# Bounds on Gorenstein Dimensions and
Exceptional Complete Intersection Maps

Hossein Faridian  Hossein Faridian, School of Mathematical and Statistical
Sciences, Clemson University, Clemson, SC 29634, USA<EMAIL_ADDRESS>
###### Abstract.

We prove that if $f:R\rightarrow S$ is a local homomorphism of noetherian
local rings of finite flat dimension and $M$ is a non-zero finitely generated
$S$-module whose Gorenstein flat dimension over $R$ is bounded by the
difference of the embedding dimensions of $R$ and $S$, then $M$ is a totally
reflexive $S$-module and $f$ is an exceptional complete intersection map. This
is an extension of a result of Brochard, Iyengar, and Khare to Gorenstein flat
dimension. We also prove two analogues involving Gorenstein injective
dimension.

###### Key words and phrases:

Gorenstein flat dimension; Gorenstein injective dimension; exceptional
complete intersection map; G-regular ring

###### 2010 Mathematics Subject Classification:

13D05; 13C11; 13E05; 18G25.

##  1\. Introduction

An interesting and fruitful theme in commutative algebra is the study of those
properties of rings that can be characterized or identified by prescribed
conditions on their modules. This has gradually been extended to the relative
situation where one investigates certain properties of ring homomorphisms
through imposed conditions on the modules along them. An important class of
ring homomorphisms is that of complete intersection maps. A surjective
homomorphism of noetherian local rings is a complete intersection map if its
kernel is generated by a regular sequence. If in addition, the regular
sequence is part of a minimal generating set for the maximal ideal of the
source, then the homomorphism is an “exceptional” complete intersection map.
This subclass is of particular interest as it appears in various contexts; for
example, the diagonal of a smooth map is locally exceptional complete
intersection. Recently, some research has been conducted to characterize or
detect this property.

In one direction, Iyengar, Letz, Liu, and Pollitz characterize surjective
exceptional complete intersection maps in terms of lattices of thick
subcategories of bounded derived categories as well as the vanishing of
certain Atiyah classes. More precisely, they prove that if
$f:(R,\mathfrak{m},k)\rightarrow(S,\mathfrak{n},k)$ is a surjective local
homomorphism of noetherian local rings, then $f$ is an exceptional complete
intersection map if and only if $\operatorname{fd}_{R}(S)<\infty$ and the
restriction functor
$\operatorname{res}_{f}:\mathcal{D}_{\square}^{\operatorname{fl}}(S)\rightarrow\mathcal{D}_{\square}^{\operatorname{fl}}(R)$
induces an isomorphism between the corresponding lattices of thick
subcategories. Furthermore, they show that such an $f$ is an exceptional
complete intersection map if and only if $\operatorname{fd}_{R}(S)<\infty$ and
$\operatorname{at}^{f}(k)=0$; see [ILP, Theorems A and B].

In another direction, Brochard, Iyengar, and Khare detect exceptional complete
intersection maps through a certain bound on the flat dimension of a module
along them. More specifically, they prove that if $f:R\rightarrow S$ is a
local homomorphism of noetherian local rings, and $M$ is a non-zero finitely
generated $S$-module with
$\operatorname{fd}_{R}(M)\leq\operatorname{edim}(R)-\operatorname{edim}(S)$,
then $M$ is a free $S$-module and $f$ is an exceptional complete intersection
map; see [BIK, Theorem 3.1]. Dually, the author shows that if $f:R\rightarrow
S$ is a local homomorphism of noetherian local rings, and $M$ is a non-zero
finitely generated or artinian $S$-module with
$\operatorname{id}_{R}(M)\leq\operatorname{edim}(R)-\operatorname{edim}(S)$,
then $M$ is an injective $S$-module and $f$ is an exceptional complete
intersection map; see [Fa, Theorems 3.3 and 3.12].

The goal of this article is to establish some analogues of the results
mentioned in the previous paragraph for Gorenstein dimensions. In other words,
we show that under some mild assumptions, the same bound on Gorenstein
dimensions of a module along a map can detect the exceptional complete
intersection property. More specifically, we prove the following; see Theorem
3.5.

######  Theorem A.

Let $f:(R,\mathfrak{m})\rightarrow(S,\mathfrak{n})$ be a local homomorphism of
noetherian local rings with $\operatorname{fd}_{R}(S)<\infty$, and $M$ a non-
zero finitely generated $S$-module with
$\operatorname{Gfd}_{R}(M)\leq\operatorname{edim}(R)-\operatorname{edim}(S)$.
Then the following assertions hold:

  1. (i)

$M$ is a totally reflexive $S$-module.

  2. (ii)

$f$ is an exceptional complete intersection map.

  3. (iii)

$\operatorname{Gfd}_{R}(M)=\operatorname{edim}(R)-\operatorname{edim}(S)$.

One notes that Gorenstein flat dimension does not exceed flat dimension, so
the above result could be conceived as an extension of [BIK, Theorem 3.1] to
Gorenstein flat dimension. We further discuss how the finite flat dimension in
the previous theorem might be replaced by a G-regularity assumption; see
Remark 3.9.

Dually, We prove two analogues for Gorenstein injective dimension; see
Theorems 3.11 and 3.13.

######  Theorem B.

Let $f:(R,\mathfrak{m},k)\rightarrow(S,\mathfrak{n},l)$ be a local
homomorphism of noetherian local rings with $\operatorname{fd}_{R}(S)<\infty$
in which $R$ has a dualizing complex and $S$ is $\mathfrak{n}$-adically
complete, and $M$ a non-zero artinian $S$-module with
$\operatorname{Gid}_{R}(M)\leq\operatorname{edim}(R)-\operatorname{edim}(S)$.
Then the following assertions hold:

  1. (i)

$M$ is a Gorenstein injective $S$-module.

  2. (ii)

$f$ is an exceptional complete intersection map.

  3. (iii)

$\operatorname{Gid}_{R}(M)=\operatorname{edim}(R)-\operatorname{edim}(S)$.

######  Theorem C.

Let $f:(R,\mathfrak{m},k)\rightarrow(S,\mathfrak{n},l)$ be a local
homomorphism of noetherian local rings with $\operatorname{fd}_{R}(S)<\infty$
in which $R$ has a dualizing complex and $S$ is G-regular, and $M$ a non-zero
artinian $S$-module with
$\operatorname{Gid}_{R}(M)\leq\operatorname{edim}(R)-\operatorname{edim}(S)$.
Then the following assertions hold:

  1. (i)

$M$ is an injective $S$-module.

  2. (ii)

$f$ is an exceptional complete intersection map.

  3. (iii)

$\operatorname{Gid}_{R}(M)=\operatorname{edim}(R)-\operatorname{edim}(S)$.

##  2\. Basic Definitions and Observations

In what follow, all rings are assumed to be non-zero commutative with unity.
We write $(R,\mathfrak{m},k)$ to indicate $R$ is a local ring with the maximal
ideal $\mathfrak{m}$ and the residue field $k\cong R/\mathfrak{m}$. Moreover,
$\widehat{R}^{\mathfrak{m}}$ denotes its $\mathfrak{m}$-adic completion. A
ring homomorphism $f:(R,\mathfrak{m})\rightarrow(S,\mathfrak{n})$ is local if
$f(\mathfrak{m})\subseteq\mathfrak{n}$. Given an $R$-module $M$, the notations
$\operatorname{pd}_{R}(M)$, $\operatorname{id}_{R}(M)$, and
$\operatorname{fd}_{R}(M)$ are used for projective, injective, and flat
dimensions of $M$, respectively. Moreover, $0pt_{R}(\mathfrak{m},M)$,
$\dim(R)$, and $\operatorname{edim}(R)$ indicate depth, Krull dimension, and
embedding dimension, respectively.

