicode[STIX]{x1D6FA}_{R}^{\bullet }$ in the case $M=R$ ), whose objects are $R$ -modules but whose differentials are merely $k$ -linear. It is defined as follows [Reference BjörkBjö79, § 1.6]: for $0\leqslant i\leqslant n$ , $M\otimes \unicode[STIX]{x1D6FA}_{R}^{i}$ is a direct sum of $\binom{n}{i}$ copies of $M$ , indexed by $i$ -tuples $1\leqslant j_{1}<\cdots <j_{i}\leqslant n$ . The summand corresponding to such an $i$ -tuple will be written $M\,dx_{j_{1}}\wedge \cdots \wedge dx_{j_{i}}$ .

Convention 2.1. The subscript $R$ in $\unicode[STIX]{x1D6FA}_{R}^{\bullet }$ indicates over which ring the tensor products of objects are being taken. To simplify notation, we will follow this convention when de Rham complexes over different rings are being simultaneously considered.

The $k$ -linear differentials $d^{i}:M\otimes \unicode[STIX]{x1D6FA}_{R}^{i}\rightarrow M\otimes \unicode[STIX]{x1D6FA}_{R}^{i+1}$ are defined by

$$\begin{eqnarray}d^{i}(m\,dx_{j_{1}}\wedge \cdots \wedge dx_{j_{i}})=\mathop{\sum }_{s=1}^{n}\unicode[STIX]{x2202}_{s}(m)\,dx_{s}\wedge dx_{j_{1}}\wedge \cdots \wedge dx_{j_{i}},\end{eqnarray}$$

with the usual exterior algebra conventions for rearranging the wedge terms, and extended by linearity to the direct sum. The cohomology objects $h^{i}(M\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet })$ , which are $k$ -spaces, are called the de Rham cohomology spaces of the left ${\mathcal{D}}$ -module $M$ , and are denoted $H_{\text{dR}}^{i}(M)$ . In the case of a holonomic module, van den Essen proved that these spaces are finite-dimensional.

Theorem 2.2 [Reference van den EssenvdE85, Proposition 2.2].

If $M$ is a holonomic left ${\mathcal{D}}$ -module, its de Rham cohomology $H_{\text{dR}}^{i}(M)$ is a finite-dimensional $k$ -space for all $i$ .

Remark 2.3. An important family of examples of holonomic ${\mathcal{D}}$ -modules is that of local cohomology modules. Our basic references for facts about local cohomology modules are Brodmann and Sharp [Reference Brodmann and SharpBS13] and SGA2 [Reference Grothendieck and RaynaudGR05]. If $R$ is a commutative ring and $I\subset R$ is an ideal, the functor $\unicode[STIX]{x1D6E4}_{I}$ of sections with support at $I$ is a left-exact functor on the category of $R$ -modules (for an $R$ -module $M$ , $\unicode[STIX]{x1D6E4}_{I}(M)$ consists of those $m\in M$ annihilated by some power of $I$ ). The local cohomology modules $H_{I}^{i}(M)$ are the right derived functors of $\unicode[STIX]{x1D6E4}_{I}=H_{I}^{0}$ evaluated at $M$ . There is a more general sheaf-theoretic formulation, given in [Reference Grothendieck and RaynaudGR05]: if $X$ is a topological space, $Y\subset X$ is a locally closed subset, and ${\mathcal{F}}$ is a sheaf of Abelian groups on $X$ , the local cohomology groups $H_{Y}^{i}(X,{\mathcal{F}})$ are obtained by evaluating at ${\mathcal{F}}$ the right derived functors of $\unicode[STIX]{x1D6E4}_{Y}$ , the functor of global sections supported at $Y$ . The relationship between these two definitions in the case of an affine scheme $X$ is given below in Lemma 2.16. In the case of the ring $R=k[[x_{1},\ldots ,x_{n}]]$ , Lyubeznik proved that for any ideal $I$ , the local cohomology modules $H_{I}^{i}(R)$ have a natural structure of left ${\mathcal{D}}$ -module [Reference LyubeznikLyu93]; cf. [Reference MebkhoutMeb77]. Indeed, they are holonomic  ${\mathcal{D}}$ -modules [Reference LyubeznikLyu93, 2.2(d)], a fact which will repeatedly prove crucial for us.

In addition to the standard long exact cohomology sequence for derived functors, there is another useful long exact sequence for local cohomology: the Mayer–Vietoris sequence with respect to two ideals.

Proposition 2.4 [Reference Brodmann and SharpBS13, Theorem 3.2.3].

