# Exceptional complete intersection
maps of local rings

Srikanth B. Iyengar  Srikanth B. Iyengar, Department of Mathematics,
University of Utah, Salt Lake City, UT 84112, U.S.A<EMAIL_ADDRESS>,
Janina C. Letz  Janina C. Letz, Faculty of Mathematics, Bielefeld University,
PO Box 100 131, 33501 Bielefeld, Germany<EMAIL_ADDRESS>,  Jian
Liu  Jian Liu, School of Mathematical Sciences, Shanghai Jiao Tong University,
Shanghai 200240, P.R. China<EMAIL_ADDRESS>and  Josh Pollitz  Josh
Pollitz, Department of Mathematics, University of Utah, Salt Lake City, UT
84112, U.S.A<EMAIL_ADDRESS>
(Date: 22 February 2022)

###### Abstract.

This work concerns surjective maps $\varphi\colon R\to S$ of commutative
noetherian local rings with kernel generated by a regular sequence that is
part of a minimal generating set for the maximal ideal of $R$. The main result
provides criteria for detecting such exceptional complete intersection maps in
terms of the lattices of thick subcategories of the derived category of
complexes of finite length homology. A key input is a characterization of such
maps in terms of the truncated Atiyah class of $\varphi$.

###### Key words and phrases:

exceptional complete intersection, Hochschild cohomology, truncated Atiyah
class, thick subcategory

###### 2020 Mathematics Subject Classification:

13B10 (primary); 13D09, 13D03 (secondary)

This work was partly supported by National Science Foundation DMS grants
2001368 (SBI) and 2002173 (JP) and by the Alexander von Humboldt Foundation in
the framework of an Alexander von Humboldt Professorship endowed by the German
Federal Ministry of Education and Research (JCL)

## Introduction

This work is part of an ongoing project to understand properties of
commutative noetherian rings, and of maps between them, in terms of the
triangulated structure of their derived categories. Building on the work of
Dwyer, Greenlees and the first author [20], the fourth author [35] established
a homotopical description of locally complete intersection rings: These are
the commutative noetherian rings with the property that any nonzero thick
subcategory of its bounded derived category contains a nonzero perfect
complex. Similar homotopical characterizations of the complete intersection
property have since appeared in the works of Briggs, Grifo and the authors
[14, 15, 30]; the most general result is contained in [15].

A surjective map ${\varphi}\colon R\to S$ of local rings is complete
intersection if its kernel is generated by an $R$-regular sequence. The focus
of the present manuscript is on a special class of complete intersection maps
where, in addition, the regular sequence is part of a minimal generating set
for the maximal ideal of $R$. Such maps arise naturally in various contexts;
for example, the diagonal of a smooth map has this property locally. They also
arose in recent work of the first author with Brochard and Khare [16], where
they are called _exceptional complete intersection_ maps; we adopt this
terminology.

Our motivation for the present undertaking comes from [15] that raises the
prospect of defining locally complete intersection maps in the broader setting
of tensor triangulated categories. However to realize this goal requires a
homotopical description of the diagonal of a smooth map, and more generally,
of a weakly regular map. Our main result provides such a description for
general exceptional complete intersection maps; see also the discussion in
5.8. Let ${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$ be the subcategory of the
derived category of $R$ consisting of complexes with finite length homology.
The statement involves the functor
${\varphi}_{*}\colon{\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)\to{\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$
induced by restriction along ${\varphi}$.

######  Theorem A.

Let ${\varphi}\colon R\to S$ be a surjective map of commutative noetherian
local rings. Then ${\varphi}$ is exceptional complete intersection if and only
if $\operatorname{proj\,dim}_{R}S$ is finite and ${\varphi}_{*}$ induces an
isomorphism between the lattice of thick subcategories of
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$ and that of
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$.

See Corollary 5.10 for a more precise statement. The exceptional complete
intersection property does not localize and so, unsurprisingly, our
characterization is in terms of the complexes supported only at the maximal
ideals of $R$ and $S$. One cannot, in general, relax this hypothesis to allow
thick subcategories with bigger support; see Example 3.7 and Lemma 5.3. It is
also worth pointing out that the characterizations of the complete
intersection property in [14, 15, 35] use objects from
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$ in an essential way.

Questions about the thick subcategories of
${\mathsf{D}}^{\sf{f}}_{\sf{b}}(R)$, the bounded derived category of $R$, can
often be reduced to those about the thick subcategories of
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R_{\mathfrak{p}})$ as $\mathfrak{p}$ ranges
over the prime ideals in $R$; this is the upshot of local-to-global principles
from [12]. When the ring $R$ is Cohen–Macaulay, A allows one to further reduce
to the case of artinian rings. This idea is illustrated in Corollary 5.13.

To each $S$-complex $M$ there is a naturally defined morphism in
${\mathsf{D}}(S)$ called the _truncated Atiyah class of ${\varphi}$ at $M$_,
denoted $\mathsf{at}^{{\varphi}}(M)$, which was introduced in [27]; see 2.1.
The following characterization of exceptional complete intersection maps in
terms of the truncated Atiyah class of $k$ is an input in our proof of A.

######  Theorem B.

Let ${\varphi}\colon R\to S$ be a surjective map of commutative noetherian
local rings, and let $k$ be their common residue field. Then ${\varphi}$ is
exceptional complete intersection if and only if
$\operatorname{proj\,dim}_{R}S$ is finite and $\mathsf{at}^{{\varphi}}(k)=0$.

This result is part of Theorem 3.4. It is essentially contained in [3,
Proposition 2.8] and [15, 3.11], though couched in terms of actions of
Hochschild cohomology. Our contribution is to clarify the connection with the
Atiyah class; see, especially, Lemma 2.8.

B also sheds light on the faithfulness (or lack thereof) of the restriction
functor
${\varphi}_{*}\colon{\mathsf{D}}^{\sf{f}}_{\sf{b}}(S)\to{\mathsf{D}}^{\sf{f}}_{\sf{b}}(R)$,
at least when ${\varphi}$ is complete intersection. In this case,
${\varphi}_{*}$ is faithful if and only if $\mathsf{at}^{{\varphi}}(M)$ is
zero for each $M$ in ${\mathsf{D}}(S)$; see 3.8. This holds, for example, when
${\varphi}$ is an algebra retract, but not always, even for exceptional
complete intersection maps. This is connected to the “lifting problem”; see
Example 3.9. This connection also makes it clear that in B the vanishing of
$\mathsf{at}^{{\varphi}}(k)$ cannot be replaced with the stronger conclusion
that $\mathsf{at}^{{\varphi}}(M)$ is zero for each $M$ in ${\mathsf{D}}(S)$;
see Examples 3.7 and 3.9. One can thus view the non-vanishing of the truncated
Atiyah class as a homological obstruction to ${\varphi}$ admitting an algebra
retract.

##  1\. Hochschild cohomology

The Atiyah class of a map is closely connected to Hochschild cohomology of the
algebras involved. In this section we record the necessary facts on Hochschild
cohomology and their characteristic action on derived categories. Some
constructions require dg (=differential graded) algebras and the corresponding
derived category of dg modules; everything we need is explained in [4, 7].

######  1.1.

We only consider graded-commutative dg algebras, so let $A$ be such a dg
algebra. The derived category of dg $A$-modules is denoted ${\mathsf{D}}(A)$,
and its suspension functor is denoted ${\sf\Sigma}$. One has the usual bi-
functors $\operatorname{RHom}_{A}(-,-)$ and $-\otimes^{\operatorname{L}}_{A}-$
on ${\mathsf{D}}(A)$; for dg $A$-modules $M$ and $N$, we set

$\operatorname{Ext}^{*}_{A}(M,N)\colonequals\operatorname{H}^{*}(\operatorname{RHom}_{A}(M,N))\text{
and
}\operatorname{Tor}^{A}_{*}(M,N)\colonequals\operatorname{H}_{*}(M\otimes^{\operatorname{L}}_{A}N)\,.$

An element $\alpha$ in $\operatorname{Ext}_{A}^{n}(M,N)$ corresponds to a
morphism $M\to{\sf\Sigma}^{n}N$ in ${\mathsf{D}}(A)$; by a slight abuse of
notation we write also $\alpha$ for this morphism. Given $\alpha\colon
M\to{\sf\Sigma}^{n}M$ we write $\alpha^{i}$ for the composition

$M\xrightarrow{\ \alpha\
}{\sf\Sigma}^{n}M\xrightarrow{{\sf\Sigma}^{n}\alpha}\ldots\xrightarrow{{\sf\Sigma}^{n(i-1)}\alpha}{\sf\Sigma}^{ni}M$

in ${\mathsf{D}}(A)$, and say $\alpha$ is _nilpotent_ if $\alpha^{i}=0$ in
${\mathsf{D}}(A)$ for some $i\geqslant 0$.