We begin with a remark on totally reflexive modules and G-regular rings.

######  Remark 2.1.

Let $R$ be a ring, and consider the functor
$(-)^{\ast}=\operatorname{Hom}_{R}(-,R)$. A finitely generated $R$-module $G$
is said to be totally reflexive if the biduality map $\eta_{G}:G\rightarrow
G^{\ast\ast}$, given by $\eta_{G}(x)(f)=f(x)$ for every $x\in G$ and $f\in
G^{\ast}$, is an isomorphism, and
$\operatorname{Ext}_{R}^{i}(G,R)=0=\operatorname{Ext}_{R}^{i}(G^{\ast},R)$ for
every $i\geq 1$; see [Ch, Definition 1.1.2]. Given a non-zero finitely
generated $R$-module $M$, the G-dimension of $M$, denoted by
$\operatorname{Gdim}_{R}(M)$, is the least integer $n\geq 0$ for which there
exists an exact sequence

$0\rightarrow G_{n}\rightarrow\cdots\rightarrow G_{1}\rightarrow
G_{0}\rightarrow M\rightarrow 0$

where $G_{i}$ is a totally reflexive $R$-module for every $0\leq i\leq n$. We
have $\operatorname{Gdim}_{R}(M)\leq\operatorname{pd}_{R}(M)$ with equality if
$\operatorname{pd}_{R}(M)<\infty$; see [Ch, Proposition 1.2.10].

Every finitely generated free $R$-module is clearly totally reflexive.
Following Takahashi, [Ta, Definition 1.7], we say that a noetherian local ring
$R$ is G-regular if the converse holds, i.e. every totally reflexive
$R$-module is free. If $R$ is G-regular, then by [Ta, Proposition 1.8 (2)]
coupled with [Ro, Theorem 8.27 (i)], we have
$\operatorname{Gdim}_{R}(M)=\operatorname{pd}_{R}(M)=\operatorname{fd}_{R}(M)$
for every finitely generated $R$-module $M$. The class of G-regular rings is
quite large:

  1. (i)

Every regular local ring is G-regular. Indeed, if $R$ is a regular local ring
and $G$ is a totally reflexive $R$-module, then
$\operatorname{pd}_{R}(G)<\infty$, so
$\operatorname{pd}_{R}(G)=\operatorname{Gdim}_{R}(G)=0$, whence $G$ is free as
$R$ is local.

  2. (ii)

Every Golod local ring which is not a hypersurface is G-regular; see [AM,
Example 3.5 (2)]. In particular, every non-Gorenstein Cohen-Macaulay local
ring with minimal multiplicity is G-regular; see [Ta, Lemma 5.1]. As for
concrete examples of this kind, if $k$ is a field, then
$k[[X,Y]]/(X^{2},XY,Y^{2})$, $k[[X,Y,Z]]/(X^{2}-YZ,Y^{2}-XZ,Z^{2}-XY)$, and
$k[[X^{3},X^{4},X^{5}]]$ are G-regular; see [Ta, Example 5.2].

  3. (iii)

If $k$ is field, then $k[[X,Y]]/(X^{3},XY,Y^{3})$ is an artinian non-
Gorenstein G-regular local ring which does not have minimal multiplicity; see
[Ta, Examples 5.4 (1)]. Also,
$k[[X,Y,Z]]/(X^{3}-Y^{2}Z,Y^{3}-X^{2}Z,Z^{2}-XY)$ is a one-dimensional Cohen-
Macaulay non-Gorenstein G-regular local ring which does not have minimal
multiplicity; see [Ta, Examples 5.4 (2)].

  4. (iv)

Given noetherian local rings $(R,\mathfrak{m},k)$ and $(S,\mathfrak{n},k)$
with a common residue field $k$, if the pullback

${R\times_{k}S}$${S}$${R}$${k}$

is not Gorenstein, then it is G-regular; see [NS, Corollary 4.7]. On the other
hand, by [NTSV, Fact 2.2], the ring $R\times_{k}S$ is Cohen-Macaulay if and
only if $R$ and $S$ are Cohen-Macaulay with $\dim(R)=\dim(S)\leq 1$. This
indicates that there are plenty of examples of non-Cohen-Macaulay G-regular
rings; see also [DMT, Remark 2.9 (ii)]. As for concrete examples, if $k$ is a
field, then $k[[X,Y]]/(X^{2},XY)$ is a non-Cohen-Macaulay G-regular local
ring; see [Ta, Examples 5.5].

  5. (v)

Every non-Gorenstein quotient of small colength of a deeply embedded
equicharacteristic artinian Gorenstein local ring is G-regular; see [KV].

  6. (vi)

If $(R,\mathfrak{m})$ is a G-regular local ring, then the $\mathfrak{m}$-adic
completion $\widehat{R}^{\mathfrak{m}}$, the quotient $R/(a)$ by an
$R$-regular element $a\in\mathfrak{m}\setminus\mathfrak{m}^{2}$, and the power
series ring $R[[X_{1},...,X_{n}]]$ for every $n\geq 1$, are G-regular; see
[Ta, Corollary 4.7, Proposition 4.6, and Corollary 4.4].

We next recall Gorenstein dimensions which are refinements of classical
homological dimensions.

######  Remark 2.2.

Let $R$ be a ring, and $\mathcal{I}$ denote the class of injective
$R$-modules. An $R$-module $Q$ is Gorenstein flat if there exists an
$(\mathcal{I}\otimes_{R}-)$-exact exact $R$-complex

$F:\cdots\rightarrow
F_{2}\xrightarrow{\partial^{F}_{2}}F_{1}\xrightarrow{\partial^{F}_{1}}F_{0}\xrightarrow{\partial^{F}_{0}}F_{-1}\xrightarrow{\partial^{F}_{-1}}F_{-2}\rightarrow\cdots$

of flat modules such that $Q\cong\operatorname{Im}\partial^{F}_{0}$; see [EJ,
Definition 10.3.1]. Given a non-zero $R$-module $M$, the Gorenstein flat
dimension of $M$, denoted by $\operatorname{Gfd}_{R}(M)$, is the least integer
$n\geq 0$ for which there exists an exact sequence

$0\rightarrow Q_{n}\rightarrow\cdots\rightarrow Q_{1}\rightarrow
Q_{0}\rightarrow M\rightarrow 0$

where $Q_{i}$ is a Gorenstein flat $R$-module for every $0\leq i\leq n$. We
have $\operatorname{Gfd}_{R}(M)\leq\operatorname{fd}_{R}(M)$ with equality if
$\operatorname{fd}_{R}(M)<\infty$; see [CFH1, Propositions 4.8]. This shows
that Gorenstein flat dimension is finer than flat dimension. Moreover, if $R$
is noetherian and $M$ is finitely generated, then
$\operatorname{Gfd}_{R}(M)=\operatorname{Gdim}_{R}(M)$; see [CFH1, Proposition
4.24].

