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 $M\,dx_{j_{1}}\wedge \cdots \wedge dx_{j_{i}}$ .

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.

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.)

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.

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.

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 $\unicode[STIX]{x1D6FD}^{\prime p,q}\circ u_{\infty }^{p,q}=u_{p}^{p+q}\circ \unicode[STIX]{x1D6FD}^{p,q}$ .

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.

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.

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.

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.

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

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.

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 $\unicode[STIX]{x1D70B}^{\prime \prime }:R^{\prime \prime }\rightarrow A$ be the induced surjection $\unicode[STIX]{x1D70B}\,\widehat{\otimes }_{k}\,\unicode[STIX]{x1D70B}^{\prime }$ of $k$ -algebras, which gives rise to a third closed immersion $Y{\hookrightarrow}X^{\prime \prime }=\operatorname{Spec}(R^{\prime \prime })$ and a third Hodge–de Rham spectral sequence $\{E_{r,R^{\prime \prime }}^{p,q}\}$ . It suffices to show that both $E_{2,R}^{p,q}$ and $\mathbf{E}_{2,R^{\prime }}^{p+n^{\prime }-n,q+n-n^{\prime }}$ are isomorphic to $E_{2,R^{\prime \prime }}^{p+n^{\prime },q+n^{\prime }}$ . Replacing $R^{\prime }$ with $R^{\prime \prime }$ and using symmetry, we reduce to the case in which the two surjections $\unicode[STIX]{x1D70B}:R\rightarrow A$ and $\unicode[STIX]{x1D70B}^{\prime }:R^{\prime }\rightarrow A$ satisfy $\unicode[STIX]{x1D70B}^{\prime }=\unicode[STIX]{x1D70B}\circ g$ for some surjection $g:R^{\prime }\rightarrow R$ of $k$ -algebras. Let $I=\ker \unicode[STIX]{x1D70B}$ , $I^{\prime }=\ker \unicode[STIX]{x1D70B}^{\prime }$ , and $I^{\prime \prime }=\ker g$ , and suppose the dimensions of $R$ and $R^{\prime }$ are $n$ and $n^{\prime }$ respectively. As $R^{\prime }/I^{\prime \prime }\simeq R$ is regular, $I^{\prime \prime }$ is generated by $n^{\prime }-n$ elements that form part of a regular system of parameters for $R^{\prime }$ . By induction on $n^{\prime }-n$ , we reduce further to the case $n^{\prime }-n=1$ , since we can factor the closed immersion $X{\hookrightarrow}X^{\prime }$ into a sequence of codimension-one immersions, and the isomorphisms on $E_{2}$ -terms compose while the bidegree shifts add. Therefore we assume $\ker g$ is a principal ideal, of the form $(f)$ where $x_{1},\ldots ,x_{n},f$ is a regular system of parameters for $R^{\prime }$ . By Cohen’s structure theorem, the complete regular local $k$ -algebra $R^{\prime }$ takes the form $k[[x_{1},\ldots ,x_{n},z]]$ ; making a change of variables if necessary, we may assume $f=z$ . By Proposition 2.5, this change of variables does not affect de Rham cohomology. Thus $R=k[[x_{1},\ldots ,x_{n}]]$ , $R^{\prime }=R[[z]]$ , and $g$ is the surjection carrying $z$ to $0$ , so that $I^{\prime }=IR^{\prime }+(z)$ . We state this special case, to which we have reduced the proposition, in the form of the following lemma. Because of the amount of new notation needed to define them, we have not specified the isomorphisms in the statement of Lemma 2.18, only asserted their existence. However, the maps themselves will be needed later, and so we will refer not only to Lemma 2.18 but also to its proof.

Lemma 2.18. Let $R=k[[x_{1},\ldots ,x_{n}]]$ and let $I$ be an ideal of $R$ . Let $R^{\prime }=R[[z]]$ and $I^{\prime }=IR^{\prime }+(z)$ . Then for all $p$ and $q$ , we have an isomorphism

$$\begin{eqnarray}H_{\text{dR}}^{p}(H_{I}^{q}(R))\simeq H_{\text{dR}}^{p+1}(H_{I^{\prime }}^{q+1}(R^{\prime }))\end{eqnarray}$$

of $k$ -spaces, where the de Rham cohomology is computed by regarding $H_{I}^{q}(R)$ as a ${\mathcal{D}}(R,k)$ -module and $H_{I^{\prime }}^{q+1}(R^{\prime })$ as a ${\mathcal{D}}(R^{\prime },k)$ -module.

We first give a definition.

We will frequently refer to this functor as the ‘ $+$ -operation’ on $R$ -modules or $k$ -spaces. In the case of an $R$ -module $M$ , this definition coincides with the ‘key functor’ $G(M)=M\otimes _{R}R_{z}^{\prime }/R^{\prime }$ of Núñez–Betancourt and Witt [Reference Núñez-Betancourt and WittNBW14, § 3]. A special case of one of their results will be useful for us.