######  1.2.

Let ${\varphi}\colon R\to S$ be a map of commutative rings. Set
${S}_{R}^{\operatorname{e}}\colonequals S\otimes^{\operatorname{L}}_{R}S$, the
(derived) enveloping algebra of $S$ over $R$, and let
$\mu_{R}^{S}\colon{S}_{R}^{\operatorname{e}}\to S$ be the multiplication map.
We can model ${S}_{R}^{\operatorname{e}}$ as $S\otimes_{R}A$ where ${\varphi}$
factors as maps of dg algebras

$R\longrightarrow A\xrightarrow{\ \epsilon\ }S$

with $A$ a dg $R$-algebra such that each $R$-module $A_{i}$ is flat, and
$\epsilon$ is a quasi-isomorphism. The multiplication map $\mu_{R}^{S}$ is
modeled by $1\otimes\epsilon\colon S\otimes_{R}A\to S$. This is independent of
the choice of factorization of ${\varphi}$ in the following sense: If $R\to
A^{\prime}\to S$ is another such factorization of ${\varphi}$, then
$A^{\prime}\otimes_{R}S\simeq A\otimes_{R}S$ as dg algebras augmented to $S$.
In particular, $S$ is fixed under the equivalence
${\mathsf{D}}(S\otimes_{R}A)\equiv{\mathsf{D}}(S\otimes_{R}A^{\prime})$
induced by the aforementioned quasi-isomorphism of dg algebras.

We view $S$ as a dg ${S}_{R}^{\operatorname{e}}$-module through the
multiplication map $\mu^{S}_{R}$.

######  1.3.

The Hochschild cohomology of $S$ over $R$ is the graded-commutative
$S$-algebra

$\operatorname{HH}^{*}(S/R)\colonequals\operatorname{Ext}_{{S}_{R}^{\operatorname{e}}}^{*}(S,S)\,.$

For each $M$ in ${\mathsf{D}}(S)$ there is a morphism of graded $S$-algebras

$\chi_{M}\colon\operatorname{HH}^{*}(S/R)\longrightarrow\operatorname{Ext}_{S}^{*}(M,M)\,$

induced by the functor
$-\otimes^{\operatorname{L}}_{S}M\colon{\mathsf{D}}({S}^{\operatorname{e}})\to{\mathsf{D}}(S)$.
We call $\chi_{M}$ the _characteristic map of $M$_. It defines a central
action of $\operatorname{HH}^{*}(S/R)$ on ${\mathsf{D}}(S)$: for $N$ in
${\mathsf{D}}(S)$ and maps $\alpha$ in $\operatorname{HH}^{*}(S/R)$ and
$\beta$ in $\operatorname{Ext}_{S}(M,N)$ one has

$\chi_{N}(\alpha)\beta=(-1)^{|\alpha||\beta|}\beta\chi_{M}(\alpha)\,.$

######  1.4.

A subset $U$ of $\operatorname{HH}^{*}(S/R)$ acts _trivially_ , respectively,
_nilpotently_ , on $M$ if for each $\alpha\in U$, the element
$\chi_{M}(\alpha)$ in $\operatorname{Ext}_{S}(M,M)$ is zero, respectively,
nilpotent. For example, $\operatorname{HH}^{\geqslant 1}(S/R)$ acts trivially
on the $S$-complex $S\otimes^{\operatorname{L}}_{R}L$ for any $R$-complex $L$.
The subcategory of ${\mathsf{D}}(S)$ consisting of $S$-complexes on which $U$
acts nilpotently is thick; that is to say, it is a triangulated subcategory of
${\mathsf{D}}(S)$ closed under direct summands. These observation will be used
often in what follows.

##  2\. The truncated Atiyah class

In this section we recall the construction of the truncated Atiyah class,
introduced by Huybrechts and Thomas in [27], and discuss its connection to
Hochschild cohomology. While the truncated Atiyah class, like the Atiyah class
itself, can be defined for any map of rings, we only need it for surjective
maps.

Throughout ${\varphi}\colon R\to S$ is a surjective map of commutative
noetherian rings, with kernel $I$. The discussion below is extracted from
[15].

######  2.1.

The multiplication map $\mu^{S}_{R}$ from 1.2 extends in an exact triangle

${S}_{R}^{\operatorname{e}}\xrightarrow{\ \mu\ }S\to{\sf\Sigma}J\to$

in ${\mathsf{D}}({S}_{R}^{\operatorname{e}})$. From the triangle above, it is
clear that

$\operatorname{H}_{i}(J)=\begin{cases}\operatorname{Tor}_{\geqslant
1}^{R}(S,S)&i\geqslant 1;\\\ 0&i\leqslant 0\,.\end{cases}$

Thus the canonical truncation yields a morphism $J\to{\sf\Sigma}I/I^{2}$ in
${\mathsf{D}}({S}_{R}^{\operatorname{e}})$. The _truncated Atiyah class_ of
${\varphi}$, denoted $\mathsf{at}^{{\varphi}}$, is the composition of
morphisms

$S\to{\sf\Sigma}J\to{\sf\Sigma}^{2}I/I^{2}$

in ${\mathsf{D}}({S}_{R}^{\operatorname{e}})$. Consider the map sending
$\alpha$ in $\operatorname{Hom}_{S}(I/I^{2},S)$ to the composition

${S}$${{\sf\Sigma}^{2}I/I^{2}}$${{\sf\Sigma}^{2}S}$$\scriptstyle{\mathsf{at}^{{\varphi}}}$$\scriptstyle{{\sf\Sigma}^{2}\alpha}$

in ${\mathsf{D}}({S}_{R}^{\operatorname{e}})$. By [15, Lemma 2.8], this is an
isomorphism

$\delta^{\varphi}\colon\operatorname{Hom}_{S}(I/I^{2},S)\xrightarrow{\cong}\operatorname{HH}^{2}(S/R)\,.$

For each $M$ in ${\mathsf{D}}(S)$ and $\alpha$ in
$\operatorname{Hom}_{S}(I/I^{2},S)$, applying
$-\otimes^{\operatorname{L}}_{S}M$ yields the following commutative diagram in
${\mathsf{D}}(S)$

(2.1.1)
${M}$${{\sf\Sigma}^{2}I/I^{2}\otimes^{\operatorname{L}}_{S}M}$${{\sf\Sigma}^{2}M\,.}$$\scriptstyle{\mathsf{at}^{{\varphi}}\otimes^{\operatorname{L}}_{S}M}$$\scriptstyle{\chi_{M}(\delta^{\varphi}(\alpha))}$$\scriptstyle{{\sf\Sigma}^{2}\alpha\otimes
1}$

The horizontal map in this diagram is the _truncated Atiyah class of
${\varphi}$ at M_, denoted $\mathsf{at}^{{\varphi}}(M)$. Thus we can interpret
$\mathsf{at}^{{\varphi}}(M)$ as a class in
$\operatorname{Ext}_{S}^{2}(M,I/I^{2}\otimes^{\operatorname{L}}_{S}M)$.