Dually, an $R$-module $J$ is Gorenstein injective if there exists a
$\operatorname{Hom}_{R}(\mathcal{I},-)$-exact exact $R$-complex

$I:\cdots\rightarrow
I_{2}\xrightarrow{\partial^{I}_{2}}I_{1}\xrightarrow{\partial^{I}_{1}}I_{0}\xrightarrow{\partial^{I}_{0}}I_{-1}\xrightarrow{\partial^{I}_{-1}}I_{-2}\rightarrow\cdots$

of injective modules such that $J\cong\operatorname{Im}\partial^{I}_{0}$; see
[EJ, Definition 10.1.1]. Given a non-zero $R$-module $M$, the Gorenstein
injective dimension of $M$, denoted by $\operatorname{Gid}_{R}(M)$, is the
least integer $n\geq 0$ for which there exists an exact sequence

$0\rightarrow M\rightarrow J_{0}\rightarrow J_{-1}\rightarrow\cdots\rightarrow
J_{-n}\rightarrow 0$

where $J_{-i}$ is a Gorenstein injective $R$-module for every $0\leq i\leq n$.
We have $\operatorname{Gid}_{R}(M)\leq\operatorname{id}_{R}(M)$ with equality
if $\operatorname{id}_{R}(M)<\infty$; see [CFH1, Propositions 3.10]. This
shows that Gorenstein injective dimension is finer than injective dimension.

We finally recall the concept of an (exceptional) complete intersection map.

######  Remark 2.3.

Let $f:(R,\mathfrak{m},k)\rightarrow(S,\mathfrak{n},l)$ be a local
homomorphism of noetherian local rings. Then $f$ admits a Cohen factorization,
i.e. it fits into a commutative diagram

${R}$${S}$${R^{\prime}}$${\widehat{S}^{\mathfrak{n}}}$$\scriptstyle{f}$$\scriptstyle{\dot{f}}$$\scriptstyle{f^{\prime}}$

of noetherian local rings in which $\dot{f}$ is flat with regular closed fiber
$R^{\prime}/\mathfrak{m}R^{\prime}$, $R^{\prime}$ is complete with respect to
its maximal ideal, and $f^{\prime}$ is surjective; see [AFHe, Theorem 1.1].
Accordingly, $f$ is said to be a complete intersection map if there is a Cohen
factorization of $f$ in which $\operatorname{Ker}(f^{\prime})$ is generated by
a regular sequence on $R^{\prime}$. This property is independent of the choice
of Cohen factorization; see [Av, Theorem 1.2]. In particular, $f$ is complete
intersection if and only if $f^{\prime}$ is complete intersection. Moreover,
the class of complete intersection maps is closed under composition and flat
base change.

When $f$ is complete intersection, there is an inequality

$\operatorname{edim}(R)-\dim(R)\leq\operatorname{edim}(S)-\dim(S);$

see [BIK, Lemma 2.13]. Then $f$ is said to be an exceptional complete
intersection map if equality holds above. When $f$ is surjective, this
property is equivalent to the condition that $\operatorname{Ker}(f)$ can be
generated by a regular sequence whose image in $\mathfrak{m}/\mathfrak{m}^{2}$
is a linearly independent set over $k$; see [ILP, Lemma 3.2]. Considering a
Cohen factorization of $f$ as above and applying [BIK, 2.6] to $\dot{f}$, we
obtain

$\operatorname{edim}(R^{\prime})-\operatorname{edim}(R)=\operatorname{edim}\left(R^{\prime}/\mathfrak{m}R^{\prime}\right)=\dim(R^{\prime})-\dim(R),$

so
$\operatorname{edim}(R^{\prime})-\dim(R^{\prime})=\operatorname{edim}(R)-\dim(R)$.
It follows that $f$ is exceptional complete intersection if and only if
$f^{\prime}$ is exceptional complete intersection.

Exceptional complete intersection maps are prevalent. Here are a few examples:

  1. (i)

If $f:(R,\mathfrak{m})\rightarrow(S,\mathfrak{n})$ is a flat local
homomorphism of noetherian local rings whose closed fiber $S/\mathfrak{m}S$ is
regular, then $f$ is an exceptional complete intersection map; see [BIK,
2.14].

  2. (ii)

If $f:R\rightarrow S$ is a local homomorphism of regular local rings, then $f$
is an exceptional complete intersection map; see [BIK, Example 2.15].

  3. (ii)

If $f:R\rightarrow S$ is a flat local homomorphism of noetherian local rings
which is essentially of finite type and smooth, then the diagonal map
$\mu:S\otimes_{R}S\rightarrow S$, given by $\mu(a\otimes b)=ab$ for every
$a,b\in S$, is an exceptional complete intersection map; see [Iy1, Theorem
9.9].

For more information on (exceptional) complete intersection maps, refer to
[Av], [BIK], [ILP], [BILP], and [Iy1].

##  3\. Main Results

In this section, we prove our main results. The following lemma generalizes
the Auslander-Buchsbaum formula to Gorenstein flat dimension.

######  Lemma 3.1.

Let $(R,\mathfrak{m})$ be a noetherian local ring, and $M$ a non-zero finitely
generated or $\mathfrak{m}$-adically complete $R$-module with
$\operatorname{Gfd}_{R}(M)<\infty$. Then we have:

$\operatorname{Gfd}_{R}(M)=0pt(\mathfrak{m},R)-0pt_{R}(\mathfrak{m},M)$

###### Proof.

If $M$ is finitely generated, then by remark 2.2, we have
$\operatorname{Gfd}_{R}(M)=\operatorname{Gdim}_{R}(M)$, so the result is
[CFH1, Theorem 1.25]. Now assume that $M$ is $\mathfrak{m}$-adically complete.
Then [CFH2, Theorem 13.1.30] implies that
$M\simeq\textrm{L}\Lambda^{\mathfrak{m}}(M)$ in the derive category
$\mathcal{D}(R)$, so by [CFH2, Theorem 14.4.18],
$\operatorname{cosupp}_{R}(M)\subseteq\operatorname{Var}(\mathfrak{m})=\\{\mathfrak{m}\\}$.
However, $M\neq 0$, so [CFH2, Theorem 14.3.35] yields
$\operatorname{cosupp}_{R}(M)\neq\emptyset$. It follows that
$\operatorname{cosupp}_{R}(M)=\\{\mathfrak{m}\\}$. Therefore, we conclude from
[CFH2, Theorem 19.4.9] that:

$\begin{split}\operatorname{Gfd}_{R}(M)&=\sup\left\\{0pt\left(\mathfrak{p}R_{\mathfrak{p}},R_{\mathfrak{p}}\right)-0pt_{R_{\mathfrak{p}}}\left(\mathfrak{p}R_{\mathfrak{p}},\operatorname{RHom}_{R}\left(R_{\mathfrak{p}},M\right)\right)\;|\;\mathfrak{p}\in\operatorname{cosupp}_{R}(M)\right\\}\\\
&=0pt\left(\mathfrak{m}R_{\mathfrak{m}},R_{\mathfrak{m}}\right)-0pt_{R_{\mathfrak{m}}}\left(\mathfrak{m}R_{\mathfrak{m}},\operatorname{RHom}_{R}\left(R_{\mathfrak{m}},M\right)\right)\\\
&=0pt(\mathfrak{m},R)-0pt_{R}\left(\mathfrak{m},\operatorname{RHom}_{R}\left(R,M\right)\right)\\\
&=0pt(\mathfrak{m},R)-0pt_{R}\left(\mathfrak{m},M\right)\end{split}$

∎

The next lemma can be viewed as a generalization of [AFHa, Lemma 2.7].