Let $I$ and $J$ be ideals of a commutative ring $R$ . There is a long exact sequence of $R$ -modules

$$\begin{eqnarray}\cdots \rightarrow H_{I\cap J}^{i-1}(M)\rightarrow H_{I+J}^{i}(M)\rightarrow H_{I}^{i}(M)\oplus H_{J}^{i}(M)\rightarrow H_{I\cap J}^{i}(M)\rightarrow H_{I+J}^{i+1}(M)\rightarrow \cdots\end{eqnarray}$$

for every $R$ -module $M$ , functorial in $M$ .

Finally, we recall the well-known fact that the de Rham complex of a ${\mathcal{D}}$ -module is independent of the coordinates $x_{1},\ldots ,x_{n}$ for $R$ . (See [Reference SwitalaSwi16, Proposition 2.5] for a proof.)

Proposition 2.5. If $R=k[[x_{1},\ldots ,x_{n}]]$ and ${\mathcal{D}}={\mathcal{D}}(R,k)$ , the de Rham complex of any left ${\mathcal{D}}$ -module $M$ is independent of the chosen regular system of parameters $x_{1},\ldots ,x_{n}$ for $R$ .

2.2 Preliminaries on spectral sequences

As we will be working with morphisms of spectral sequences, we collect some basic facts and definitions in this subsection concerning them. References for this material include Weibel [Reference WeibelWei94, ch. 5] and EGA [Reference Grothendieck and DieudonnéGD61, § 11]. We will not need to consider convergence issues for unbounded spectral sequences and hence make no mention of such issues here.

Definition 2.6. Let ${\mathcal{C}}$ be an Abelian category. A (cohomological) spectral sequence consists of the following data: a family $\{E_{r}^{p,q}\}$ of objects of ${\mathcal{C}}$ (where $p,q\in \mathbb{Z}$ and $r\geqslant 1$ or ${\geqslant}2$ ; with $r$ fixed and $p,q$ varying, we obtain the $E_{r}$ -term of the spectral sequence), and morphisms (the differentials) $d_{r}^{p,q}:E_{r}^{p,q}\rightarrow E_{r}^{p+r,q-r+1}$ for all $p,q,r$ such that $d_{r}^{p,q}\circ d_{r}^{p-r,q+r-1}=0$ and $\ker (d_{r}^{p,q})/\text{im}(d_{r}^{p-r,q+r-1})\xrightarrow[{}]{{\sim}}E_{r+1}^{p,q}$ : a family of such isomorphisms, denoted $\unicode[STIX]{x1D6FC}_{r}^{p,q}$ , is part of the data of the spectral sequence.

Let $E$ be a spectral sequence in an Abelian category ${\mathcal{C}}$ , and suppose that for all $l$ and for all $r$ , there are only finitely many non-zero objects $E_{r}^{p,q}$ with $p+q=l$ . Such a spectral sequence is called bounded. (For example, this occurs if $E_{r}^{p,q}=0$ whenever $p$ or $q$ is negative, in which case $E$ is called a first-quadrant spectral sequence.) If $E$ is a bounded spectral sequence, for every pair $(p,q)$ , there exists $r_{0}$ such that for all $r\geqslant r_{0}$ , $d_{r}^{p,q}$ has zero target, $d_{r}^{p-r,q+r-1}$ has zero source, and so $E_{r+1}^{p,q}\simeq E_{r}^{p,q}$ . We denote this stable object by $E_{\infty }^{p,q}$ . We can now define the abutment of such a spectral sequence.

Definition 2.7. Let $E$ be a bounded spectral sequence in an Abelian category ${\mathcal{C}}$ . Suppose we are given a family $E^{m}$ of objects of ${\mathcal{C}}$ , all endowed with a finite decreasing filtration $E^{m}=E_{s}^{m}\supset E_{s+1}^{m}\supset \cdots \supset E_{t}^{m}=0$ , and for all $p$ , an isomorphism $\unicode[STIX]{x1D6FD}^{p,m-p}:E_{\infty }^{p,m-p}\xrightarrow[{}]{{\sim}}E_{p}^{m}/E_{p+1}^{m}$ . Then we say that the spectral sequence abuts or converges to $\{E^{m}\}$ (the abutment), and write $E_{1}^{p,q}\Rightarrow E^{m}$ or $E_{2}^{p,q}\Rightarrow E^{m}$ .

For example, if $E$ is a first-quadrant spectral sequence with abutment $\{E^{m}\}$ , every $E^{m}$ has a filtration of length $m+1$ (we take $s=0$ and $t=m+1$ in the definition above), with $E^{m}/E_{1}^{m}\simeq E_{\infty }^{0,m}$ and $E_{m}^{m}\simeq E_{\infty }^{m,0}$ .