Proposition 2.20 [Reference Núñez-Betancourt and WittNBW14, Lemma 3.9].

With $R$ , $R^{\prime }$ , $I$ , and $I^{\prime }$ as in the statement of Lemma 2.18, we have isomorphisms

$$\begin{eqnarray}(H_{I}^{p}(R))_{+}\simeq H_{I^{\prime }}^{p+1}(R^{\prime })\end{eqnarray}$$

of $R^{\prime }$ -modules, for all $p$ (these isomorphisms are functorial in $R$ ).

We need one final ingredient before giving the proof of Lemma 2.18: a short exact sequence relating the ‘full’ de Rham complex $\unicode[STIX]{x1D6FA}_{R^{\prime }}^{\bullet }$ of $R^{\prime }$ with its ‘partial’ de Rham complex $R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet }$ defined using the derivations $\unicode[STIX]{x2202}_{1},\ldots ,\unicode[STIX]{x2202}_{n}$ but omitting $\unicode[STIX]{x2202}_{z}=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}z$ .

Definition 2.21. With $R$ and $R^{\prime }$ as in the statement of Lemma 2.18, we define a short exact sequence of complexes

$$\begin{eqnarray}0\rightarrow R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet }[-1]\xrightarrow[{}]{\unicode[STIX]{x1D704}}\unicode[STIX]{x1D6FA}_{R^{\prime }}^{\bullet }\xrightarrow[{}]{\unicode[STIX]{x1D70B}}R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet }\rightarrow 0,\end{eqnarray}$$

where the map $\unicode[STIX]{x1D704}$ is simply the wedge product with $dz$ , and so its image is precisely the direct sum of those summands of $\unicode[STIX]{x1D6FA}_{R^{\prime }}^{\bullet }$ with a $dz$ wedge factor (thus $\unicode[STIX]{x1D70B}$ corresponds to setting $dz=0$ ). The sheaf-theoretic analogue, a short exact sequence of complexes of sheaves on $X^{\prime }$ , is constructed similarly.

This short exact sequence of complexes gives rise to a long exact sequence of cohomology:

$$\begin{eqnarray}\displaystyle & & \displaystyle \cdots \rightarrow h^{p-1}(R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet })\xrightarrow[{}]{c^{p-1}}h^{p}(R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet }[-1])\rightarrow h^{p}(\unicode[STIX]{x1D6FA}_{R^{\prime }}^{\bullet })\nonumber\\ \displaystyle & & \displaystyle \qquad \rightarrow h^{p}(R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet })\xrightarrow[{}]{c^{p}}h^{p+1}(R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet }[-1])\rightarrow \cdots \,,\nonumber\end{eqnarray}$$

where c denotes the connecting homomorphism. After accounting for the shift of $-1$ , we see that $c^{p}$ is a map from the ${\mathcal{D}}(k[[z]],k)$ -module $h^{p}(R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet })$ to itself. We will need precisely to identify the maps $c^{p}$ .

Proof. We use the explicit construction of the connecting homomorphism given, for example, in [Reference RotmanRot09, Proposition 6.9]. Denote by $d_{R^{\prime }}^{\bullet }$ the differentials in the de Rham complex $\unicode[STIX]{x1D6FA}_{R^{\prime }}^{\bullet }$ . Given an element of $h^{p}(R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet })$ , which is the cohomology class of some cocycle $\unicode[STIX]{x1D714}\in R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{p}$ , the image under $c^{p}$ of this class is taken to be the class of the cocycle $(\unicode[STIX]{x1D704}^{p+1})^{-1}(d_{R^{\prime }}^{p}((\unicode[STIX]{x1D70B}^{p})^{-1}(\unicode[STIX]{x1D714})))$ , where the superscript $-1$ means ‘choose any preimage’: this definition is independent of all choices made. We proceed to calculate this composite, making convenient choices for the preimages.

We can write $\unicode[STIX]{x1D714}$ as a sum

$$\begin{eqnarray}\unicode[STIX]{x1D714}=\mathop{\sum }_{i_{1},\ldots ,i_{p}}\unicode[STIX]{x1D70C}_{i_{1}\cdots i_{p}}\,dx_{i_{1}}\wedge \cdots \wedge dx_{i_{p}},\end{eqnarray}$$