######  2.2.

As discussed in [27], the truncated Atiyah class $\mathsf{at}^{{\varphi}}(M)$
can be recovered from the universal Atiyah class

${\sf At}^{\varphi}\colon S\to{\sf\Sigma}{\rm L}({\varphi})$

where ${\rm L}({\varphi})$ is the cotangent complex of ${\varphi}$; cf. [28,
36]. Namely, $\mathsf{at}^{{\varphi}}(M)$ factors as

$M\xrightarrow{{\sf
At}^{\varphi}\otimes^{\operatorname{L}}_{S}M}{\sf\Sigma}{\rm
L}({\varphi})\otimes^{\operatorname{L}}_{S}M\to{\sf\Sigma}^{2}I/I^{2}\otimes^{\operatorname{L}}_{S}M$

where the second map is induced by the soft truncation using that
$\operatorname{H}_{0}({\rm L}({\varphi}))=0$ and $\operatorname{H}_{1}({\rm
L}({\varphi}))=I/I^{2}$. The cotangent complex and universal Atiyah class of
${\varphi}$ are defined via simplicial resolutions and since this machinery is
not needed in this article we direct the interested reader to _loc. cit._ for
further details.

######  2.3.

Suppose now that the surjective map ${\varphi}\colon R\to S$ is complete
intersection of codimension $c$; that is to say, that the ideal $I$ is
generated by an $R$-regular sequence of length $c$. Then the $S$-module
$I/I^{2}$ is a free $S$-module of rank $c$, and the map $\delta^{\varphi}$
induces the isomorphism of $S$-algebras

$\mathrm{Sym}(\delta^{\varphi})\colon\mathrm{Sym}_{S}(\operatorname{Hom}_{S}(I/I^{2},S))\xrightarrow{\
\cong\ }\operatorname{HH}^{*}(S/R)$

where $\operatorname{Hom}_{S}(I/I^{2},S)$ sits in cohomological degree 2. See
[6, Section 3] for a proof. Choosing a basis for the $S$-module $I/I^{2}$
gives rise to a description of $\operatorname{HH}^{*}(S/R)$ as a polynomial
ring over $S$, in $c$ variables of cohomological degree $2$. These are the
operators of Gulliksen [24] and Eisenbud [21]; cf. [11].

######  Lemma 2.4.

Let ${\varphi}\colon R\to S$ be a surjective complete intersection map, and
$M$ an $S$-complex. One has $\mathsf{at}^{{\varphi}}(M)=0$ if and only if
$\operatorname{HH}^{2}(S/R)$ acts trivially on $M$.

###### Proof.

Set $I\colonequals\operatorname{Ker}{\varphi}$ and consider the composition in
${\mathsf{D}}(S)$

$M\xrightarrow{\mathsf{at}^{{\varphi}}(M)}{\sf\Sigma}^{2}I/I^{2}\otimes^{\operatorname{L}}_{S}M\simeq{\sf\Sigma}^{2}I/I^{2}\otimes_{S}M\,.$

The desired result follows from this, Eq. 2.1.1, and the freeness of $I/I^{2}$
over $S$. ∎

######  2.5.

Assume ${\varphi}$ is complete intersection of codimension one. By 2.3,
$\operatorname{HH}^{2}(S/R)$ is isomorphic to $S$, and so choosing a basis
element $\eta$ for the free $S$-module $\operatorname{HH}^{2}(S/R)$ identifies
$\mathsf{at}^{{\varphi}}(M)$ with $\chi_{M}(\eta)$ in
$\operatorname{Ext}_{S}^{2}(M,M)$. Finally, from the triangle defining the
Atiyah class one obtains the exact triangle in ${\mathsf{D}}(S)$

$M\xrightarrow{\chi_{M}(\eta)}{\sf\Sigma}^{2}M\to{\sf\Sigma}S\otimes^{\operatorname{L}}_{R}{\varphi}_{*}(M)\to$

where ${\varphi}_{*}\colon{\mathsf{D}}(S)\to{\mathsf{D}}(R)$ is the
restriction functor; see, for example, [3].

The arguments in the next section use some basic results on André–Quillen
cohomology and homotopy Lie algebras of local rings. We recall the relevant
points below; for further details see [29, 32] and [4, Section 10].

######  2.6.

Let ${\varphi}\colon R\to S$ be a surjective map of local rings, and let $k$
be their common residue field. We write $\operatorname{D}^{i}(S/R;k)$ for the
$i$th André–Quillen cohomology module of ${\varphi}$ with coefficients in $k$.
We need elementary properties whose proofs are contained in the references
above.

First, one has the following natural identification

$\operatorname{D}^{1}(S/R;k)\cong\operatorname{Hom}_{k}(I/\mathfrak{m}_{R}I,k)$

where $\mathfrak{m}_{R}$ is the maximal ideal of $R$ and $I$ is the kernel of
${\varphi}$. Second, there is a canonically induced exact sequence of
$k$-spaces

$0\longrightarrow\operatorname{D}^{1}(k/S;k)\longrightarrow\operatorname{D}^{1}(k/R;k)\longrightarrow\operatorname{D}^{1}(S/R;k)\xrightarrow{\eth}\operatorname{D}^{2}(k/S;k)\longrightarrow\ldots$

called the Jacobi–Zariski sequence corresponding to the composition
$R\xrightarrow{{\varphi}}S\to k$.

######  2.7.

Fix a local ring $R$ with residue field $k$. Let $\pi(R)$ denote the homotopy
Lie algebra of $R$. It is a graded Lie algebra over $k$ whose universal
enveloping algebra is $\operatorname{Ext}_{R}(k,k)$. By the
Poincaré–Birkhoff–Witt theorem, a $k$-basis for $\operatorname{Ext}_{R}(k,k)$
consists of all compositions of the form

$\zeta_{1}^{i_{1}}\zeta_{2}^{i_{2}}\ldots\zeta_{n}^{i_{n}}$

where $\zeta_{j}$ is in $\pi(R)$ with $i_{j}\geqslant 0$ when $|\zeta_{j}|$ is
even and $i_{j}=0,1$ when $|\zeta_{j}|$ is odd; see, for example, [4, Theorem
10.2.1]. In particular, a nonzero element in $\pi(R)$ of even degree is not
nilpotent when regarded as an element of $\operatorname{Ext}_{R}(k,k)$.

Let ${\varphi}\colon R\to S$ be a surjective map of local rings. There is a
naturally defined map of graded Lie algebras

$\pi({\varphi})\colon\pi(S)\to\pi(R)\,.$

An important point is that for $i=1,2$, the map ${\varphi}$ determines a
commutative diagram

(2.7.1)
${\operatorname{D}^{i}(k/S;k)}$${\pi^{i}(S)}$${\operatorname{Ext}_{S}^{i}(k,k)\,\
}$${\operatorname{D}^{i}(k/R;k)}$${\pi^{i}(R)}$${\operatorname{Ext}_{R}^{i}(k,k)\,.}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{D}^{i}(k/{\varphi};k)}$$\scriptstyle{\pi^{i}({\varphi})}$$\scriptstyle{\operatorname{Ext}_{\varphi}^{i}(k,k)}$$\scriptstyle{\cong}$

Consider the diagram of $S$-modules

(2.7.2)
${\operatorname{HH}^{2}(S/R)}$${\operatorname{Hom}_{S}(I/I^{2},S)}$${\operatorname{Hom}_{S}(I/I^{2},k)}$${\operatorname{D}^{1}(S/R;k)}$${\operatorname{Ext}_{S}^{2}(k,k)}$${\pi^{2}(S)}$${\operatorname{D}^{2}(k/S;k)\,}$$\scriptstyle{\chi_{k}}$$\scriptstyle{\delta^{\varphi}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{\psi}$$\scriptstyle{\eth}$$\scriptstyle{\iota}$$\scriptstyle{\cong}$

where $\iota$ is the canonical inclusion and the unlabelled horizontal arrow
is induced by $S\to k$; the map $\psi$ is obtained by applying
$\operatorname{Hom}_{{\mathsf{D}}(S)}(-,{\sf\Sigma}^{2}k)$ to the composition

$k\xrightarrow{\mathsf{at}^{{\varphi}}(k)}{\sf\Sigma}^{2}I/I^{2}\otimes^{\operatorname{L}}_{S}k\to{\sf\Sigma}^{2}I/I^{2}\otimes_{S}k$

where the second map is the augmentation.