Given two spectral sequences, there is a natural notion of a morphism between them, which consists of morphisms between the objects in the $E_{r}$ -terms for all $r$ , each of which induces its successor on cohomology. There is also a natural notion of morphisms between bounded spectral sequences with given abutments.

Definition 2.8. Let $E$ and $E^{\prime }$ be two spectral sequences in ${\mathcal{C}}$ with respective differentials $d$ and  $d^{\prime }$ . A morphism  $u:E\rightarrow E^{\prime }$ is a family of morphisms $u_{r}^{p,q}:E_{r}^{p,q}\rightarrow E_{r}^{\prime p,q}$ such that the $u_{r}^{p,q}$ are compatible with the differentials ( $d_{r}^{\prime p,q}\circ u_{r}^{p,q}=u_{r}^{p+r,q-r+1}\circ d_{r}^{p,q}$ for all $p,q,r$ ) and the morphisms

$$\begin{eqnarray}\overline{u}_{r}^{p,q}:\ker (d_{r}^{p,q})/\text{im}(d_{r}^{p-r,q+r-1})\rightarrow \ker (d_{r}^{\prime p,q})/\text{im}(d_{r}^{\prime p-r,q+r-1})\end{eqnarray}$$

induced by $u_{r}^{p,q}$ commute with the given isomorphisms $\unicode[STIX]{x1D6FC}_{r}^{p,q}$ (that is, $\unicode[STIX]{x1D6FC}_{r}^{\prime p,q}\circ \overline{u}_{r}^{p,q}=\overline{u}_{r+1}^{p,q}\circ \unicode[STIX]{x1D6FC}_{r}^{p,q}$ ) so that, in the appropriate sense, $u_{r+1}^{p,q}$ is the morphism induced by $u_{r}^{p,q}$ . If $E$ and $E^{\prime }$ are bounded spectral sequences with abutments $\{E^{m}\}$ and $\{{E^{\prime }}^{m}\}$ , a morphism with abutments between the spectral sequences is a morphism $u:E\rightarrow E^{\prime }$ as just defined together with a family of morphisms $u^{m}:E^{m}\rightarrow {E^{\prime }}^{m}$ compatible with the filtrations on $E^{m}$ and ${E^{\prime }}^{m}$ such that, if we denote by $u_{\infty }^{p,q}$ the map induced by $u_{1}^{p,q}$ (or $u_{2}^{p,q}$ ) between the stable objects $E_{\infty }^{p,q}$ and $E_{\infty }^{\prime p,q}$ , this map must commute with the isomorphisms $\unicode[STIX]{x1D6FD}^{p,q}$ : if we denote by $u_{p}^{m}$ the morphism $E_{p}^{m}/E_{p+1}^{m}\rightarrow E_{p}^{\prime m}/E_{p+1}^{\prime m}$ induced by $u^{m}$ , which is required to be filtration-compatible, we must have $\unicode[STIX]{x1D6FD}^{\prime p,q}\circ u_{\infty }^{p,q}=u_{p}^{p+q}\circ \unicode[STIX]{x1D6FD}^{p,q}$ .

Convention 2.9. For the remainder of this paper, every spectral sequence will be a bounded spectral sequence with abutment, and every morphism of spectral sequences will be a morphism with abutments. Consequently, we suppress the phrase ‘with abutment’.

To show that two spectral sequences are isomorphic, it suffices to construct a morphism between them which is an isomorphism on the objects of the initial ( $r=1$ or $r=2$ ) terms. This result is crucial to our work in both this section and in § 7, so we record a version here.

Proposition 2.10 [Reference WeibelWei94, Theorem 5.2.12].

Let ${\mathcal{C}}$ be an Abelian category, and let $u=(u_{r}^{p,q},u^{n})$ be a morphism between two spectral sequences $E$ , $E^{\prime }$ in ${\mathcal{C}}$ . If there exists $r$ such that $u_{r}^{p,q}$ is an isomorphism for all $p$ and $q$ , then $u_{s}^{p,q}$ is an isomorphism for all $p$ and $q$ and all $s\geqslant r$ , and $u^{m}$ is an isomorphism for all $m$ . It follows that the abutments of $E$ and $E^{\prime }$ are isomorphic as filtered objects.

There is also a notion of a degree-shifted morphism of spectral sequences and a degree-shifted analogue of Proposition 2.10, which we will make use of in this paper. Again, for us, all spectral sequences will be bounded and all morphisms will be morphisms with abutments.