where all $\unicode[STIX]{x1D70C}_{i_{1}\cdots i_{p}}\in R^{\prime }$ . One choice of preimage $(\unicode[STIX]{x1D70B}^{p})^{-1}(\unicode[STIX]{x1D714})$ is $\unicode[STIX]{x1D714}$ itself, since none of its terms contain $dz$ wedge factors and hence all are left fixed by $\unicode[STIX]{x1D70B}^{p}$ . Therefore $d_{R^{\prime }}^{p}((\unicode[STIX]{x1D70B}^{p})^{-1}(\unicode[STIX]{x1D714}))=d_{R^{\prime }}^{p}(\unicode[STIX]{x1D714})$ . Since $\unicode[STIX]{x1D714}$ is a cocycle in $R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{p}$ , its image under the $p$ th de Rham differential with respect to $dx_{1},\ldots ,dx_{n}$ is zero, and so the only terms in the definition of $d_{R^{\prime }}^{p}(\unicode[STIX]{x1D714})$ that survive are those involving $dz$ . That is, we have

$$\begin{eqnarray}d_{R^{\prime }}^{p}(\unicode[STIX]{x1D714})=\mathop{\sum }_{i_{1},\ldots ,i_{p}}\unicode[STIX]{x2202}_{z}(\unicode[STIX]{x1D70C}_{i_{1}\cdots i_{p}})\,dz\wedge dx_{i_{1}}\wedge \cdots \wedge dx_{i_{p}},\end{eqnarray}$$

which, by rearranging the wedge terms, is equal to

$$\begin{eqnarray}\mathop{\sum }_{i_{1},\ldots ,i_{p}}(-1)^{p}\unicode[STIX]{x2202}_{z}(\unicode[STIX]{x1D70C}_{i_{1}\cdots i_{p}})\,dx_{i_{1}}\wedge \cdots \wedge dx_{i_{p}}\wedge dz.\end{eqnarray}$$

Finally, a choice of preimage under $\unicode[STIX]{x1D704}^{p+1}$ (which is simply the map $\wedge \,dz$ ) for the above sum is

$$\begin{eqnarray}(\unicode[STIX]{x1D704}^{p+1})^{-1}(d_{R^{\prime }}^{p}((\unicode[STIX]{x1D70B}^{p})^{-1}(\unicode[STIX]{x1D714})))=\mathop{\sum }_{i_{1},\ldots ,i_{p}}(-1)^{p}\unicode[STIX]{x2202}_{z}(\unicode[STIX]{x1D70C}_{i_{1}\cdots i_{p}})\,dx_{i_{1}}\wedge \cdots \wedge dx_{i_{p}}=(-1)^{p}\unicode[STIX]{x2202}_{z}(\unicode[STIX]{x1D714}),\end{eqnarray}$$

from which the lemma follows. (The same calculation works for arbitrary ${\mathcal{D}}(R^{\prime },k)$ -modules $M$ , replacing each $\unicode[STIX]{x1D70C}_{i_{1}\cdots i_{p}}$ with an element $m_{i_{1}\cdots i_{p}}$ of $M$ .)◻

We can now prove Lemma 2.18.

Proof of Lemma 2.18.

The differentials in the complexes of Definition 2.21 are merely $k$ -linear, but in every degree $p$ , the short exact sequence

$$\begin{eqnarray}0\rightarrow R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{p-1}\xrightarrow[{}]{\unicode[STIX]{x1D704}^{p}}\unicode[STIX]{x1D6FA}_{R^{\prime }}^{p}\rightarrow R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{p}\rightarrow 0\end{eqnarray}$$

is a split exact sequence of finite free $R^{\prime }$ -modules. As local cohomology commutes with direct sums, this sequence remains split exact after applying the functor $H_{I^{\prime }}^{q}$ for any $q$ :

$$\begin{eqnarray}0\rightarrow H_{I^{\prime }}^{q}(R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{p-1})\xrightarrow[{}]{\unicode[STIX]{x1D704}_{q}^{p}}H_{I^{\prime }}^{q}(\unicode[STIX]{x1D6FA}_{R^{\prime }}^{p})\rightarrow H_{I^{\prime }}^{q}(R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{p})\rightarrow 0.\end{eqnarray}$$

Fixing $q$ but varying $p$ , we obtain a short exact sequence of complexes of $k$ -spaces

$$\begin{eqnarray}0\rightarrow H_{I^{\prime }}^{q}(R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet }[-1])\xrightarrow[{}]{\unicode[STIX]{x1D704}_{q}^{\bullet }}H_{I^{\prime }}^{q}(\unicode[STIX]{x1D6FA}_{R^{\prime }}^{\bullet })\rightarrow H_{I^{\prime }}^{q}(R^{\prime }\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet })\rightarrow 0\end{eqnarray}$$

which we can rewrite (replacing $q$ with $q+1$ ) as

$$\begin{eqnarray}0\rightarrow H_{I^{\prime }}^{q+1}(R^{\prime })\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet }[-1]\xrightarrow[{}]{\unicode[STIX]{x1D704}_{q+1}^{\bullet }}H_{I^{\prime }}^{q+1}(R^{\prime })\otimes \unicode[STIX]{x1D6FA}_{R^{\prime }}^{\bullet }\rightarrow H_{I^{\prime }}^{q+1}(R^{\prime })\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet }\rightarrow 0\end{eqnarray}$$