######  Lemma 3.2.

Let $f:(R,\mathfrak{m})\rightarrow(S,\mathfrak{n})$ be a flat local
homomorphism of complete noetherian local rings with Gorenstein closed fiber
$S/\mathfrak{m}S$, and $M$ a finitely generated $S$-module. Then we have:

$\operatorname{Gfd}_{S}(M)\leq\operatorname{Gfd}_{R}(M)+0pt\left(\mathfrak{n}/\mathfrak{m}S,S/\mathfrak{m}S\right)$

###### Proof.

If $\operatorname{Gfd}_{R}(M)=\infty$, then there is nothing to prove, so we
may assume that $\operatorname{Gfd}_{R}(M)<\infty$. Since $R$ is
$\mathfrak{m}$-adically complete, Cohen’s Structure Theorem implies that $R$
is a homomorphic image of a regular hence Gorenstein local ring, thereby $R$
has a dualizing complex, say $D$; see [Ka]. As
$\operatorname{Gfd}_{R}(M)<\infty$, [CFH1, Theorem 9.2] implies that
$M\in\mathcal{A}_{D}(R)$ where $\mathcal{A}_{D}(R)$ is the Auslander class of
$R$ with respect to $D$. Since $f$ is flat and $S/\mathfrak{m}S$ is
Gorenstein, we conclude that $f$ is Gorenstein, so $D\otimes_{R}S$ is a
dualizing complex for $S$ and $M\in\mathcal{A}_{D\otimes_{R}S}(S)$; see [AF2,
Paragraph before (2.11) and Proposition 3.7 (b)]. Therefore, [CFH1, Theorem
9.2] yields $\operatorname{Gfd}_{S}(M)<\infty$. Now Lemma 3.1 gives:

$\operatorname{Gfd}_{S}(M)=0pt(\mathfrak{n},S)-0pt_{S}(\mathfrak{n},M)$

On the other hand, $S$ is $\mathfrak{n}$-adically complete and $M$ is a
finitely generated $S$-module, so we conclude that $M\cong M\otimes_{S}S\cong
M\otimes_{S}\widehat{S}^{\mathfrak{n}}\cong\widehat{M}^{\mathfrak{n}}$, whence
$M$ is an $\mathfrak{n}$-adically complete $S$-module. But $f$ is local, so
$\mathfrak{m}S\subseteq\mathfrak{n}$, whence [SS, 2.2.9 (b)] implies that $M$
is $\mathfrak{m}S$-adically complete. However,
$(\mathfrak{m}S)^{n}M=\mathfrak{m}^{n}SM=\mathfrak{m}^{n}M$ for every $n\geq
1$, so we notice that:

$M\cong\widehat{M}^{\mathfrak{m}S}=\underset{n\geq
1}{\varprojlim}M/(\mathfrak{m}S)^{n}M=\underset{n\geq
1}{\varprojlim}M/\mathfrak{m}^{n}M=\widehat{M}^{\mathfrak{m}}$

As $\mathfrak{m}$ is finitely generated, [SS, Theorem 2.2.2] implies that
$\widehat{M}^{\mathfrak{m}}$ is an $\mathfrak{m}$-adically complete
$R$-module, so the above display shows that $M$ is an $\mathfrak{m}$-adically
complete $R$-module. Therefore, Lemma 3.1 gives:

$\operatorname{Gfd}_{R}(M)=0pt(\mathfrak{m},R)-0pt_{R}(\mathfrak{m},M)$

Moreover, [BH, Proposition 1.2.16] implies that:

$0pt(\mathfrak{n},S)=0pt(\mathfrak{m},R)+0pt\left(\mathfrak{n}/\mathfrak{m}S,S/\mathfrak{m}S\right)$

Also, $0pt_{R}(\mathfrak{m},M)\leq 0pt_{S}(\mathfrak{n},M)$ by [Iy2,
Proposition 5.2 (2)]. Putting everything together, we get:

$\begin{split}\operatorname{Gfd}_{S}(M)&=0pt(\mathfrak{n},S)-0pt_{S}(\mathfrak{n},M)\\\
&\leq 0pt(\mathfrak{n},S)-0pt_{R}(\mathfrak{m},M)\\\
&=0pt(\mathfrak{m},R)-0pt_{R}(\mathfrak{m},M)+0pt\left(\mathfrak{n}/\mathfrak{m}S,S/\mathfrak{m}S\right)\\\
&=\operatorname{Gfd}_{R}(M)+0pt\left(\mathfrak{n}/\mathfrak{m}S,S/\mathfrak{m}S\right)\end{split}$

∎

######  Corollary 3.3.

Let $f:(R,\mathfrak{m})\rightarrow(S,\mathfrak{n})$ be a flat local
homomorphism of complete noetherian local rings with regular closed fiber
$S/\mathfrak{m}S$, and $M$ a finitely generated $S$-module. Then we have:

$\operatorname{Gfd}_{S}(M)\leq\operatorname{Gfd}_{R}(M)+\operatorname{edim}(S)-\operatorname{edim}(R)$

###### Proof.

The closed fiber $S/\mathfrak{m}S$ is regular, hence Gorenstein, so Lemma 3.2
together with [BIK, 2.6] yield:

$\begin{split}\operatorname{Gfd}_{S}(M)&\leq\operatorname{Gfd}_{R}(M)+0pt\left(\mathfrak{n}/\mathfrak{m}S,S/\mathfrak{m}S\right)\\\
&=\operatorname{Gfd}_{R}(M)+\operatorname{edim}\left(S/\mathfrak{m}S\right)\\\
&=\operatorname{Gfd}_{R}(M)+\operatorname{edim}(S)-\operatorname{edim}(R)\end{split}$

∎

######  Lemma 3.4.

Let $f:(R,\mathfrak{m})\rightarrow(S,\mathfrak{n})$ be an epimorphism of
noetherian local rings with $\operatorname{fd}_{R}(S)<\infty$. If $f$ is not
an isomorphism, then $\operatorname{Ker}(f)$ contains a regular element on
$R$.

###### Proof.

Since $f$ is surjective, we have
$\operatorname{pd}_{R}(S)=\operatorname{fd}_{R}(S)<\infty$. In addition,
$\operatorname{Ker}(f)\neq 0$, so [AB, Corollary 6.3] yields a non-zerodivisor
$a\in\operatorname{Ker}(f)$. But $\operatorname{Ker}(f)\subseteq\mathfrak{m}$,
so $a$ is regular on $R$. ∎

We are now ready to prove our first main result which is Theorem A from the
introduction.

######  Theorem 3.5.

Let $f:(R,\mathfrak{m})\rightarrow(S,\mathfrak{n})$ be a local homomorphism of
noetherian local rings with $\operatorname{fd}_{R}(S)<\infty$, and $M$ a non-
zero finitely generated $S$-module with
$\operatorname{Gfd}_{R}(M)\leq\operatorname{edim}(R)-\operatorname{edim}(S)$.
Then the following assertions hold:

  1. (i)

$M$ is a totally reflexive $S$-module.

  2. (ii)

$f$ is an exceptional complete intersection map.

  3. (iii)

$\operatorname{Gfd}_{R}(M)=\operatorname{edim}(R)-\operatorname{edim}(S)$.

In particular, if $M$ is Gorenstein flat as an $R$-module, then
$\operatorname{edim}(R)=\operatorname{edim}(S)$.