Definition 2.11. Let $E$ and $E^{\prime }$ be two spectral sequences in ${\mathcal{C}}$ with respective differentials $d$ and $d^{\prime }$ . If $a,b\in \mathbb{Z}$ , a morphism  $u:E\rightarrow E^{\prime }$ of bidegree  $(a,b)$ is a family of morphisms $u_{r}^{p,q}:E_{r}^{p,q}\rightarrow E_{r}^{\prime p+a,q+b}$ such that the $u_{r}^{p,q}$ are compatible with the differentials ( $d_{r}^{\prime p+a,q+b}\circ u_{r}^{p,q}=u_{r}^{p+r,q-r+1}\circ d_{r}^{p,q}$ for all $p,q,r$ ) and $u_{r+1}^{p,q}$ is induced on cohomology by $u_{r}^{p,q}$ . If $E$ and $E^{\prime }$ are bounded spectral sequences with abutments $\{E^{m}\}$ and $\{{E^{\prime }}^{m}\}$ , a morphism with abutments between the spectral sequences of bidegree  $(a,b)$ is a morphism $u:E\rightarrow E^{\prime }$ of bidegree $(a,b)$ as just defined together with a family of morphisms $u^{m}:E^{m}\rightarrow E^{\prime m+a+b}$ such that $u^{m}(E_{p}^{m})\subset E_{p+a}^{\prime m+a+b}$ for all $p$ and satisfying the obvious compatibility conditions analogous to those in the non-degree-shifted definition.

The degree-shifted analogue of Proposition 2.10 is proved in exactly the same way, but in the conclusion (that the abutments are isomorphic as filtered objects), it is worth recording precisely which filtrations are being compared and what the corresponding degree shifts are.

Proposition 2.12. Let ${\mathcal{C}}$ be an Abelian category, and let $u=(u_{r}^{p,q},u^{n})$ be a morphism of bidegree $(a,b)$ between two spectral sequences $E$ , $E^{\prime }$ in ${\mathcal{C}}$ . If there exists $r$ such that $u_{r}^{p,q}$ is an isomorphism for all $p$ and $q$ , then $u_{s}^{p,q}$ is an isomorphism for all $p$ and $q$ and all $s\geqslant r$ , and $u^{m}$ is an isomorphism for all $m$ . This has the following consequence for the abutments: for all $m$ , $E^{m}$ , endowed with the filtration $\{E_{p}^{m}\}_{p=0}^{m+1}$ where $E_{p}^{m}/E_{p+1}^{m}\simeq E_{\infty }^{p,m-p}$ , is isomorphic (as a filtered object) to $E^{\prime m+a+b}$ , endowed with the filtration $\{E_{p+a}^{\prime m+a+b}\}_{p=0}^{m+1}$ , where $E_{p+a}^{\prime m+a+b}/E_{p+a+1}^{\prime m+a+b}\simeq E_{\infty }^{\prime p+a,m+b-p}$ .

Double complexes are a common source of spectral sequences: the cohomology of the totalization of a double complex can be approximated, and in some cases even computed, by the objects in the early terms of either of two spectral sequences associated with the double complex. To be precise, let $K^{\bullet ,\bullet }$ be a double complex in an Abelian category ${\mathcal{C}}$ , which we think of abusively as the ‘ $E_{0}$ -term’ of a spectral sequence, and let $T^{\bullet }$ be its totalization. Our conventions for double complexes are those of EGA: the horizontal ( $d_{h}^{\bullet ,\bullet }$ ) and vertical ( $d_{v}^{\bullet ,\bullet }$ ) differentials of $K^{\bullet ,\bullet }$ commute, we define $T^{i}=\bigoplus _{p+q=i}K^{p,q}$ , and the differentials of $T^{\bullet }$ require signs, namely $d(x)=d_{h}(x)+(-1)^{p}d_{v}(x)$ for $x\in K^{p,q}$ . The two spectral sequences associated with $K^{\bullet ,\bullet }$ [Reference Grothendieck and DieudonnéGD61, § 11.3] are the column-filtered (‘vertical differentials first’) spectral sequence, for which $E_{1}^{p,q}=h_{v}^{q}(K^{p,\bullet })$ (and the differentials are those induced on vertical cohomology by the maps $d_{h}^{p,q}$ ), and the row-filtered (‘horizontal differentials first’) spectral sequence, for which $E_{1}^{p,q}=h_{h}^{p}(K^{\bullet ,q})$ (and the differentials are those induced on horizontal cohomology by the maps $d_{v}^{p,q}$ ). Both have $h^{p+q}(T^{\bullet })$ , the cohomology of the totalization, for their abutment. A morphism $K^{\bullet ,\bullet }\rightarrow K^{\prime \bullet ,\bullet }$ of double complexes induces morphisms between their column-filtered spectral sequences as well as between their row-filtered spectral sequences [Reference Grothendieck and DieudonnéGD61, p. 30].