since $\unicode[STIX]{x1D6FA}_{R}^{i}$ (respectively $\unicode[STIX]{x1D6FA}_{R^{\prime }}^{i}$ ) is a finite free $R$ - (respectively $R^{\prime }$ -) module. This short exact sequence of complexes gives rise to a long exact sequence of cohomology which (accounting for the shift of $-1$ ) takes the form

$$\begin{eqnarray}\displaystyle \cdots \rightarrow h^{p}(H_{I^{\prime }}^{q+1}(R^{\prime })\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet }) & \xrightarrow[{}]{\unicode[STIX]{x2202}_{z}} & \displaystyle h^{p}(H_{I^{\prime }}^{q+1}(R^{\prime })\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet })\xrightarrow[{}]{\overline{\unicode[STIX]{x1D704}}_{q+1}^{p}}h^{p+1}(H_{I^{\prime }}^{q+1}(R^{\prime })\otimes \unicode[STIX]{x1D6FA}_{R^{\prime }}^{\bullet })\nonumber\\ \displaystyle & \rightarrow & \displaystyle h^{p+1}(H_{I^{\prime }}^{q+1}(R^{\prime })\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet })\xrightarrow[{}]{\unicode[STIX]{x2202}_{z}}h^{p+1}(H_{I^{\prime }}^{q+1}(R^{\prime })\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet })\rightarrow \cdots \,,\nonumber\end{eqnarray}$$

where we know by Lemma 2.22 that, up to a sign, the connecting homomorphism is $\unicode[STIX]{x2202}_{z}$ . Now by Lemma 2.20, we know that $H_{I^{\prime }}^{q+1}(R^{\prime })\simeq (H_{I}^{q}(R))_{+}$ as $R^{\prime }$ -modules. The differentials in the complex $H_{I^{\prime }}^{q+1}(R^{\prime })\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet }$ do not involve $z$ or $dz$ , and so the $+$ -operation passes to its cohomology, since cohomology commutes with direct sums: we have

$$\begin{eqnarray}h^{p}(H_{I^{\prime }}^{q+1}(R^{\prime })\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet })\simeq (h^{p}(H_{I}^{q}(R)\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet }))_{+}\end{eqnarray}$$

as $k$ -spaces for all $p$ . For any $k$ -space $M$ , the action of $\unicode[STIX]{x2202}_{z}$ on $M_{+}$ is given in Definition 2.19, and it is clear from this definition (since $\operatorname{char}(k)=0$ ) that $\ker (\unicode[STIX]{x2202}_{z}:M_{+}\rightarrow M_{+})=0$ and $\operatorname{coker}(\unicode[STIX]{x2202}_{z}:M_{+}\rightarrow M_{+})\simeq M$ , the latter corresponding to the $1/z$ -component of $M_{+}$ . Returning to the displayed portion of the long exact sequence (with $M=h^{p}(H_{I}^{q}(R)\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet })$ ), the second $\unicode[STIX]{x2202}_{z}$ is injective, and so by exactness the unlabeled arrow is the zero map; this implies that $\overline{\unicode[STIX]{x1D704}}_{q+1}^{p}$ is surjective, inducing an isomorphism between $h^{p+1}(H_{I^{\prime }}^{q+1}(R^{\prime })\otimes \unicode[STIX]{x1D6FA}_{R^{\prime }}^{\bullet })=H_{\text{dR}}^{p+1}(H_{I^{\prime }}^{q+1}(R^{\prime }))$ and the cokernel of the first $\unicode[STIX]{x2202}_{z}$ . Since this cokernel is isomorphic to $h^{p}(H_{I}^{q}(R)\otimes \unicode[STIX]{x1D6FA}_{R}^{\bullet })=H_{\text{dR}}^{p}(H_{I}^{q}(R))$ , the proof of Lemma 2.18, and hence of Proposition 2.17(b), is complete.◻

Proposition 2.17 gives a new set of invariants for complete local rings in equicharacteristic zero, namely, the (finite) dimensions of the $E_{2}$ -objects, with the bidegree shift taken into account.

By Proposition 2.17, $\unicode[STIX]{x1D70C}_{p,q}$ is finite and depends only on $A$ and a choice of coefficient field $k\subset A$ . We note the similarity of the definition of the invariants $\unicode[STIX]{x1D70C}_{p,q}$ to the Lyubeznik numbers $\unicode[STIX]{x1D706}_{p,q}$ [Reference LyubeznikLyu93, Theorem–Definition 4.1], although our $\unicode[STIX]{x1D70C}_{p,q}$ appear to be well defined only in the characteristic zero case. One way to define $\unicode[STIX]{x1D706}_{p,q}$ is as the dimension of the socle of $H_{\mathfrak{m}}^{p}(H_{I}^{n-q}(R))$ , where $\mathfrak{m}\subset R$ is the maximal ideal [Reference LyubeznikLyu06, Lemma 2.2]. To define the $\unicode[STIX]{x1D70C}_{p,q}$ , we use de Rham cohomology instead of iterated local cohomology; furthermore, note the difference in the indices.