###### Proof.

We first reduce to the case where $R$ is $\mathfrak{m}$-adically complete and
$S$ is $\mathfrak{n}$-adically complete. Note that
$M\otimes_{S}\widehat{S}^{\mathfrak{n}}$ is a non-zero finitely generated
$\widehat{S}^{\mathfrak{n}}$-module. Since $\operatorname{Gfd}_{R}(M)<\infty$,
we can invoke [CI, Corollary 4.8] to observe that:

$\begin{split}\operatorname{Gfd}_{\widehat{R}^{\mathfrak{m}}}\left(M\otimes_{S}\widehat{S}^{\mathfrak{n}}\right)&=\operatorname{Gfd}_{R}(M)\\\
&\leq\operatorname{edim}(R)-\operatorname{edim}(S)\\\
&=\operatorname{edim}\left(\widehat{R}^{\mathfrak{m}}\right)-\operatorname{edim}\left(\widehat{S}^{\mathfrak{n}}\right)\end{split}$

On the other hand, [CFH1, Proposition 1.26] gives
$\operatorname{Gdim}_{S}(M)=\operatorname{Gdim}_{\widehat{S}^{\mathfrak{n}}}\left(M\otimes_{S}\widehat{S}^{\mathfrak{n}}\right)$,
so it follows that $M$ is a totally reflexive $S$-module if and only if
$M\otimes_{S}\widehat{S}^{\mathfrak{n}}$ is a totally reflexive
$\widehat{S}^{\mathfrak{n}}$-module. In view of Remark 2.3, consider the
commutative diagram

${R}$${S}$${\widehat{R}^{\mathfrak{m}}}$${\widehat{S}^{\mathfrak{n}}}$${R^{\prime}}$${\widehat{S}^{\mathfrak{n}}}$$\scriptstyle{f}$$\scriptstyle{\widehat{f}}$$\scriptstyle{\dot{f}}$$\scriptstyle{f^{\prime}}$

in which the bottom square provides a Cohen factorization of $\widehat{f}$.
Then it is clear that the combined rectangle provides a Cohen factorization of
$f$. As a result, $f$ is exceptional complete intersection if and only if
$f^{\prime}$ is exceptional complete intersection if and only if $\widehat{f}$
is exceptional complete intersection. Finally, since the combined rectangle
gives a Cohen factorization of $f$, and $\operatorname{fd}_{R}(S)<\infty$,
[AFHe, 3.3] implies that
$\operatorname{fd}_{R^{\prime}}\left(\widehat{S}^{\mathfrak{n}}\right)<\infty$.
But then [AF1, Corollary 4.2(b)(F)] yields

$\operatorname{fd}_{\widehat{R}^{\mathfrak{m}}}\left(\widehat{S}^{\mathfrak{n}}\right)\leq\operatorname{fd}_{R^{\prime}}\left(\widehat{S}^{\mathfrak{n}}\right)+\operatorname{fd}_{\widehat{R}^{\mathfrak{m}}}(R^{\prime})=\operatorname{fd}_{R^{\prime}}\left(\widehat{S}^{\mathfrak{n}}\right)<\infty.$

All in all, we can replace $f$ with $\widehat{f}$, $M$ with
$M\otimes_{S}\widehat{S}^{\mathfrak{n}}$, and assume accordingly that $R$ is
$\mathfrak{m}$-adically complete and $S$ is $\mathfrak{n}$-adically complete.

We next reduce to the case where $f$ is surjective. By Remark 2.3, $f$ has a
Cohen factorization

$R\xrightarrow{\dot{f}}R^{\prime}\xrightarrow{f^{\prime}}S$

in which $\dot{f}$ is flat with regular closed fiber
$R^{\prime}/\mathfrak{m}R^{\prime}$, $R^{\prime}$ is complete, and
$f^{\prime}$ is surjective. Since $M$ is a finitely generated $S$-module, we
conclude that $M$ is a finitely generated $R^{\prime}$-module as well.
Moreover, $f$ is exceptional complete intersection if and only if $f^{\prime}$
is exceptional complete intersection. Using Corollary 3.3 and the hypothesis,
we obtain:

$\begin{split}\operatorname{Gfd}_{R^{\prime}}(M)&\leq\operatorname{Gfd}_{R}(M)+\operatorname{edim}(R^{\prime})-\operatorname{edim}(R)\\\
&\leq\operatorname{edim}(R)-\operatorname{edim}(S)+\operatorname{edim}(R^{\prime})-\operatorname{edim}(R)\\\
&=\operatorname{edim}(R^{\prime})-\operatorname{edim}(S)\end{split}$

If
$\operatorname{Gfd}_{R^{\prime}}(M)=\operatorname{edim}(R^{\prime})-\operatorname{edim}(S)$,
then the above display shows that
$\operatorname{Gfd}_{R}(M)=\operatorname{edim}(R)-\operatorname{edim}(S)$.
Finally, since $\operatorname{fd}_{R}(S)<\infty$, [AFHe, 3.3] implies that
$\operatorname{fd}_{R^{\prime}}(S)<\infty$. Hence it suffices to prove the
result for $f^{\prime}$. Therefore, we can assume that $f$ is surjective, so
that $M$ is a finitely generated $R$-module as well. Now by Remark 2.2,
$\operatorname{Gfd}_{R}(M)=\operatorname{Gdim}_{R}(M)$.

We finally reduce to the case where
$\operatorname{Ker}(f)\subseteq\mathfrak{m}^{2}$. Assume to the contrary that
$\operatorname{Ker}(f)\nsubseteq\mathfrak{m}^{2}$. In particular, $f$ is not
an isomorphism, so Lemma 3.4 implies that $\operatorname{Ker}(f)$ contains a
regular element on $R$, thereby
$\operatorname{Ker}(f)\nsubseteq\mathcal{Z}(R)=\bigcup_{\mathfrak{p}\in\operatorname{Ass}(R)}\mathfrak{p}$.
By the Prime Avoidance Lemma (see [Sh, Theorem 3.61]), we infer that
$\operatorname{Ker}(f)\nsubseteq\mathfrak{m}^{2}\cup\bigcup_{\mathfrak{p}\in\operatorname{Ass}(R)}\mathfrak{p}$,
so there exists a regular element
$a\in\operatorname{Ker}(f)\backslash\mathfrak{m}^{2}$. We then get the
factorization

$R\rightarrow R/(a)\xrightarrow{f^{\prime}}S$

of $f$. We note that $\operatorname{Ker}(f)$ is generated by a regular
sequence if and only if $\operatorname{Ker}(f^{\prime})$ is generated by a
regular sequence. Hence $f$ is complete intersection if and only $f^{\prime}$
is complete intersection. In addition,
$\operatorname{edim}\left(R/(a)\right)=\operatorname{edim}(R)-1$ and
$\dim\left(R/(a)\right)=\dim(R)-1$, so
$\operatorname{edim}\left(R/(a)\right)-\dim\left(R/(a)\right)=\operatorname{edim}(R)-\dim(R)$.
It follows that $f$ is exceptional complete intersection if and only
$f^{\prime}$ is exceptional complete intersection. On the other hand,
$\operatorname{Gdim}_{R/(a)}(M)=\operatorname{Gdim}_{R}(M)-1$ by [BM, Theorem
4.1], so we get
$\operatorname{Gdim}_{R/(a)}(M)\leq\operatorname{edim}\left(R/(a)\right)-\operatorname{edim}(S)$.
Finally, $a$ is a regular element on $R$ that is not contained in
$\mathfrak{m}^{2}$, so [Na, Corollary 27.5] yields
$\operatorname{fd}_{R/(a)}(S)=\operatorname{pd}_{R/(a)}(S)=\operatorname{pd}_{R}(S)-1<\infty$.
As a result, we may replace $f$ by $f^{\prime}$, and continue this process.
Since $\operatorname{Ker}(f)$ cannot contain any infinite regular sequence, we
cannot continue this process indefinitely, so at some finite step, the kernel
of the resulting map will be contained in $\mathfrak{m}^{2}$. Therefore, we
can assume that $\operatorname{Ker}(f)\subseteq\mathfrak{m}^{2}$.