The spectral sequences of a double complex are useful for computing hyperderived functors of left-exact functors between Abelian categories. Suppose ${\mathcal{A}},{\mathcal{B}}$ are Abelian categories, ${\mathcal{A}}$ has enough injective objects, and $F:{\mathcal{A}}\rightarrow {\mathcal{B}}$ is a left-exact additive functor. If $K^{\bullet }$ is a complex with differential $d$ in ${\mathcal{A}}$ , the (right) hyperderived functors of $F$ evaluated at $K^{\bullet }$ are defined as follows [Reference Grothendieck and DieudonnéGD61, § 11.4]: if $K^{\bullet }\rightarrow I^{\bullet }$ is a quasi-isomorphism and $I^{\bullet }$ is a complex of injective objects in ${\mathcal{A}}$ , then $\mathbf{R}^{i}F(K^{\bullet })=h^{i}(F(I^{\bullet }))$ , and the objects of ${\mathcal{B}}$ thus obtained are independent of the choice of  $I^{\bullet }$ . Such a complex $I^{\bullet }$ can be produced as the totalization of a Cartan–Eilenberg resolution of  $K^{\bullet }$ , which is a double complex $J^{\bullet ,\bullet }$ with differentials $d_{h},d_{v}$ such that every $J^{p,q}$ is an injective object of ${\mathcal{A}}$ and, for all $p$ , $J^{p,\bullet }$ (respectively $\ker (d_{v}^{p,\bullet })$ , $\operatorname{im}(d_{v}^{p,\bullet })$ , $h_{v}^{\bullet }(J^{p,\bullet })$ ) is an injective resolution of $K^{p}$ (respectively $\ker (d^{p})$ , $\operatorname{im}(d^{p})$ , $h^{p}(K^{\bullet })$ ). It follows that $\mathbf{R}^{i}F(K^{\bullet })$ is the cohomology of the totalization of the double complex $F(J^{\bullet ,\bullet })$ , and so, by the previous paragraph, we have two spectral sequences whose abutment is this cohomology. For example, the column-filtered spectral sequence begins $E_{1}^{p,q}=h_{v}^{q}(F(J^{p,\bullet }))$ and has abutment $\mathbf{R}^{p+q}F(K^{\bullet })$ . But since $J^{p,\bullet }$ is an injective resolution of $K^{p}$ , we see that $h_{v}^{q}(F(J^{p,\bullet }))=R^{q}F(K^{p})$ , the ordinary $q$ th right derived functor of $F$ applied to $K^{p}$ ; this is the form in which the ‘first’ hyperderived functor spectral sequence is usually given [Reference Grothendieck and DieudonnéGD61, 11.4.3.1].

Now suppose $K^{\bullet }$ , $K^{\prime \bullet }$ are complexes in ${\mathcal{A}}$ with respective Cartan–Eilenberg resolutions $J^{\bullet ,\bullet }$ , $J^{\prime \bullet ,\bullet }$ . A morphism of complexes $f:K^{\bullet }\rightarrow K^{\prime \bullet }$ induces a morphism of double complexes $J^{\bullet ,\bullet }\rightarrow J^{\prime \bullet ,\bullet }$ which is unique up to homotopy [Reference Grothendieck and DieudonnéGD61, p. 33]. This implies that $f$ induces a well-defined morphism between the spectral sequences for the hyperderived functors of $F$ evaluated at $K^{\bullet }$ and at $K^{\prime \bullet }$ [Reference Grothendieck and DieudonnéGD61, p. 30], since two double complex morphisms that are chain homotopic induce the same morphisms on horizontal and vertical cohomology, hence the same spectral sequence morphisms. By taking $K^{\prime \bullet }=K^{\bullet }$ and $f$ to be the identity, we see that the isomorphism class of the spectral sequence for $F$ evaluated at $K^{\bullet }$ is independent of the Cartan–Eilenberg resolution.

Later in this section, we will need to build a spectral sequence for hyperderived functors using a double complex that is not a Cartan–Eilenberg resolution of the original complex, and for this purpose, the following comparison lemma will be useful.

Lemma 2.13. Let ${\mathcal{A}}$ be an Abelian category with enough injective objects, ${\mathcal{B}}$ another Abelian category, and $F:{\mathcal{A}}\rightarrow {\mathcal{B}}$ a left-exact additive functor. Suppose that $K^{\bullet }$ is a complex in ${\mathcal{A}}$ , concentrated in degrees $p\geqslant 0$ , and that $L^{\bullet ,\bullet }$ is a double complex in ${\mathcal{A}}$ whose objects $L^{p,q}$ are all $F$ -acyclic and such that, for all $p\geqslant 0$ , $L^{p,\bullet }$ is a resolution of $K^{p}$ . Then the first spectral sequence for the hyperderived functors of $F$ applied to $K^{\bullet }$ , which begins $E_{1}^{p,q}=R^{q}F(K^{p})$ and has $\mathbf{R}^{p+q}F(K^{\bullet })$ for its abutment, is isomorphic to the column-filtered spectral sequence of the double complex $F(L^{\bullet ,\bullet })$ .