We now prove the full statement of Theorem A(a) (we have already proved part (b) above). Our goal is to construct a bidegree-shifted morphism between the Hodge–de Rham spectral sequences arising from two surjections $R\rightarrow A$ and $R^{\prime }\rightarrow A$ of $k$ -algebras which, at the level of $E_{2}$ -objects, consists of the isomorphisms of Lemma 2.18: by Proposition 2.12, this is enough. The preliminary reductions given in the paragraph before Lemma 2.18 remain valid when considering the spectral sequences, so we need only address the case in which $R$ , $R^{\prime }$ , $I$ , and $I^{\prime }$ are as in the statement of that lemma. The basic strategy is the same: we use the short exact sequence Definition 2.21 of complexes as well as the $+$ -operation, which will now be applied to double complex resolutions of the de Rham complexes $\unicode[STIX]{x1D6FA}_{X}^{\bullet }$ and $\unicode[STIX]{x1D6FA}_{X^{\prime }}^{\bullet }$ . Our first task is to construct these; we work first at the level of $R$ - (respectively $R^{\prime }$ -) modules and $k$ -linear maps, and then sheafify the results.

Likewise, if we let ${\mathcal{J}}^{\bullet }$ be the minimal injective resolution of $R^{\prime }$ as an $R^{\prime }$ -module, Lemma 2.25 implies that $R^{\prime }\rightarrow {\mathcal{J}}^{\bullet }$ is a complex in the category of ${\mathcal{D}}(R^{\prime },k)$ -modules. By taking finite direct sums of the resolutions ${\mathcal{I}}^{\bullet }$ and ${\mathcal{J}}^{\bullet }$ , we construct three double complexes.

Note that these double complexes have exact sequences of $R$ - (or $R^{\prime }$ -) modules for columns, but merely complexes in the category of $k$ -spaces for rows. In the case of ${\mathcal{I}}^{\bullet ,\bullet }$ , the rows are the de Rham complexes of the ${\mathcal{D}}(R,k)$ -modules ${\mathcal{I}}^{q}$ ; in the case of ${\mathcal{J}}_{0}^{\bullet ,\bullet }$ , the rows are the de Rham complexes of the ${\mathcal{J}}^{q}$ regarded as ${\mathcal{D}}(R,k)$ -modules; and in the case of ${\mathcal{J}}^{\bullet ,\bullet }$ , the rows are the de Rham complexes of the ${\mathcal{J}}^{q}$ regarded as ${\mathcal{D}}(R^{\prime },k)$ -modules. (We recall again Convention 2.1: if we write $\otimes \,\unicode[STIX]{x1D6FA}_{R}^{\bullet }$ , the tensor products of objects are being taken over $R$ , but if we write $\otimes \,\unicode[STIX]{x1D6FA}_{R^{\prime }}^{\bullet }$ , the tensor products of objects are being taken over $R^{\prime }$ .)

Each of these three double complexes can be sheafified. Consider first the double complex ${\mathcal{I}}^{\bullet ,\bullet }$ . For all $p$ and $q$ , let $\widetilde{{\mathcal{I}}^{p,q}}$ denote the associated quasi-coherent sheaf on $X$ . The vertical differentials of ${\mathcal{I}}^{\bullet ,\bullet }$ are $R$ -linear, and so induce ${\mathcal{O}}_{X}$ -linear morphisms between the associated sheaves, and the horizontal differentials induce $k$ -linear morphisms on the associated sheaves in the same way that the de Rham complex of ${\mathcal{O}}_{X}$ is constructed. For all $p$ and $q$ , ${\mathcal{I}}^{p,q}$ is an injective $R$ -module, and so the sheaf $\widetilde{{\mathcal{I}}^{p,q}}$ is flasque [Reference HartshorneHar77, Proposition III.3.4], and hence acyclic for the functor $\unicode[STIX]{x1D6E4}_{Y}$ on the category of sheaves of $k$ -spaces on $X$ [Reference HartshorneHar67, Proposition 1.10]. Therefore we have a double complex $\widetilde{{\mathcal{I}}^{\bullet ,\bullet }}$ whose objects are all $\unicode[STIX]{x1D6E4}_{Y}$ -acyclic sheaves of $k$ -spaces on $X$ and whose columns are acyclic resolutions of the $\unicode[STIX]{x1D6FA}_{X}^{p}$ (because the associated sheaf functor is exact when applied to complexes of $R$ -modules). In the same way, we sheafify the double complexes ${\mathcal{J}}_{0}^{\bullet ,\bullet }$ and ${\mathcal{J}}^{\bullet ,\bullet }$ , obtaining double complexes $\widetilde{{\mathcal{J}}_{0}^{\bullet ,\bullet }}$ and $\widetilde{{\mathcal{J}}^{\bullet ,\bullet }}$ of sheaves of $k$ -spaces on $X^{\prime }$ which are $\unicode[STIX]{x1D6E4}_{Y}$ -acyclic.