######  Lemma 2.8.

Diagram Eq. 2.7.2 is commutative, and
$\operatorname{Im}(\psi)\supseteq\chi_{k}(\operatorname{HH}^{2}(S/R))$ with
equality when ${\varphi}$ is complete intersection.

###### Proof.

By [15, 3.11], the image of $\psi$ is $\operatorname{Ker}\pi^{2}({\varphi})$.
Now the commutativity of Eq. 2.7.2, and hence the desired containment, follows
from the observations in 2.6 and Eq. 2.7.1. When ${\varphi}$ is complete
intersection, the containment is an equality since $I/I^{2}$ is a free
$S$-module. ∎

##  3\. Exceptional complete intersections

In this section we prove B from the introduction, and present examples that
witness some, perhaps, surprising behavior.

######  3.1.

Let ${\varphi}\colon R\to S$ be a surjective map of noetherian local rings;
set $I\colonequals\operatorname{Ker}{\varphi}$. Following [16] we say
${\varphi}$ is an _exceptional complete intersection_ provided $I$ is
generated by a regular sequence that can be extended to a minimal generating
set for $\mathfrak{m}_{R}$, the maximal ideal of $R$. The latter condition
means that the natural map

$\frac{I}{\mathfrak{m}_{R}I}\longrightarrow\frac{\mathfrak{m}_{R}}{\mathfrak{m}_{R}^{2}}$

is one-to-one. We refer to [16], where this notion is introduced for not
necessarily surjective maps of local rings, for various characterizations of
exceptional complete intersection maps in terms of numerical invariants
associated to $R$ and $S$. Here is one more, extracted from the proof of [16,
Theorem 3.1].

######  Lemma 3.2.

Let ${\varphi}\colon R\to S$ be a surjective map of noetherian local rings.
The map ${\varphi}$ is exceptional complete intersection if and only if
$\operatorname{proj\,dim}_{R}S$ is finite and the natural map
$I/\mathfrak{m}_{R}I\to\mathfrak{m}_{R}/\mathfrak{m}_{R}^{2}$ is one-to-one.

###### Proof.

Since a complete intersection map has finite projective dimension, the only if
direction is clear. As to the converse: We can assume $I\neq 0$. Then the
hypothesis that $\operatorname{proj\,dim}_{R}S$ is finite means that $I$
contains a nonzero divisor, say $x$; this is by a result of Auslander and
Buchsbaum [17, Corollary 1.4.7]. Since $I\not\subseteq\mathfrak{m}^{2}_{R}$,
one can ensure that $x$ is not in $\mathfrak{m}^{2}_{R}$; this is by a
variation of the prime avoidance argument [17, Lemma 1.2.2]. Setting
$R^{\prime}\colonequals R/xR$, the map ${\varphi}$ factors as

$R\longrightarrow R^{\prime}\longrightarrow S\,.$

By the choice of $x$, the map $R\to R^{\prime}$ is exceptional complete
intersection. Thus since $\operatorname{proj\,dim}_{R}S$ is finite so is
$\operatorname{proj\,dim}_{R^{\prime}}S$; this is by Nagata’s theorem recalled
in 3.3. Moreover one can verify easily that the hypothesis on $I$ is inherited
by the kernel of the map $R^{\prime}\to S$. We can thus induce on the number
of generators of $I$ to deduce that ${\varphi}$ is exceptional complete
intersection. ∎

The result below, due to Nagata [33, Corollary 27.5], was used above; this
will be generalized later in Theorem 4.1.

######  3.3.

When ${\varphi}\colon R\to S$ is exceptional complete intersection and $M$ is
a finitely generated $S$-module, $\operatorname{proj\,dim}_{R}M$ is finite if
and only if $\operatorname{proj\,dim}_{S}M$ is finite.

The following result contains B from the introduction. Most of the groundwork
for the proof has been laid in Section 2.

######  Theorem 3.4.

Let ${\varphi}\colon R\to S$ be a surjective map of noetherian local rings
with $\operatorname{proj\,dim}_{R}S$ finite. The conditions below are
equivalent:

  1. (1)

${\varphi}$ is exceptional complete intersection;

  2. (2)

${\varphi}$ is complete intersection and $\operatorname{HH}^{2}(S/R)$ acts
trivially on $k$;

  3. (3)

${\varphi}$ is complete intersection and $\operatorname{HH}^{2}(S/R)$ acts
nilpotently on $k$;

  4. (4)

$\mathsf{at}^{{\varphi}}(k)=0$, where $k$ is the residue field of $S$.

###### Proof.

Item 1$\Rightarrow$Item 2: From the assumption and [3, Proposition 2.8] it
follows that $\operatorname{HH}^{2}(S/R)$ acts trivially on $k$.

Item 2$\Leftrightarrow$Item 3: This follows from Lemma 2.8, since nonzero
elements of $\pi^{2}(S)$ cannot be nilpotent when regarded as maps in
$\operatorname{Ext}_{S}^{2}(k,k);$ cf. 2.7.

Item 2$\Rightarrow$Item 4: This is contained in Lemma 2.4.

Item 4$\Rightarrow$Item 1: Given that $\operatorname{proj\,dim}_{R}S$ is
finite and Lemma 3.2 it suffices to prove that the natural map
$I/\mathfrak{m}_{R}I\to\mathfrak{m}_{R}/\mathfrak{m}_{R}^{2}$ is one-to-one;
equivalently, that its $k$-vector space dual is onto. By assumption the
commutative diagram Eq. 2.7.2 yields $\eth=0$ in the Jacobi–Zariski sequence
corresponding to $R\to S\to k$ from 2.6. Thus the latter reduces to an exact
sequence

$0\longrightarrow\left(\frac{\mathfrak{m}_{S}}{\mathfrak{m}_{S}^{2}}\right)^{\vee}\longrightarrow\left(\frac{\mathfrak{m}_{R}}{\mathfrak{m}_{R}^{2}}\right)^{\vee}\longrightarrow\left(\frac{I}{\mathfrak{m}_{R}I}\right)^{\vee}\longrightarrow
0$

where $(-)^{\vee}$ denotes the vector space dual and the maps are the
canonically induced ones. This is the desired conclusion. ∎

######  3.5.

Recall from 2.2 that in ${\mathsf{D}}(S)$ there is a commutative diagram

${k}$${{\sf\Sigma}\operatorname{L}({{\varphi}})\otimes_{S}k}$${k}$${{\sf\Sigma}^{2}I/I^{2}\otimes^{\operatorname{L}}_{S}k}$$\scriptstyle{{\sf{At}}^{\varphi}\otimes_{S}k}$$\scriptstyle{\mathsf{at}^{{\varphi}}(k)}$

where the map on the right is induced by the soft truncation
$\tau\colon\operatorname{L}({{\varphi}})\to{\sf\Sigma}I/I^{2}$; it is an
isomorphism when ${\varphi}$ is complete intersection. It follows from this
observation and Theorem 3.4 that ${\varphi}$ is exceptional complete
intersection if and only if $\operatorname{proj\,dim}_{R}S$ is finite and the
map ${\sf{At}}^{\varphi}\otimes_{S}k$ is zero in ${\mathsf{D}}(S)$.

######  3.6.

Let ${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$ denote the full subcategory of
${\mathsf{D}}(S)$ consisting of $S$-complexes $M$ such that the $S$-module
$\operatorname{H}(M)$ has finite length. Since
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$ is the smallest thick subcategory of
${\mathsf{D}}(S)$ containing $k$—see, for example, [20, 3.5]—it follows from
Theorem 3.4 that when ${\varphi}$ is exceptional complete intersection, the
action of $\operatorname{HH}^{2}(S/R)$ on ${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$
is nilpotent; however there may be no bound on the nilpotence degree. Moreover
$\operatorname{HH}^{2}(S/R)$ need not act nilpotently on objects not in
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$. The example below illustrates these
points.