Proof. By definition, the first spectral sequence is the column-filtered spectral sequence of the double complex $F(J^{\bullet ,\bullet })$ , where $J^{\bullet ,\bullet }$ is a choice of Cartan–Eilenberg resolution of $K^{\bullet }$ in ${\mathcal{A}}$ . The assertion of the lemma is that we can replace $J^{\bullet ,\bullet }$ with the resolution $L^{\bullet ,\bullet }$ , which is generally not a Cartan–Eilenberg resolution and whose objects may not even be injective.

Our strategy will be to compare both of these double complexes to a third one. Let ${\mathcal{C}}^{+}$ denote the category of complexes in ${\mathcal{A}}$ that are concentrated in degrees $p\geqslant 0$ . Then ${\mathcal{C}}^{+}$ is an Abelian category with enough injective objects, and if $I^{\bullet }\in {\mathcal{C}}^{+}$ is injective, then $I^{p}$ is an injective object of ${\mathcal{A}}$ for all $p$ [Reference RotmanRot09, Theorems 10.42, 10.43; Remark, p. 652].

We return now to the complex $K^{\bullet }$ . Choose an injective resolution $0\rightarrow K^{\bullet }\rightarrow I^{\bullet ,\bullet }$ of $K^{\bullet }$ in  ${\mathcal{C}}^{+}$ . In particular, $I^{\bullet ,\bullet }$ is a double complex of injective objects in ${\mathcal{A}}$ . Now note that the two double complex resolutions $J^{\bullet ,\bullet }$ and $L^{\bullet ,\bullet }$ can also be regarded as resolutions of $K^{\bullet }$ in the category ${\mathcal{C}}^{+}$ . Any resolution in ${\mathcal{C}}^{+}$ can be compared with an injective one by [Reference LangLan02, Lemma XX.5.2]: there exist morphisms $J^{\bullet ,\bullet }\rightarrow I^{\bullet ,\bullet }$ and $L^{\bullet ,\bullet }\rightarrow I^{\bullet ,\bullet }$ extending the identity on $K^{\bullet }$ and unique up to homotopy as maps in ${\mathcal{C}}^{+}$ . These morphisms of double complexes induce morphisms between the column-filtered spectral sequences corresponding to the double complexes after applying the functor $F$ . To finish the proof, by Proposition 2.10, it is enough to check that these morphisms of spectral sequences are isomorphisms at the $E_{1}$ -level. We first consider the morphism $F(J^{\bullet ,\bullet })\rightarrow F(I^{\bullet ,\bullet })$ . For all $p$ , $J^{p,\bullet }\rightarrow I^{p,\bullet }$ is a morphism between two injective resolutions of $K^{p}$ extending the identity on $K^{p}$ , which induces an isomorphism $h^{q}(F(J^{p,\bullet }))\xrightarrow[{}]{{\sim}}h^{q}(F(I^{p,\bullet }))$ , both sides being equal to $R^{q}F(K^{p})$ by definition and being the $E_{1}^{p,q}$ -terms of the respective spectral sequences. In the case of the morphism $F(L^{\bullet ,\bullet })\rightarrow F(I^{\bullet ,\bullet })$ , we do not have injective resolutions of $K^{p}$ (only $F$ -acyclic ones) on the left-hand side, but by [Reference LangLan02, Theorem XX.6.2], this is enough: the $L^{p,\bullet }\rightarrow I^{p,\bullet }$ also give rise to isomorphisms after applying $F$ and taking cohomology. We conclude that the three column-filtered spectral sequences corresponding to the double complexes $F(J^{\bullet ,\bullet })$ , $F(I^{\bullet ,\bullet })$ , and $F(L^{\bullet ,\bullet })$ are isomorphic beginning with their $E_{1}$ -terms, completing the proof.◻

We will need one more type of spectral sequence, the Grothendieck composite-functor spectral sequence.

Proposition 2.14 [Reference WeibelWei94, Theorem 5.8.3].

Let ${\mathcal{A}}$ , ${\mathcal{B}}$ , and ${\mathcal{C}}$ be Abelian categories, and suppose ${\mathcal{A}}$ and ${\mathcal{B}}$ have enough injective objects. Let $F:{\mathcal{A}}\rightarrow {\mathcal{B}}$ and $G:{\mathcal{B}}\rightarrow {\mathcal{C}}$ be left-exact additive functors. Suppose that for every injective object $I$ of ${\mathcal{A}}$ , the object $F(I)$ of ${\mathcal{B}}$ is acyclic for $G$ . Then for every object $A$ of ${\mathcal{A}}$ , there is a spectral sequence which begins $E_{2}^{p,q}=(R^{p}G)((R^{q}F)(A))$ and abuts to $R^{p+q}(G\circ F)(A)$ .