By Lemma 2.13, we know that $E_{\bullet ,R}^{\bullet ,\bullet }$ coincides with the Hodge–de Rham spectral sequence for the complex $\unicode[STIX]{x1D6FA}_{X}^{\bullet }$ , and that $\mathbf{E}_{\bullet ,R^{\prime }}^{\bullet ,\bullet }$ coincides with the Hodge–de Rham spectral sequence for the complex $\unicode[STIX]{x1D6FA}_{X^{\prime }}^{\bullet }$ , so there is no ambiguity of notation. The ‘intermediate’ spectral sequence ${\mathcal{E}}_{\bullet }^{\bullet ,\bullet }$ , which by Lemma 2.13 coincides with the hypercohomology spectral sequence for the complex ${\mathcal{O}}_{X^{\prime }}\otimes \unicode[STIX]{x1D6FA}_{X}^{\bullet }$ , will be used to relate the Hodge–de Rham spectral sequences for $X$ and $X^{\prime }$ via the $+$ -operation. The first step in this process is the following lemma.

Lemma 2.28. Let $R$ , $R^{\prime }$ , $I$ , and $I^{\prime }$ be as in the statement of Lemma 2.18, and let ${\mathcal{I}}^{\bullet }$ (respectively ${\mathcal{J}}^{\bullet }$ ) be the minimal injective resolution of $R$ (respectively $R^{\prime }$ ) in the category of $R$ -modules (respectively $R^{\prime }$ -modules) as above. Then for all $q$ , we have an isomorphism

$$\begin{eqnarray}(\unicode[STIX]{x1D6E4}_{I}({\mathcal{I}}^{q-1}))_{+}\simeq \unicode[STIX]{x1D6E4}_{I^{\prime }}({\mathcal{J}}^{q})\end{eqnarray}$$

of ${\mathcal{D}}(R^{\prime },k)$ -modules.

Proof. Fix $q\geqslant 0$ . As $R^{\prime }$ is a Gorenstein local ring, the structure of its minimal injective resolution ${\mathcal{J}}^{\bullet }$ is well known [Reference MatsumuraMat86, Theorem 18.8]: ${\mathcal{J}}^{q}=\bigoplus _{\operatorname{ht}\mathfrak{p}=q}E(R^{\prime }/\mathfrak{p})$ , where $E(R^{\prime }/\mathfrak{p})$ is the $R^{\prime }$ -injective hull of $R^{\prime }/\mathfrak{p}$ . In particular, ${\mathcal{J}}^{q}=0$ for $q>n+1$ and ${\mathcal{J}}^{n+1}$ is the Matlis dualizing module $E_{R^{\prime }}$ . Applying the functor $\unicode[STIX]{x1D6E4}_{I^{\prime }}$ amounts to discarding those summands corresponding to prime ideals outside the closed subscheme $V(I^{\prime })\subset X^{\prime }$ [Reference Brodmann and SharpBS13, Example 10.1.11]: that is, we have the equality

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{I^{\prime }}({\mathcal{J}}^{q})=\underset{\operatorname{ht}\mathfrak{p}=q,I^{\prime }\subset \mathfrak{p}}{\bigoplus }E_{R^{\prime }}(R^{\prime }/\mathfrak{p}),\end{eqnarray}$$

where again only prime ideals of height $q$ appear in the decomposition, since ${\mathcal{J}}^{q}=\bigoplus _{\operatorname{ht}\mathfrak{p}=q}E(R^{\prime }/\mathfrak{p})$ . Since $I^{\prime }=IR^{\prime }+(z)$ , there is a one-to-one correspondence between prime ideals $\mathfrak{p}$ of $R^{\prime }$ containing $I^{\prime }$ and prime ideals $\mathfrak{q}$ of $R=R^{\prime }/(z)$ containing $I$ (indeed, any such $\mathfrak{p}$ takes the form $\mathfrak{q}R^{\prime }+(z)$ ). If $\operatorname{ht}\mathfrak{p}=q$ , then $\operatorname{ht}\mathfrak{q}=q-1$ , so we have the decomposition