######  Example 3.7.

Let $k$ be a field, consider the hypersurface ring

$R\colonequals\frac{k\llbracket x,y,z\rrbracket}{(x^{2}-yz)}$

and the exceptional complete intersection map

${\varphi}\colon R\to S\colonequals R/(z)\cong\frac{k\llbracket
x,y\rrbracket}{(x^{2})}\,.$

Let $\eta$ denote the class dual to the class of $z$ in $(z)/(z^{2})$, viewed
as an element of $\operatorname{HH}^{2}(S/R)$ via the isomorphism
$\delta^{\varphi}$.

Set $M\colonequals S/(x)$, regarded as an $S$-module via the surjection $S\to
S/(x)$. The minimal resolution of $M$ over $S$ is
$\ldots\xrightarrow{x}S\xrightarrow{x}S\to 0$. From there it is clear that
$\operatorname{Ext}_{S}^{i}(M,M)=M$ for all $i\geqslant 0$. Moreover using the
construction of cohomology operators from [21, Section 1], one gets that
multiplication by $\chi_{M}(\eta)$ on $\operatorname{Ext}_{S}(M,M)$ is
represented by

$M\xrightarrow{y\cdot}M\,.$

In particular $\operatorname{HH}^{2}(S/R)$ does not act nilpotently on $M$.

In the same vein for $N\colonequals S/(x,y^{n})$, where $n$ is any positive
integer, one gets

$\operatorname{Ext}_{S}^{i}(N,N)=\begin{cases}N&i=0\\\ N^{\oplus
2}&i>0\end{cases}$

and the action of $\chi_{N}(\eta)$ is represented by

$N^{\oplus 2}\xrightarrow{\begin{pmatrix}y&0\\\ 0&y\end{pmatrix}}N^{\oplus
2}.$

Thus the $\operatorname{HH}^{i}(S/R)$ action on $N$ is nontrivial for
$0\leqslant i<2n$ and trivial for $i\geqslant 2n$.

######  3.8.

Let ${\varphi}\colon R\to S$ be complete intersection. Consider the
restriction functor

${\varphi}_{*}\colon{\mathsf{D}}(S)\to{\mathsf{D}}(R)\,.$

It is straightforward from Lemmas 2.4 and 2.5 to see the following are
equivalent:

  1. (1)

${\varphi}_{*}$ is faithful on morphisms;

  2. (2)

$\operatorname{HH}^{2}(S/R)$ acts trivially on each object of
${\mathsf{D}}(S)$;

  3. (3)

$\mathsf{at}^{{\varphi}}(M)$ is zero for each object $M$ in ${\mathsf{D}}(S)$.

When ${\varphi}$ admits an algebra section—that is to say, a map of rings
$\sigma\colon S\to R$ such that ${\varphi}\sigma=\operatorname{id}^{S}$—the
restriction functor ${\varphi}_{*}$ is faithful and so
$\mathsf{at}^{{\varphi}}(M)$ vanishes for each $M$ in ${\mathsf{D}}(S)$.

Contrast this with the exceptional complete intersection map in Example 3.7
where a class of modules demonstrate nonzero, even non-nilpotent, truncated
Atiyah class. Hence, when ${\varphi}$ is exceptional complete intersection,
non-vanishing of truncated Atiyah classes serve as homological obstructions to
the regular sequence generating $\operatorname{Ker}{\varphi}$ being, loosely
speaking, independent power series variables in $R$. These can also be viewed
as obstructions to the notion of weak liftability discussed in [2, 37].

######  Example 3.9.

Assume ${\varphi}\colon R\to S$ is exceptional complete intersection with $R$,
and hence $S$, a complete regular local ring. Whether ${\varphi}_{*}$ is
faithful depends on the characteristic of $R$. When $R$ is equicharacteristic,
${\varphi}$ admits an algebra section; hence, by 3.8 we conclude that
${\varphi}_{*}$ is faithful.

In contrast, in [25, Example 1], Hochster presented an example which settled
Grothendieck’s lifting problem in the negative. By combining a calculation of
Dao [19, Example 3.5] and a result of Yoshino [37, Theorem 3.11], the module
considered by Hochster has non-vanishing truncated Atiyah class over an
exceptional complete intersection map whose base ring is a mixed
characteristic regular local ring.

##  4\. Homological invariants

In this section we track the change of certain homological invariants along
exceptional complete intersection maps. The results obtained suggest that,
notwithstanding the remarks in 3.8 and 3.9, homological properties of the
source and target of an exceptional complete intersection map are tightly
linked.

As before, given a local ring $S$, we write
${\mathsf{D}}^{\sf{f}}_{\sf{b}}(S)$ for the full subcategory of
${\mathsf{D}}(S)$ consisting of objects with finitely generated homology.

######  Theorem 4.1.

Let ${\varphi}\colon R\to S$ be exceptional complete intersection and $M,N$ in
${\mathsf{D}}^{\sf{f}}_{\sf{b}}(S)$. Then the following hold:

  1. (1)

$\operatorname{Ext}_{R}^{i}(M,N)=0$ for $i\gg 0$ if and only if
$\operatorname{Ext}_{S}^{i}(M,N)=0$ for $i\gg 0$;

  2. (2)

$\operatorname{Tor}^{R}_{i}(M,N)=0$ for $i\gg 0$ if and only if
$\operatorname{Tor}^{S}_{i}(M,N)=0$ for $i\gg 0$.

###### Proof.

We first reduce to the case when $\operatorname{H}(M)$ has finite length: Let
$K$ be the Koszul complex on a system of parameters for $S$ and consider the
$S$-complex $K\otimes_{S}M$. Then the length of the $S$-module
$\operatorname{H}(K\otimes_{S}M)$ is finite. Moreover, since $K\otimes_{S}M$
can be realized as an iterated mapping cone of maps defined by multiplication
by an element of $S$, a straightforward argument yields that

$\displaystyle\operatorname{Ext}_{R}^{i}(M,N)=0\quad\text{for $i\gg
0$}\iff\operatorname{Ext}_{R}^{i}(K\otimes_{S}M,N)=0\quad\text{for $i\gg 0$}$
$\displaystyle\operatorname{Ext}_{S}^{i}(M,N)=0\quad\text{for $i\gg
0$}\iff\operatorname{Ext}_{S}^{i}(K\otimes_{S}M,N)=0\quad\text{for $i\gg 0$;}$

see, for instance, [20, 2.9]. Thus replacing $M$ by $K\otimes_{S}M$ we can
assume the length of $\operatorname{H}(M)$ is finite.

(1) Since the length of $\operatorname{H}(M)$ is finite, the
$\operatorname{HH}^{*}(S/R)$-action on $\operatorname{Ext}_{S}(M,M)$, and
hence also on $\operatorname{Ext}_{S}(M,N)$, is nilpotent; see 3.6. The
desired equivalence is now immediate from [24, Theorem 3.1] and [9, Theorem
4.2] that state the $R$-module $\operatorname{Ext}_{R}(M,N)$ is finitely
generated if and only if the $\operatorname{HH}^{*}(S/R)$-module
$\operatorname{Ext}_{S}(M,N)$ is finitely generated.

(2) With $(-)^{\vee}$ denoting Matlis duality of $S$-modules, one has
isomorphisms

$\operatorname{Tor}^{R}_{i}(M,N)^{\vee}\cong\operatorname{Ext}_{R}^{i}(M^{\vee},N)\quad\text{and}\quad\operatorname{Tor}^{S}_{i}(M,N)^{\vee}\cong\operatorname{Ext}_{S}^{i}(M^{\vee},N)\,.$

Since $\operatorname{H}(M)$ has finite length, so does
$\operatorname{H}(M^{\vee})$; the desired equivalence now follows from (1) and
the faithfulness of Matlis duality. ∎

Setting $N\colonequals k$ in either part of Theorem 4.1 recovers Nagata’s
theorem 3.3, whilst setting $M\colonequals k$ in LABEL:{thm:nagata+}Item 1
yields an analogous result for injective dimension. Here is a more precise
statement; see [8] for the definitions of the projective dimension and
injective dimension of a complex.