With these reductions, we end up with a local epimorphism $f:R\rightarrow S$
of noetherian local rings with $\operatorname{fd}_{R}(S)<\infty$ and
$\operatorname{Ker}(f)\subseteq\mathfrak{m}^{2}$, and a finitely generated
$S$-module $M$ such that
$\operatorname{Gdim}_{R}(M)\leq\operatorname{edim}(R)-\operatorname{edim}(S)$.
But then the assumption $\operatorname{Ker}(f)\subseteq\mathfrak{m}^{2}$
implies that $\operatorname{edim}(R)=\operatorname{edim}(S)$, so
$\operatorname{Gdim}_{R}(M)=0$. If $f$ is not an isomorphism, then as in the
previous paragraph, we can find a regular element $a\in\operatorname{Ker}(f)$
so that $\operatorname{Gdim}_{R/(a)}(M)=\operatorname{Gdim}_{R}(M)-1=-1$ which
is a contradiction. Therefore, $f$ is an isomorphism, so in particular, it is
an exceptional complete intersection map. Furthermore,
$\operatorname{Gdim}_{S}(M)=\operatorname{Gdim}_{R}(M)=0=\operatorname{edim}(R)-\operatorname{edim}(S)$,
so $M$ is a totally reflexive $S$-module, and the equality in (iii) trivially
holds. ∎

######  Corollary 3.6.

Let $f:(R,\mathfrak{m})\rightarrow(S,\mathfrak{n})$ be a local homomorphism of
noetherian local rings with $\operatorname{fd}_{R}(S)<\infty$, and $M$ a non-
zero finitely generated $S$-module. Then we have

$\operatorname{Gfd}_{R}(M)\geq\operatorname{edim}(R)-\operatorname{edim}(S),$

and if equality holds above, then $f$ is an exceptional complete intersection
map.

###### Proof.

We note that if equality holds above, then by Theorem 3.5, $f$ is an
exceptional complete intersection map. Thus we are left with establishing the
inequality. If $\operatorname{Gfd}_{R}(M)=\infty$, then there is nothing to
prove, so we may assume that $\operatorname{Gfd}_{R}(M)<\infty$.

We first handle the special case where $f$ is surjective. Then $M$ is a
finitely generated $R$-module as well, so by Remark 2.2,
$\operatorname{Gfd}_{R}(M)=\operatorname{Gdim}_{R}(M)$. We argue by induction
on $\operatorname{edim}(R)-\operatorname{edim}(S)$. If
$\operatorname{edim}(R)-\operatorname{edim}(S)=0$, then there is nothing to
prove. Now suppose that $\operatorname{edim}(R)-\operatorname{edim}(S)\geq 1$.
Then $f$ is not an isomorphism, so Lemma 3.4 implies that
$\operatorname{Ker}(f)$ contains a regular element on $R$, so
$\operatorname{Ker}(f)\nsubseteq\mathcal{Z}(R)=\bigcup_{\mathfrak{p}\in\operatorname{Ass}(R)}\mathfrak{p}$.
On the other hand,
$\operatorname{edim}(S)=\operatorname{edim}\left(R/\operatorname{Ker}(f)\right)\leq\operatorname{edim}(R)$
with equality if and only if $\operatorname{Ker}(f)\subseteq\mathfrak{m}^{2}$.
But $\operatorname{edim}(R)-\operatorname{edim}(S)\geq 1$, so
$\operatorname{Ker}(f)\nsubseteq\mathfrak{m}^{2}$. By the Prime Avoidance
Lemma (see [Sh, Theorem 3.61]), we infer that
$\operatorname{Ker}(f)\nsubseteq\mathfrak{m}^{2}\cup\bigcup_{\mathfrak{p}\in\operatorname{Ass}(R)}\mathfrak{p}$,
so there exists a regular element
$a\in\operatorname{Ker}(f)\backslash\mathfrak{m}^{2}$. As in the proof of
Theorem 3.5, we notice that $\operatorname{fd}_{R/(a)}(S)<\infty$. Thus in
light of [BM, Theorem 4.1]) and the induction hypothesis applied to
$\bar{f}:R/\langle a\rangle\rightarrow S$, we see that:

$\begin{split}\operatorname{Gdim}_{R}(M)&=\operatorname{Gdim}_{R/\langle
a\rangle}(M)+1\\\ &\geq\operatorname{edim}\left(R/\langle
a\rangle\right)-\operatorname{edim}(S)+1\\\
&=\operatorname{edim}(R)-\operatorname{edim}(S)\end{split}$

We next consider the general case. It is clear that
$M\otimes_{S}\widehat{S}^{\mathfrak{n}}$ is a non-zero finitely generated
$\widehat{S}^{\mathfrak{n}}$-module. Also, as we observed in the proof of
Theorem 3.5, we have
$\operatorname{Gfd}_{R}(M)=\operatorname{Gfd}_{\widehat{R}^{\mathfrak{m}}}\left(M\otimes_{S}\widehat{S}^{\mathfrak{n}}\right)$,
$\operatorname{edim}(R)=\operatorname{edim}\left(\widehat{R}^{\mathfrak{m}}\right)$,
$\operatorname{edim}(S)=\operatorname{edim}\left(\widehat{S}^{\mathfrak{n}}\right)$,
and
$\operatorname{fd}_{R}(S)=\operatorname{fd}_{\widehat{R}^{\mathfrak{m}}}\left(\widehat{S}^{\mathfrak{n}}\right)$.
Therefore, we can assume that $R$ is $\mathfrak{m}$-adically complete and $S$
is $\mathfrak{n}$-adically complete. Then by Remark 2.3, $f$ has a Cohen
factorization

$R\xrightarrow{\dot{f}}R^{\prime}\xrightarrow{f^{\prime}}S$

in which $\dot{f}$ is flat with regular closed fiber
$R^{\prime}/\mathfrak{m}R^{\prime}$, $R^{\prime}$ is complete, and
$f^{\prime}$ is surjective. Besides,
$\operatorname{fd}_{R^{\prime}}(S)<\infty$. By Corollary 3.3, we have:

$\operatorname{Gfd}_{R^{\prime}}(M)\leq\operatorname{Gfd}_{R}(M)+\operatorname{edim}(R^{\prime})-\operatorname{edim}(R)$

Applying the special case to $f^{\prime}$, we get:

$\begin{split}\operatorname{Gfd}_{R}(M)&\geq\operatorname{Gfd}_{R^{\prime}}(M)+\operatorname{edim}(R)-\operatorname{edim}(R^{\prime})\\\
&\geq\operatorname{edim}(R^{\prime})-\operatorname{edim}(S)+\operatorname{edim}(R)-\operatorname{edim}(R^{\prime})\\\
&=\operatorname{edim}(R)-\operatorname{edim}(S)\end{split}$

Therefore, we are done. ∎

######  Corollary 3.7.

Let $f:(R,\mathfrak{m})\rightarrow(S,\mathfrak{n})$ be a local homomorphism of
noetherian local rings. If
$\operatorname{fd}_{R}(S)=\operatorname{edim}(R)-\operatorname{edim}(S)$, then
$f$ is an exceptional complete intersection map.