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{I^{\prime }}({\mathcal{J}}^{q})=\underset{I\subset \mathfrak{q}\in \operatorname{Spec}(R),\operatorname{ht}\mathfrak{q}=q-1}{\bigoplus }E_{R^{\prime }}(R^{\prime }/(\mathfrak{q}R^{\prime }+(z)))\end{eqnarray}$$

as $R^{\prime }$ -modules. Note that we have $R^{\prime }$ -module isomorphisms

$$\begin{eqnarray}R^{\prime }/(\mathfrak{q}R^{\prime }+(z))\simeq (R^{\prime }/(z))/((\mathfrak{q}R^{\prime }+(z))/(z))\simeq R/\mathfrak{q},\end{eqnarray}$$

where $R/\mathfrak{q}$ is viewed as an $R$ -module upon which $z\in R^{\prime }$ acts trivially. But by [Reference Núñez-Betancourt and WittNBW14, Proposition 3.11], the $R^{\prime }$ -module $E_{R^{\prime }}(R^{\prime }/(\mathfrak{q}R^{\prime }+(z)))$ is obtained from the $R$ -module $E_{R}(R/\mathfrak{q})$ by the $+$ -operation. (This identification holds at the level of ${\mathcal{D}}(R^{\prime },k)$ -modules.) We then have isomorphisms

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{I^{\prime }}({\mathcal{J}}^{q})\simeq \underset{I\subset \mathfrak{q}\in \operatorname{Spec}(R),\operatorname{ht}\mathfrak{q}=q-1}{\bigoplus }(E_{R}(R/\mathfrak{q}))_{+}=(\unicode[STIX]{x1D6E4}_{I}({\mathcal{I}}^{q-1}))_{+},\end{eqnarray}$$

of ${\mathcal{D}}(R^{\prime },k)$ -modules, where we have again used [Reference Brodmann and SharpBS13, Example 10.1.11].◻

Lemma 2.29. Let $E_{\bullet ,R}^{\bullet ,\bullet }$ and ${\mathcal{E}}_{\bullet }^{\bullet ,\bullet }$ be the spectral sequences of Definition 2.27. There is an isomorphism

$$\begin{eqnarray}(E_{\bullet ,R}^{\bullet ,\bullet })_{+}\xrightarrow[{}]{{\sim}}{\mathcal{E}}_{\bullet }^{\bullet ,\bullet }[(0,1)],\end{eqnarray}$$

where the object on the left-hand side is obtained by applying the $+$ -operation to all the objects and differentials of the spectral sequence $E_{\bullet ,R}^{\bullet ,\bullet }$ (this notation means that the morphism of spectral sequences has bidegree $(0,1)$ , as in Definition 2.11).

Proof. We consider the objects of the double complexes giving rise to these spectral sequences. For all $p$ and $q$ , we have

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{Y}(X^{\prime },\widetilde{{\mathcal{J}}_{0}^{p,q+1}})\simeq \unicode[STIX]{x1D6E4}_{I^{\prime }}({\mathcal{J}}_{0}^{p,q+1})=\unicode[STIX]{x1D6E4}_{I^{\prime }}({\mathcal{J}}^{q+1}\otimes \unicode[STIX]{x1D6FA}_{R}^{p})\simeq \unicode[STIX]{x1D6E4}_{I^{\prime }}({\mathcal{J}}^{q+1})\otimes \unicode[STIX]{x1D6FA}_{R}^{p}\end{eqnarray}$$

and similarly

$$\begin{eqnarray}(\unicode[STIX]{x1D6E4}_{Y}(X,\widetilde{{\mathcal{I}}^{p,q}}))_{+}\simeq (\unicode[STIX]{x1D6E4}_{I}({\mathcal{I}}^{p,q}))_{+}=(\unicode[STIX]{x1D6E4}_{I}({\mathcal{I}}^{q}\otimes \unicode[STIX]{x1D6FA}_{R}^{p}))_{+}\simeq (\unicode[STIX]{x1D6E4}_{I}({\mathcal{I}}^{q})\otimes \unicode[STIX]{x1D6FA}_{R}^{p})_{+}\end{eqnarray}$$

by Lemma 2.16 and the fact that $\unicode[STIX]{x1D6E4}_{I}=H_{I}^{0}$ and $\unicode[STIX]{x1D6E4}_{I^{\prime }}=H_{I^{\prime }}^{0}$ commute with direct sums. By Lemma 2.28, we have $\unicode[STIX]{x1D6E4}_{I^{\prime }}({\mathcal{J}}^{q+1})\simeq (\unicode[STIX]{x1D6E4}_{I}({\mathcal{I}}^{q}))_{+}$ for all $q$ . Therefore, for all $p$ and $q$ , we have