######  Corollary 4.2.

If ${\varphi}\colon R\to S$ is exceptional complete intersection, then for
each $M$ in ${\mathsf{D}}^{\sf{f}}_{\sf{b}}(S)$ there are equalities

$\displaystyle\operatorname{proj\,dim}_{R}M$
$\displaystyle=\operatorname{proj\,dim}_{S}M+\dim R-\dim S$
$\displaystyle\mathrm{inj\,dim}_{R}M$
$\displaystyle=\mathrm{inj\,dim}_{S}M+\dim R-\dim S\,.$

###### Proof.

As noted above, from Theorem 4.1 we know $\operatorname{proj\,dim}_{R}M$ is
finite if and only if $\operatorname{proj\,dim}_{S}M$ is; so assume both
values are finite. Now we can apply a generalization of the
Auslander–Buchsbaum formula to complexes in [23, Theorem II] to obtain the
first and third equalities below

$\displaystyle\operatorname{proj\,dim}_{R}M$
$\displaystyle=\operatorname{depth}R-\operatorname{depth}_{R}M$
$\displaystyle=\operatorname{depth}R-\operatorname{depth}S+\operatorname{depth}S-\operatorname{depth}_{S}M$
$\displaystyle=\operatorname{depth}R-\operatorname{depth}S+\operatorname{proj\,dim}_{S}M$
$\displaystyle=\dim R-\dim S+\operatorname{proj\,dim}_{S}M\,.$

The last equality holds because ${\varphi}$ is complete intersection.

For the equality involving injective dimensions, from Theorem 4.1, we can
assume $\mathrm{inj\,dim}_{R}M$ and $\mathrm{inj\,dim}_{S}M$ are both finite.
Now observe that

$\displaystyle\mathrm{inj\,dim}_{R}M$
$\displaystyle=\operatorname{depth}R-\inf\\{i\in\mathbb{Z}:\operatorname{H}_{i}(M)\neq
0\\}$
$\displaystyle=\operatorname{depth}R-\operatorname{depth}S+\operatorname{depth}S-\inf\\{i\in\mathbb{Z}:\operatorname{H}_{i}(M)\neq
0\\}$
$\displaystyle=\operatorname{depth}R-\operatorname{depth}S+\mathrm{inj\,dim}_{S}M$
$\displaystyle=\dim R-\dim S+\mathrm{inj\,dim}_{S}M$

where the first and third equalities hold by [10, Lemma 1.4]. ∎

##  5\. Homotopical characterization

In this section ${\varphi}\colon R\to S$ is a surjective map of commutative
noetherian local rings with (common) residue field $k$. Set
$I\colonequals\operatorname{Ker}{\varphi}$. The main result in this section is
Theorem 5.6. An essential point is that when ${\varphi}$ is complete
intersection information regarding finite building, in the sense recalled
below, is related to nilpotence of the action of $\operatorname{HH}^{2}(S/R)$.

Given an $S$-complex $M$ we write $\operatorname{thick}_{S}(M)$ for the
smallest thick subcategory of ${\mathsf{D}}(S)$ containing $M$; see [13,
Section 2], or [7, Section 2]. Any $S$-complex $N$ in
$\operatorname{thick}_{S}(M)$ is said to be _finitely built_ from $M$; we
write $M\models_{S}N$ to indicate that this is so.

Hopkins [26] and Neeman [34] proved that if $M$ and $N$ in
${\mathsf{D}}^{\sf{f}}_{\sf{b}}(S)$ are $S$-complexes of finite projective
dimension, and
$\operatorname{supp}_{S}\operatorname{H}(M)\supseteq\operatorname{supp}_{S}\operatorname{H}(N)$,
then $M\models_{S}N$. Here $\operatorname{supp}_{S}(-)$ denotes support. This
result has the following consequence.

######  5.1.

Let $M$ in ${\mathsf{D}}^{\sf{f}}_{\sf{b}}(S)$ be an $S$-complex of finite
projective dimension and $\operatorname{H}(M)\neq 0$. If $K$ is the Koszul
complex on finite set of elements that generate an $\mathfrak{m}_{S}$-primary
ideal, then $M\models_{S}K$.

######  Lemma 5.2.

If $\operatorname{proj\,dim}_{R}S$ is finite, then
$\operatorname{thick}_{R}(M)=\operatorname{thick}_{R}(S\otimes^{\operatorname{L}}_{R}M)$
for any $M$ in ${\mathsf{D}}(S)$.

###### Proof.

The desired result translates to
$M\models_{R}S\otimes^{\operatorname{L}}_{R}M$ and
$S\otimes^{\operatorname{L}}_{R}M\models_{R}M$.

As $\operatorname{proj\,dim}_{R}S$ is finite, $R\models_{R}S$, and so applying
$-\otimes^{\operatorname{L}}_{R}M$ yields
$M\models_{R}S\otimes^{\operatorname{L}}_{R}M$.

Let $K$ be the Koszul complex on a generating set for the ideal $I$. As
$\operatorname{proj\,dim}_{R}S$ is finite it follows that $S\models_{R}K;$ see
5.1. Applying $-\otimes^{\operatorname{L}}_{R}M$ one sees that

$S\otimes^{\operatorname{L}}_{R}M\mathrel{\mathop{\models}\limits_{\vbox
to3.5pt{\kern-3.0pt\hbox{$\scriptstyle\,\,R$}\vss}}}K\otimes_{R}M\,.$

It remains to observe that since $R$ acts on $M$ through $S$, in
${\mathsf{D}}(R)$ the complex $K\otimes_{R}M$ is isomorphic to a finite direct
sum of suspensions of $M$. ∎

Our interest is in the statement corresponding to Lemma 5.2 in
${\mathsf{D}}(S)$. Let $M$ be an object of ${\mathsf{D}}(S)$. In what follows
$S\otimes^{\operatorname{L}}_{R}M$ is always regarded as an $S$-complex via
the left $S$-action. That is, $S\otimes^{\operatorname{L}}_{R}M$ denotes
$S\otimes^{\operatorname{L}}_{R}{\varphi}_{*}(M)$ as an object of
${\mathsf{D}}(S)$.

By [20, Theorem 8.3], when ${\varphi}\colon R\to S$ is a surjective complete
intersection homomorphism,

(5.2.1) $M\mathrel{\mathop{\models}\limits_{\vbox
to3.5pt{\kern-3.0pt\hbox{$\scriptstyle\,\,S$}\vss}}}S\otimes^{\operatorname{L}}_{R}M$

but the reverse building is not guaranteed. The obstruction is the non-
nilpotence of the action of $\operatorname{HH}^{2}(S/R)$ on $M$; this is the
content of the result below.

######  Lemma 5.3.

Let ${\varphi}\colon R\to S$ be a surjective complete intersection map. For an
$S$-complex $N$ the following conditions are equivalent:

  1. (1)

$\operatorname{HH}^{2}(S/R)$ acts nilpotently on $N$;

  2. (2)

$S\otimes^{\operatorname{L}}_{R}N$ finitely builds $N$ in ${\mathsf{D}}(S)$;

  3. (3)

for any $S$-complex $M$ one has $M\models_{R}N\iff M\models_{S}N$.

###### Proof.

Item 1 $\Rightarrow$ Item 2: As $\operatorname{HH}^{2}(S/R)$ acts nilpotently
on $N$, any factorization of ${\varphi}$ into a composition of complete
intersection maps

$R\to R^{\prime}\to S$

has the property that $\operatorname{HH}^{2}(R^{\prime}/R)$ and
$\operatorname{HH}^{2}(S/R^{\prime})$ act nilpotently on $N$ in
${\mathsf{D}}(R^{\prime})$ and ${\mathsf{D}}(S),$ respectively; this follows
from [11, 3.1], or [3, Proposition 2.3]. Furthermore there is an isomorphism

$S\otimes^{\operatorname{L}}_{R^{\prime}}(R^{\prime}\otimes^{\operatorname{L}}_{R}N)\simeq
S\otimes^{\operatorname{L}}_{R}N$

in ${\mathsf{D}}(S).$ Hence, by induction on the codimension of ${\varphi}$,
it is enough to show Item 2 holds when ${\varphi}$ is complete intersection of
codimension one.