Example 2.15. For our purposes, the most important example of a composite-functor spectral sequence is the spectral sequence for iterated local cohomology. Let $R$ be a Noetherian ring, and let $I$ and $J$ be ideals of $R$ . If ${\mathcal{I}}$ is an injective $R$ -module, then $\unicode[STIX]{x1D6E4}_{J}({\mathcal{I}})$ is again injective [Reference HartshorneHar77, Lemma III.3.2], hence acyclic for the functor $\unicode[STIX]{x1D6E4}_{I}$ . It follows that the left-exact functors $F=\unicode[STIX]{x1D6E4}_{J}$ and $G=\unicode[STIX]{x1D6E4}_{I}$ satisfy the conditions of Proposition 2.14. Since $R$ is Noetherian, $\unicode[STIX]{x1D6E4}_{I}\circ \unicode[STIX]{x1D6E4}_{J}=\unicode[STIX]{x1D6E4}_{I+J}$ . For any $R$ -module $M$ , the corresponding spectral sequence for the derived (local cohomology) functors begins $E_{2}^{p,q}=H_{I}^{p}(H_{J}^{q}(M))$ and abuts to $H_{I+J}^{p+q}(M)$ .

2.3 The proof of Theorem A

We now recall from [Reference HartshorneHar75] Hartshorne’s definition of de Rham homology for a complete local ring. Let $k$ be a field of characteristic zero and $A$ be a complete local ring with coefficient field $k$ . By Cohen’s structure theorem, there exists a surjection of $k$ -algebras $\unicode[STIX]{x1D70B}:R\rightarrow A$ where $R=k[[x_{1},\ldots ,x_{n}]]$ for some $n$ . Let $I\subset R$ be the kernel of this surjection. We have a corresponding closed immersion $Y{\hookrightarrow}X$ where $Y=\operatorname{Spec}(A)$ and $X=\operatorname{Spec}(R)$ . The de Rham homology of the local scheme $Y$ is defined as $H_{i}^{\text{dR}}(Y)=\mathbf{H}_{Y}^{2n-i}(X,\unicode[STIX]{x1D6FA}_{X}^{\bullet })$ , the hypercohomology (with support at  $Y$ ) of the continuous de Rham complex of sheaves of $k$ -spaces on $X$ . Here, the sheaf $\unicode[STIX]{x1D6FA}_{X}^{1}$ is free of rank $n$ with basis $dx_{1},\ldots ,dx_{n}$ , and the other sheaves in the complex are its corresponding exterior powers. In fact, the complex $\unicode[STIX]{x1D6FA}_{X}^{\bullet }$ is the sheafified version of the de Rham complex $\unicode[STIX]{x1D6FA}_{R}^{\bullet }$ of the left ${\mathcal{D}}(R,k)$ -module $R$ as defined in § 2.1.

The de Rham homology spaces defined above are known to be independent of the choice of $R$ and $\unicode[STIX]{x1D70B}$ [Reference HartshorneHar75, Proposition III.1.1] and to be finite-dimensional $k$ -spaces [Reference HartshorneHar75, Theorem III.2.1]. In this section we give arguments for the embedding-independence and the finiteness which are purely local and provide new information. Recall from § 1 that the Hodge–de Rham spectral sequence for homology has $E_{1}$ -term given by $E_{1}^{n-p,n-q}=H_{Y}^{n-q}(X,\unicode[STIX]{x1D6FA}_{X}^{n-p})$ and abuts to $H_{p+q}^{\text{dR}}(Y)$ . (When needed, we will write $\{E_{r,R}^{n-p,n-q}\}$ for this spectral sequence, recording the dependence on the base ring $R$ .) The assertion of Theorem A is that, beginning with the $E_{2}$ -term, this spectral sequence consists of finite-dimensional $k$ -spaces and its isomorphism class is independent (up to a bidegree shift) of $R$ and $\unicode[STIX]{x1D70B}$ ; this immediately recovers the embedding-independence and finiteness for the abutment $H_{\ast }^{\text{dR}}(Y)$ . To make the line of argument clearer, we give the proof first for the $E_{2}$ -term only (Proposition 2.17), then explain the additional steps needed to make the basic strategy work for the rest of the spectral sequence.