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{I^{\prime }}({\mathcal{J}}^{q+1})\otimes \unicode[STIX]{x1D6FA}_{R}^{p}\simeq (\unicode[STIX]{x1D6E4}_{I}({\mathcal{I}}^{q}))_{+}\otimes \unicode[STIX]{x1D6FA}_{R}^{p}\simeq (\unicode[STIX]{x1D6E4}_{I}({\mathcal{I}}^{q})\otimes \unicode[STIX]{x1D6FA}_{R}^{p})_{+},\end{eqnarray}$$

so the objects of the double complexes are isomorphic with the indicated bidegree shift. Finally, we observe that the differentials in the complex $\unicode[STIX]{x1D6FA}_{R}^{\bullet }$ do not involve $z$ or $dz$ , so the isomorphisms $(\unicode[STIX]{x1D6E4}_{I}({\mathcal{I}}^{q}))_{+}\otimes \unicode[STIX]{x1D6FA}_{R}^{p}\simeq (\unicode[STIX]{x1D6E4}_{I}({\mathcal{I}}^{q})\otimes \unicode[STIX]{x1D6FA}_{R}^{p})_{+}$ commute with the differentials of the double complex and thus assemble to an isomorphism of double complexes. An isomorphism of double complexes induces an isomorphism between the corresponding column-filtered spectral sequences, and the lemma follows.◻

We are now ready to complete the proof of Theorem A.

Proof of Theorem A(a).

As already described, we need only prove the result in the special case of Lemma 2.18. We retain the notation of that lemma. Consider again the short exact sequence of Definition 2.21 and its sheafified version

$$\begin{eqnarray}0\rightarrow {\mathcal{O}}_{X^{\prime }}\otimes \unicode[STIX]{x1D6FA}_{X}^{\bullet }[-1]\xrightarrow[{}]{\unicode[STIX]{x1D704}}\unicode[STIX]{x1D6FA}_{X^{\prime }}^{\bullet }\rightarrow {\mathcal{O}}_{X^{\prime }}\otimes \unicode[STIX]{x1D6FA}_{X}^{\bullet }\rightarrow 0.\end{eqnarray}$$

As described in § 2.2, the morphism of complexes $\unicode[STIX]{x1D704}$ induces a morphism between the corresponding spectral sequences for hypercohomology supported at $Y$ . These spectral sequences were identified as ${\mathcal{E}}_{\bullet }^{\bullet ,\bullet }[(-1,0)]$ (respectively, $\mathbf{E}_{\bullet ,R^{\prime }}^{\bullet ,\bullet }$ ) in the paragraph following Definition 2.27. Accounting for the shift of $-1$ , we see that this induced morphism has the following form:

$$\begin{eqnarray}\unicode[STIX]{x1D704}_{\bullet }^{\bullet ,\bullet }:{\mathcal{E}}_{\bullet }^{\bullet ,\bullet }\rightarrow \mathbf{E}_{\bullet ,R^{\prime }}^{\bullet ,\bullet }[(1,0)].\end{eqnarray}$$

Identifying first ${\mathcal{E}}_{\bullet }^{\bullet ,\bullet }[(0,1)]$ with $(E_{\bullet ,R}^{\bullet ,\bullet })_{+}$ (by Lemma 2.29) and then $E_{\bullet ,R}^{\bullet ,\bullet }$ with the $1/z$ -component of $(E_{\bullet ,R}^{\bullet ,\bullet })_{+}$ , we see that this further induces a morphism

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{\bullet }^{\bullet ,\bullet }:E_{\bullet ,R}^{\bullet ,\bullet }\rightarrow \mathbf{E}_{\bullet ,R^{\prime }}^{\bullet ,\bullet }[(1,1)],\end{eqnarray}$$

given in every degree by the inclusion of $E_{r,R}^{\bullet ,\bullet }$ as the $1/z$ -component of $(E_{r,R}^{\bullet ,\bullet })_{+}\simeq {\mathcal{E}}_{r}^{\bullet ,\bullet }[(0,1)]$ followed by $\unicode[STIX]{x1D704}_{\bullet }^{\bullet ,\bullet }$ . If $r=2$ , the maps $\unicode[STIX]{x1D719}_{2}^{p,q}$ are precisely the isomorphisms $H_{\text{dR}}^{p}(H_{I}^{q}(R))\xrightarrow[{}]{{\sim}}H_{\text{dR}}^{p+1}(H_{I^{\prime }}^{q+1}(R^{\prime }))$ appearing in the proof of Lemma 2.18, which were induced by the morphism of complexes $\unicode[STIX]{x1D704}$ and the inclusion of $E_{2,R}^{p,q}=H_{\text{dR}}^{p}(H_{I}^{q}(R))$ as the $1/z$ -component of $(E_{2,R}^{p,q})_{+}$ . Therefore the morphism $\unicode[STIX]{x1D719}_{\bullet }^{\bullet ,\bullet }$ of spectral sequences is an isomorphism at the $E_{2}$ -level. By Proposition 2.12, it follows that $\unicode[STIX]{x1D719}$ is an isomorphism at all later levels, including the abutments. The proof is complete.◻