In this case, there is an exact triangle

$N\xrightarrow{\chi_{N}(\eta)}{\sf\Sigma}^{2}N\to{\sf\Sigma}S\otimes^{\operatorname{L}}_{R}N\to$

where $\eta$ is a basis element for the free $S$-module
$\operatorname{HH}^{2}(S/R)$; cf. 2.5. Iteratively applying the octahedral
axiom yields

(5.3.1)
$S\otimes^{\operatorname{L}}_{R}N\mathrel{\mathop{\models}\limits_{\vbox
to3.5pt{\kern-3.0pt\hbox{$\scriptstyle\,\,S$}\vss}}}{\rm
cone}(\chi_{N}(\eta)^{n})$

for any $n\geqslant 1$. The assumption that $\operatorname{HH}^{2}(S/R)$ acts
nilpotently on $N$ means there is some $n\geqslant 1$ such that
$\chi_{N}(\eta)^{n}=0$ and so

${\rm cone}(\chi_{N}(\eta)^{n})\simeq{\sf\Sigma}N\oplus{\sf\Sigma}^{2n}N$

in ${\mathsf{D}}(S)$. Combining this with Eq. 5.3.1 we obtain the desired
result.

Item 2 $\Rightarrow$ Item 3: Evidently, if $M\models_{S}N$ then
$M\models_{R}N$. When $M\models_{R}N$, by applying the functor
$S\otimes^{\operatorname{L}}_{R}-$ we obtain

$S\otimes^{\operatorname{L}}_{R}M\mathrel{\mathop{\models}\limits_{\vbox
to3.5pt{\kern-3.0pt\hbox{$\scriptstyle\,\,S$}\vss}}}S\otimes^{\operatorname{L}}_{R}N\,;$

from this and the assumption in Item 2, we obtain that
$S\otimes^{\operatorname{L}}_{R}M\models_{S}N$. Finally

$M\mathrel{\mathop{\models}\limits_{\vbox
to3.5pt{\kern-3.0pt\hbox{$\scriptstyle\,\,S$}\vss}}}S\otimes^{\operatorname{L}}_{R}M$

by Eq. 5.2.1, so combining this with the already established building
$S\otimes^{\operatorname{L}}_{R}M\models_{S}N$, we conclude that
$M\models_{S}N$ as desired.

Item 3 $\Rightarrow$ Item 1: From Lemma 5.2 and the hypothesis in Item 3 we
get $S\otimes^{\operatorname{L}}_{R}N\models_{S}N$. As discussed in 1.3 the
full subcategory of ${\mathsf{D}}(S)$ consisting of objects for which
$\operatorname{HH}^{2}(S/R)$ is nilpotent is a thick subcategory of
${\mathsf{D}}(S).$ In particular, this subcategory is closed under finite
building. Note that $\operatorname{HH}^{2}(S/R)$ has trivial action on
$S\otimes^{\operatorname{L}}_{R}N$ and so $\operatorname{HH}^{2}(S/R)$ acts
nilpotently on $N$. ∎

######  5.4.

Let ${\varphi}_{*}\colon{\mathsf{D}}(S)\to{\mathsf{D}}(R)$ be the restriction
functor and ${\mathsf{S}}$ a subcategory of ${\mathsf{D}}(S)$. We say that
${\varphi}_{*}$ has _ascent of finite building on ${\mathsf{S}}$_ if whenever
$M$ and $N$ are objects of ${\mathsf{S}}$ with
${\varphi}_{*}(M)\models_{R}{\varphi}_{*}(N)$ then $M\models_{S}N$. By a
slight abuse of notation we will say ${\varphi}$ has ascent of finite building
on ${\mathsf{S}}$.

As discussed in the introduction much of this work is motivated by the
homotopical characterizations of surjective complete intersection maps in [14,
15, 30, 35]. Before continuing to the main result of the article, we recall a
method from [15] used to detect the complete intersection property among
surjective maps of finite projective dimension.

######  5.5.

Let $K$ denote the Koszul complex on a minimal generating set for
$\mathfrak{m}_{S}$ over $S$. If $\operatorname{proj\,dim}_{R}S$ is finite,
then there exist $M_{1},\ldots,M_{n}$ in ${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$
satisfying the following properties:

  1. (1)

each $M_{i}$ finitely builds $K$ in ${\mathsf{D}}(R)$;

  2. (2)

if each $M_{i}$ finitely builds $K$ in ${\mathsf{D}}(S)$, then ${\varphi}$ is
complete intersection.

Indeed explicitly one can take the $M_{i}$ as mapping cones in
${\mathsf{D}}(S)$ on a generating set for
$\operatorname{Ker}\pi^{2}({\varphi})$; see 2.7. The first point is justified
by [15, Lemma 2.5 & Lemma 3.6] while the second condition holds by [15,
Theorem 3.12].

######  Theorem 5.6.

A surjective map ${\varphi}\colon R\to S$ of commutative noetherian local
rings is exceptional complete intersection if and only if
$\operatorname{proj\,dim}_{R}S$ is finite and ${\varphi}$ has ascent of finite
building on ${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$.

###### Proof.

$(\Rightarrow)$: As ${\varphi}$ is exceptional complete intersection, 3.6
implies $\operatorname{HH}^{2}(S/R)$ acts nilpotently on each object of
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$. Now the desired result follows from
Lemma 5.3.

$(\Leftarrow)$: Since $\operatorname{proj\,dim}_{R}S$ is finite, there exist
$M_{1},\ldots,M_{n}$ in ${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$ from 5.5. Using
the assumption that ${\varphi}$ has ascent of building it follows from the two
properties of the $M_{i}$ from 5.5 that ${\varphi}$ is complete intersection.

As $\operatorname{proj\,dim}_{R}S$ is finite,
$S\otimes^{\operatorname{L}}_{R}k\models_{R}k$ in
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$. Since ${\varphi}$ has ascent of building
on ${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$, we conclude that

$S\otimes^{\operatorname{L}}_{R}k\mathrel{\mathop{\models}\limits_{\vbox
to3.5pt{\kern-3.0pt\hbox{$\scriptstyle\,\,S$}\vss}}}k\,.$

In particular $\operatorname{HH}^{2}(S/R)$ acts nilpotently on $k$, for
$\operatorname{HH}^{2}(S/R)$ acts trivially on
$S\otimes^{\operatorname{L}}_{R}k$. Thus ${\varphi}$ is exceptional complete
intersection, by Theorem 3.4. ∎

######  Remark 5.7.

Instead of invoking Theorem 3.4 in the proof of $(\Leftarrow)$ of Theorem 5.6
one can argue as follows: Use the same argument as before that ${\varphi}$ is
complete intersection. Next factor $R\to E\to S$ where $E$ is the Koszul
complex over $R$ on a minimal generating set for
$\operatorname{Ker}{\varphi}.$ There is the following commutative diagram
obtained by restricting scalars

${{\mathsf{D}}(R)}$${{\mathsf{D}}(E)}$${{\mathsf{D}}(S)}$$\scriptstyle{\equiv}$

where the vertical map is an equivalence since $E\xrightarrow{\simeq}S.$ Now
as ${\varphi}$ has ascent of building on ${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$,
so does $R\to E$ on ${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(E)$.

The main point is that the Koszul complex over $R$ on a minimal generating set
for $\mathfrak{m}_{R}$, denoted $K^{R}$, is an object of
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(E)$ with the property

$E\otimes_{R}K^{R}\mathrel{\mathop{\models}\limits_{\vbox
to3.5pt{\kern-3.0pt\hbox{$\scriptstyle\,\,E$}\vss}}}K^{R}$

if and only if ${\varphi}$ is exceptional complete intersection; this can be
verified by a direct calculation. Now as $R\to E$ has ascent of building on
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(E)$ and one always has the finite building

$E\otimes_{R}K^{R}\mathrel{\mathop{\models}\limits_{\vbox
to3.5pt{\kern-3.0pt\hbox{$\scriptstyle\,\,R$}\vss}}}K^{R}\,,$

we obtain the desired conclusion from the previous remark.