Lemma 2.16. Let the surjection $\unicode[STIX]{x1D70B}:R=k[[x_{1},\ldots ,x_{n}]]\rightarrow A$ (and the associated objects $I,X,Y$ ) be as above, and $\{E_{r}^{p,q}\}$ the corresponding Hodge–de Rham spectral sequence for homology.

  1. (a) For all $q$ , $H_{Y}^{q}(X,{\mathcal{O}}_{X})\simeq H_{I}^{q}(R)$ as $R$ -modules; indeed, if $M$ is any $R$ -module and ${\mathcal{F}}$ the associated quasi-coherent sheaf on $X$ , we have $H_{Y}^{q}(X,{\mathcal{F}})\simeq H_{I}^{q}(M)$ .

  2. (b) For all $p$ and $q$ , we have

    $$\begin{eqnarray}E_{2}^{p,q}\simeq H_{\text{dR}}^{p}(H_{Y}^{q}(X,{\mathcal{O}}_{X}))\simeq H_{\text{dR}}^{p}(H_{I}^{q}(R))\end{eqnarray}$$
    as $k$ -spaces, where the ${\mathcal{D}}$ -module structure on $H_{I}^{q}(R)\simeq H_{Y}^{q}(X,{\mathcal{O}}_{X})$ is defined as in Remark 2.3.

Proof. As $X$ is affine, part (a) is well known (e.g. [Reference Grothendieck and RaynaudGR05, Exp. II, Proposition 5]). Now consider the $E_{1}$ -term of the spectral sequence. Its differentials are horizontal and so its $E_{2}$ -objects are the cohomology objects of its rows. Fix such a row, say the $q$ th row, which takes the form $E_{1}^{\bullet ,q}=H_{Y}^{q}(X,\unicode[STIX]{x1D6FA}_{X}^{\bullet })$ . The $\unicode[STIX]{x1D6FA}_{X}^{p}$ are finite free sheaves on $X$ and local cohomology $H_{Y}^{q}$ commutes with direct sums, so for all $p$ , $E_{1}^{p,q}\simeq H_{Y}^{q}(X,{\mathcal{O}}_{X})\otimes \unicode[STIX]{x1D6FA}_{R}^{p}$ . As $p$ varies, we obtain the complex $H_{Y}^{q}(X,{\mathcal{O}}_{X})\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet }$ , whose $k$ -linear maps are the de Rham differentials of the ${\mathcal{D}}$ -module $H_{Y}^{q}(X,{\mathcal{O}}_{X})$ . Therefore its cohomology objects, the $E_{2}$ -objects of the spectral sequence, are of the stated form.◻

Proposition 2.17. Let the surjection $\unicode[STIX]{x1D70B}:R=k[[x_{1},\ldots ,x_{n}]]\rightarrow A$ (and the associated objects $I,X,Y$ ) be as above.

  1. (a) For all $p$ and $q$ , the $k$ -space $E_{2}^{p,q}=H_{\text{dR}}^{p}(H_{I}^{q}(R))$ is finite-dimensional.

  2. (b) Suppose we have another surjection of $k$ -algebras $\unicode[STIX]{x1D70B}^{\prime }:R^{\prime }=k[[x_{1},\ldots ,x_{n^{\prime }}]]\rightarrow A$ with kernel $I^{\prime }$ . Write $\{E_{r,R}^{p,q}\}$ (respectively $\{\mathbf{E}_{r,R^{\prime }}^{p,q}\}$ ) for the Hodge–de Rham spectral sequence for homology defined using $\unicode[STIX]{x1D70B}$ (respectively $\unicode[STIX]{x1D70B}^{\prime }$ ). Then for all $p$ and $q$ , the $k$ -spaces $E_{2,R}^{p,q}$ and $\mathbf{E}_{2,R^{\prime }}^{p+n^{\prime }-n,q+n-n^{\prime }}$ are isomorphic. (That is, the $E_{2}$ -term is independent of $R$ and $\unicode[STIX]{x1D70B}$ , up to a bidegree shift.)

Proof of part (a).

For all $q$ , the ${\mathcal{D}}$ -module $H_{I}^{q}(R)$ is holonomic [Reference LyubeznikLyu93, 2.2(d)], so its de Rham cohomology spaces are finite-dimensional by Theorem 2.2. This proves part (a).◻

A proof of part (b) is considerably longer. We first reduce it to Lemma 2.18 below and then prove Lemma 2.18.

Write $X^{\prime }$ for the spectrum of $R^{\prime }$ . The surjection $\unicode[STIX]{x1D70B}^{\prime }$ induces a closed immersion $Y{\hookrightarrow}X^{\prime }$ . Form the complete tensor product $R^{\prime \prime }=R\,\widehat{\otimes }_{k}\,R^{\prime }$ [Reference SerreSer00, V.B.2], again a complete regular $k$ -algebra, and let