###### Proof.

Set $M=S$ in Corollary 3.6 and note that
$\operatorname{Gfd}_{R}(S)=\operatorname{fd}_{R}(S)=\operatorname{edim}(R)-\operatorname{edim}(S)$.
∎

######  Remark 3.8.

If $f:(R,\mathfrak{m})\rightarrow(S,\mathfrak{n})$ is a complete intersection
map, then by Remark 2.3, there is a Cohen factorization

${R}$${S}$${R^{\prime}}$${\widehat{S}^{\mathfrak{n}}}$$\scriptstyle{f}$$\scriptstyle{\dot{f}}$$\scriptstyle{f^{\prime}}$

of $f$ in which $\operatorname{Ker}(f^{\prime})$ is generated by a regular
sequence on $R^{\prime}$. This implies that
$\operatorname{fd}_{R^{\prime}}\left(\widehat{S}^{\mathfrak{n}}\right)<\infty$,
so [AFHe, 3.3] yields $\operatorname{fd}_{R}(S)<\infty$. As a result, Theorem
3.5 can in particular be applied to a complete intersection map to show that a
certain bound on the Gorenstein flat dimension of a module along a complete
intersection map can identify it as being exceptional.

######  Remark 3.9.

Given a flat local homomorphism
$\dot{f}:(R,\mathfrak{m})\rightarrow(R^{\prime},\mathfrak{m}^{\prime})$ of
noetherian local rings with regular closed fiber
$R^{\prime}/\mathfrak{m}R^{\prime}$, we ask the following question:

If $R$ is G-regular, then is $R^{\prime}$ G-regular?

This is a special case of a question of Takahashi; see [Ta, Question 6.1]. If
the answer to this question is positive, then one can replace the finite flat
dimension assumption on $f$ in Theorem 3.5 with the assumption that $R$ is
G-regular. Having done so, one can even show that $M$ is a free $S$-module.
The argument goes as follows.

As in the proof of Theorem 3.5, we can reduce to the case where $R$ is
$\mathfrak{m}$-adically complete and $S$ is $\mathfrak{n}$-adically complete.
We just need to further note that $R$ is G-regular if and only if
$\widehat{R}^{\mathfrak{m}}$ is G-regular by [Ta, Corollary 4.7]. Also, we
have

$\operatorname{pd}_{S}(M)=\operatorname{fd}_{S}(M)=\operatorname{fd}_{\widehat{S}^{\mathfrak{n}}}\left(M\otimes_{S}\widehat{S}^{\mathfrak{n}}\right)=\operatorname{pd}_{\widehat{S}^{\mathfrak{n}}}\left(M\otimes_{S}\widehat{S}^{\mathfrak{n}}\right),$

which conspires with the fact that $S$ and $\widehat{S}^{\mathfrak{n}}$ are
local to imply that $M$ is a free $S$-module if and only if
$M\otimes_{S}\widehat{S}^{\mathfrak{n}}$ is a free
$\widehat{S}^{\mathfrak{n}}$-module. Similarly, we can reduce to the case
where $f$ is surjective. We just need to further note that given a Cohen
factorization

$R\xrightarrow{\dot{f}}R^{\prime}\xrightarrow{f^{\prime}}S$

of $f$, $R^{\prime}/\mathfrak{m}R^{\prime}$ is regular, so assuming a positive
answer to the above question, we can conclude that $R^{\prime}$ is G-regular.
Finally, with $f$ being surjective, $M$ becomes a finitely generated
$R$-module as well, so by Remark 2.2,
$\operatorname{Gfd}_{R}(M)=\operatorname{Gdim}_{R}(M)$. On the other hand, $R$
is G-regular, so by Remark 2.1,
$\operatorname{Gdim}_{R}(M)=\operatorname{fd}_{R}(M)$. Now the hypothesis
reads
$\operatorname{fd}_{R}(M)=\operatorname{Gfd}_{R}(M)\leq\operatorname{edim}(R)-\operatorname{edim}(S)$,
so [BIK, Theorem 3.1] implies that $M$ is a free $S$-module, $f$ is an
exceptional complete intersection map, and
$\operatorname{Gfd}_{R}(M)=\operatorname{fd}_{R}(M)=\operatorname{edim}(R)-\operatorname{edim}(S)$.

We next prove two analogues of Theorem 3.5 involving Gorenstein injective
dimension which are Theorems B and C from the introduction. We first need a
lemma which might be of independent interest.

######  Lemma 3.10.

Let $f:(R,\mathfrak{m},k)\rightarrow(S,\mathfrak{n},l)$ be a homomorphism of
noetherian local rings, $M$ an $S$-module, and
$M^{\vee}=\operatorname{Hom}_{S}\left(M,E_{S}(l)\right)$ where $E_{S}(l)$ is
the injective envelope of the $S$-module $l$. Suppose that either $R$ has a
dualizing complex or $M$ is an artinian $S$-module. Then
$\operatorname{Gfd}_{R}(M)=\operatorname{Gid}_{R}\left(M^{\vee}\right)$.

###### Proof.

First suppose that $R$ has a dualizing complex, say $D$. Consider the
Auslander class $\mathcal{A}_{D}(R)$ and the Bass class $\mathcal{B}_{D}(R)$
with respect to $D$; see [CFH1, Definitions 9.1 and 9.4]. Using the Hom-Tensor
Adjunction and the injectivity of $E_{S}(l)$, We have

$\begin{split}H_{-i}\left(\operatorname{RHom}_{R}\left(D,M^{\vee}\right)\right)&=H_{-i}\left(\operatorname{RHom}_{R}\left(D,\operatorname{Hom}_{S}\left(M,E_{S}(l)\right)\right)\right)\\\
&=H_{-i}\left(\operatorname{RHom}_{R}\left(D,\operatorname{RHom}_{S}\left(M,E_{S}(l)\right)\right)\right)\\\
&\cong
H_{-i}\left(\operatorname{RHom}_{S}\left(D\otimes_{R}^{\textrm{L}}M,E_{S}(l)\right)\right)\\\
&=H_{-i}\left(\operatorname{Hom}_{S}\left(D\otimes_{R}^{\textrm{L}}M,E_{S}(l)\right)\right)\\\
&\cong\operatorname{Hom}_{S}\left(H_{i}\left(D\otimes_{R}^{\textrm{L}}M\right),E_{S}(l)\right)\end{split}$

for every $i\in\mathbb{Z}$. As $E_{S}(l)$ is faithfully injective, the above
display shows that for any $i\in\mathbb{Z}$, we have
$H_{-i}\left(\operatorname{RHom}_{R}\left(D,M^{\vee}\right)\right)=0$ if and
only if $H_{i}\left(D\otimes_{R}^{\textrm{L}}M\right)=0$. On the other hand,
the diagram

${D\otimes_{R}^{\textrm{L}}\operatorname{RHom}_{S}\left(D\otimes_{R}^{\textrm{L}}M,E_{S}(l)\right)}$${D\otimes_{R}^{\textrm{L}}\operatorname{RHom}_{R}\left(D,\operatorname{RHom}_{S}\left(M,E_{S}(l)\right)\right)}$${\operatorname{RHom}_{S}\left(\operatorname{RHom}_{R}\left(D,D\otimes_{R}^{\textrm{L}}M\right),E_{S}(l)\right)}$${\operatorname{RHom}_{S}\left(M,E_{S}(l)\right)=M^{\vee}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\delta_{M}}$$\scriptstyle{\delta_{M}}$$\scriptstyle{\operatorname{RHom}_{S}\left(\varepsilon_{M},E_{S}(l)\right)}$

in the derived category $\mathcal{D}(R)$, is commutative where
$\varepsilon_{M}$ and $\delta_{M}$ are the natural morphisms in the
definitions of the Auslander and Bass classes, respectively; see [CFH1,
Definitions 9.1 and 9.4]. The above diagram shows that $\varepsilon_{M}$ is an
isomorphism if and only if $\delta_{M}$ is an isomorphism. As a consequence,
$M\in\mathcal{A}_{D}(R)$ if and only if $M^{\vee}\in\mathcal{B}_{D}(R)$. Now
in light of [CFH1, Theorems 9.5 and 9.2], we see that that
$\operatorname{Gfd}_{R}(M)<\infty$ if and only if
$\operatorname{Gid}_{R}\left(M^{\vee}\right)<\infty$. Hence we can assume that
both quantities are simultaneously finite.