Note that the argument sketched above justifies that when ${\varphi}$ is
complete intersection, ${\varphi}$ is exceptional complete intersection if and
only if $S\otimes^{\operatorname{L}}_{E}K^{R}$ has finite projective dimension
over $S$.

######  5.8.

Theorem 5.6 is a characterization of exceptional complete intersection maps
purely in terms of the structure of ${\mathsf{D}}(R)$ and ${\mathsf{D}}(S)$ as
triangulated categories. Namely, for a triangulated category ${\mathsf{T}}$
one can consider its thick subcategory of compact objects, denoted
${\mathsf{T^{c}}}$. Define ${\mathsf{T^{fl}_{b}}}$ to be the full subcategory
of ${\mathsf{T}}$ consisting of objects $X$ such that
$\operatorname{Hom}_{{\mathsf{T}}}(C,X)$ is a finite length
$\operatorname{Hom}_{{\mathsf{T}}}(C,C)$-module for each $C$ in
${\mathsf{T^{c}}}$. In the context of ${\mathsf{D}}(R)$, it is well known an
$R$-complex belongs to ${\mathsf{D}}(R)^{\sf c}$ precisely when it has finite
projective dimension over $R$, and ${\mathsf{D}}(R)_{\sf b}^{\sf
fl}={\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$.

Let $\operatorname{Thick}{\mathsf{T}}$ be the lattice of thick subcategories
of a triangulated category ${\mathsf{T}}$.

######  5.9.

Let ${\varphi}\colon R\to S$ be a surjective map of local rings, and
${\varphi}_{*}\colon{\mathsf{D}}(S)\to{\mathsf{D}}(R)$ the restriction
functor. For a thick subcategory ${\mathsf{U}}$ of
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$, we write $F({\mathsf{U}})$ to denote the
smallest thick subcategory containing all objects $M$ in
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$ such that ${\varphi}_{*}(M)$ is in
${\mathsf{U}}$. If ${\mathsf{S}}$ is a thick subcategory of
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$, then $G({\mathsf{S}})$ is the smallest
thick subcategory containing all objects ${\varphi}_{*}(M)$ in
${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$ such that $M$ is in ${\mathsf{S}}$. Thus
one has maps of lattices

$\operatorname{Thick}{\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)\stackrel{{\scriptstyle[}}{{G}}]{F}{\rightleftarrows}\operatorname{Thick}{\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)\,.$

The following result is Theorem A.

######  Corollary 5.10.

The map ${\varphi}$ is exceptional complete intersection if and only if
$\operatorname{proj\,dim}_{R}S$ is finite and $F$ and $G$ are mutually inverse
lattice isomorphisms.

###### Proof.

$(\Leftarrow)$: This follows immediately from Theorem 5.6.

$(\Rightarrow):$ From 3.6 and Lemma 5.3, $FG$ is the identity on
$\operatorname{Thick}{\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S).$

Fix a thick subcategory ${\mathsf{U}}$ in
$\operatorname{Thick}{\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$. To show
$GF({\mathsf{U}})={\mathsf{U}}$, it suffices to show for each object $L$ in
${\mathsf{U}}$ one has

(5.10.1)
$\operatorname{thick}_{R}L=\operatorname{thick}_{R}{\varphi}_{*}(M)\qquad\text{where
$M=S\otimes^{\operatorname{L}}_{R}L$.}$

Since $\operatorname{proj\,dim}_{R}S$ is finite,
$L\models_{R}{\varphi}_{*}(M)$, by Lemma 5.2. Let $K$ be the Koszul complex on
a minimal generating set for $\mathfrak{m}_{R}$ and $E$ the Koszul complex on
a minimal generating set for $\operatorname{Ker}({\varphi})$. By 5.1 one has
that $E\models_{R}K$ and so

${\varphi}_{*}(M)\simeq E\otimes_{R}L\mathrel{\mathop{\models}\limits_{\vbox
to3.5pt{\kern-3.0pt\hbox{$\scriptstyle\,\,R$}\vss}}}K\otimes_{R}L\,.$

Furthermore, as $\operatorname{H}(L)$ is finite length over $R$, it follows
that $K\otimes_{R}L\models_{R}L$ and so ${\varphi}_{*}(M)\models_{R}L$; thus,
Eq. 5.10.1 holds. ∎

We end this article with two applications; the first should be compared with
the factorization theorems in [5, 5.7.1] and [15, Corollary 4.2]

######  Corollary 5.11.

Consider surjective maps $R\xrightarrow{{\varphi}}S\xrightarrow{\psi}T$ of
commutative noetherian local rings. If ${\varphi}$ and $\psi$ are exceptional
complete intersection, then $\psi{\varphi}$ is exceptional complete
intersection. The converse holds when $\operatorname{proj\,dim}_{S}T$ is
finite.

###### Proof.

It is clear from the definition that the class of exceptional complete
intersection maps is closed under composition. It also follows trivially from
the characterization in Theorem 5.6.

Assume that $\operatorname{proj\,dim}_{S}T$ is finite and $\psi{\varphi}$ is
exceptional complete intersection. It can be argued as in [15, Corollary 4.2]
that $\operatorname{proj\,dim}_{R}S$ is finite. Also, it is clear that $\psi$
has ascent of finite building on ${\mathsf{D}}^{\sf{fl}}_{\sf{b}}(S)$ since
$\psi{\varphi}$ does. Hence, $\psi$ is exceptional complete intersection. Now
as $\psi$ and $\psi{\varphi}$ have ascent of finite building on finite length
objects, we can apply Corollary 5.10 to deduce ${\varphi}$ is exceptional
complete intersection. ∎

Another application of Corollary 5.10 is provided when $R$ is an
equicharacteristic Cohen–Macaulay ring of _minimal multiplicity_. When $R$ is
Cohen–Macaulay Abhyankar [1] proved the Hilbert–Samuel multiplicity of $R$ is
bounded below by ${\rm codim}R+1$; when equality holds $R$ is said to be of
minimal multiplicity.

######  5.12.

For the rest of the section, assume $R$ is an equicharacteristic
Cohen–Macaulay ring of minimal multiplicity and infinite residue field $k$.
There exists an exceptional complete intersection map $R\to S$ such that
$\mathfrak{m}_{S}^{2}=0$; cf. [17, 4.6.14]. As $S$ is equicharacteristic, it
has the form $k[x_{1},\ldots,x_{n}]/(x_{1},\ldots,x_{n})^{2}$ for some
$n\geqslant 0.$

The structure of the lattice
$\mathcal{T}=\operatorname{Thick}{\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$ is
especially simply when $R$ is also Gorenstein; this forces $n\leqslant 1$.

If $n=0$, then $R$ is regular and so $\mathcal{T}$ is simply
$0\subsetneq{\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$.

If $n=1,$ then the lattice $\mathcal{T}$ is

$0\subsetneq\operatorname{thick}_{R}(K)\subsetneq{\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$

where $K$ is a Koszul complex on any system of parameters for $R$. This is a
consequence of Corollary 5.10 and [18, 31].

When $R$ is not Gorenstein,
$\operatorname{Thick}{\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$ possesses a far
wilder structure.

######  Corollary 5.13.

Let $R$ be an equicharacteristic Cohen–Macaulay non-Gorenstein ring of minimal
multiplicity with infinite residue field $k$. Then there exists an infinite
descending binary tree of finitely generated thick subcategories in
$\operatorname{Thick}{\mathsf{D}}^{\sf{fl}}_{\sf{b}}(R)$.

###### Proof.

The assumptions on $R$ force $S$, from 5.12, to be
$k[x_{1},\ldots,x_{n}]/(x_{1},\ldots,x_{n})^{2}$ with $n\geqslant 2$. Now the
stated result is a consequence of Corollary 5.10 and the work of Elagin and
Lunts [22, Theorem A]. ∎

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