Let $I$ be an injective $R$-module, and $L$ a free resolution of $I$. Then we
get

$\begin{split}\operatorname{Tor}_{i}^{R}(I,M)^{\vee}&=\operatorname{Hom}_{S}\left(H_{i}\left(L\otimes_{R}M\right),E_{S}(l)\right)\\\
&\cong
H_{-i}\left(\operatorname{Hom}_{S}\left(L\otimes_{R}M,E_{S}(l)\right)\right)\\\
&\cong
H_{-i}\left(\operatorname{Hom}_{R}\left(L,\operatorname{Hom}_{S}\left(M,E_{S}(l)\right)\right)\right)\\\
&=\operatorname{Ext}_{R}^{i}\left(I,M^{\vee}\right)\end{split}$

for every $i\in\mathbb{Z}$. Thus for any $i\in\mathbb{Z}$, we have
$\operatorname{Tor}_{i}^{R}(I,M)=0$ if and only if
$\operatorname{Ext}_{R}^{i}\left(I,M^{\vee}\right)=0$. By [Ho, Theorems 2.22
and 3.14], we get:

$\begin{split}\operatorname{Gfd}_{R}(M)&=\sup\left\\{i\geq
0\;|\;\operatorname{Tor}_{i}^{R}(I,M)\neq 0\textrm{ for some injective
}R\textrm{-module }I\right\\}\\\ &=\sup\left\\{i\geq
0\;|\;\operatorname{Ext}_{R}^{i}\left(I,M^{\vee}\right)\neq 0\textrm{ for some
injective }R\textrm{-module }I\right\\}\\\
&=\operatorname{Gid}_{R}\left(M^{\vee}\right)\end{split}$

This proves the result in the first case.

Next assume that $M$ is an artinian $S$-module. Then $M$ is
$\mathfrak{n}$-torsion, so it has an $\widehat{S}^{\mathfrak{n}}$-module
structure that restricts to its original $S$-module structure via the
completion map $S\rightarrow\widehat{S}^{\mathfrak{n}}$. More specifically, if
$x\in M$ and $(a_{n}+\mathfrak{n}^{n})_{n\geq
1}\in\widehat{S}^{\mathfrak{n}}$, then since $M$ is $\mathfrak{n}$-torsion,
there is a $t\geq 1$ such that $\mathfrak{n}^{t}x=0$, so
$(a_{n}+\mathfrak{n}^{n})_{n\geq 1}x:=a_{t}x$ defines the desired
$\widehat{S}^{\mathfrak{n}}$-module structure on $M$. In addition, one has
$M\cong M\otimes_{S}\widehat{S}^{\mathfrak{n}}$ both as $S$-modules and
$\widehat{S}^{\mathfrak{n}}$-modules; see [SS, Proposition 2.1.15 and
Corollary 2.2.6]. As $f$ is local, $\mathfrak{m}S\subseteq\mathfrak{n}$, so
$M$ is $\mathfrak{m}$-torsion as well, thereby it has an
$\widehat{R}^{\mathfrak{m}}$-module structure which is restricted from its
$\widehat{S}^{\mathfrak{n}}$-module structure via
$\widehat{f}:\widehat{R}^{\mathfrak{m}}\rightarrow\widehat{S}^{\mathfrak{n}}$.
It follows that $M\cong M\otimes_{R}\widehat{R}^{\mathfrak{m}}$ as
$\widehat{S}^{\mathfrak{n}}$-modules. We thus have the following isomorphisms
in the derived category $\mathcal{D}\left(\widehat{R}^{\mathfrak{m}}\right)$:

$\begin{split}\operatorname{Hom}_{\widehat{S}^{\mathfrak{n}}}\left(M,E_{\widehat{S}^{\mathfrak{n}}}\left(\widehat{S}^{\mathfrak{n}}/\mathfrak{n}\widehat{S}^{\mathfrak{n}}\right)\right)&\simeq\operatorname{RHom}_{\widehat{S}^{\mathfrak{n}}}\left(M\otimes_{R}^{\textrm{L}}\widehat{R}^{\mathfrak{m}},E_{\widehat{S}^{\mathfrak{n}}}\left(\widehat{S}^{\mathfrak{n}}/\mathfrak{n}\widehat{S}^{\mathfrak{n}}\right)\right)\\\
&\simeq\operatorname{RHom}_{R}\left(\widehat{R}^{\mathfrak{m}},\operatorname{RHom}_{\widehat{S}^{\mathfrak{n}}}\left(M,E_{\widehat{S}^{\mathfrak{n}}}\left(\widehat{S}^{\mathfrak{n}}/\mathfrak{n}\widehat{S}^{\mathfrak{n}}\right)\right)\right)\\\
&\simeq\operatorname{RHom}_{R}\left(\widehat{R}^{\mathfrak{m}},\operatorname{RHom}_{\widehat{S}^{\mathfrak{n}}}\left(M\otimes_{S}^{\textrm{L}}\widehat{S}^{\mathfrak{n}},E_{\widehat{S}^{\mathfrak{n}}}\left(\widehat{S}^{\mathfrak{n}}/\mathfrak{n}\widehat{S}^{\mathfrak{n}}\right)\right)\right)\\\
&\simeq\operatorname{RHom}_{R}\left(\widehat{R}^{\mathfrak{m}},\operatorname{RHom}_{S}\left(M,\operatorname{RHom}_{\widehat{S}^{\mathfrak{n}}}\left(\widehat{S}^{\mathfrak{n}},E_{\widehat{S}^{\mathfrak{n}}}\left(\widehat{S}^{\mathfrak{n}}/\mathfrak{n}\widehat{S}^{\mathfrak{n}}\right)\right)\right)\right)\\\
&\simeq\operatorname{RHom}_{R}\left(\widehat{R}^{\mathfrak{m}},\operatorname{RHom}_{S}\left(M,E_{S}(l)\right)\right)\\\
&=\operatorname{RHom}_{R}\left(\widehat{R}^{\mathfrak{m}},M^{\vee}\right)\end{split}$

Note that $\widehat{R}^{\mathfrak{m}}$ is $\mathfrak{m}$-adically complete, so
it has a dualizing complex. Thus in view of [CFH1, Theorems 4.27 and 9.11],
the previous case, and the above isomorphisms, we achieve:

$\begin{split}\operatorname{Gfd}_{R}(M)&=\operatorname{Gfd}_{\widehat{R}^{\mathfrak{m}}}\left(M\otimes_{R}\widehat{R}^{\mathfrak{m}}\right)\\\
&=\operatorname{Gfd}_{\widehat{R}^{\mathfrak{m}}}\left(M\right)\\\
&=\operatorname{Gid}_{\widehat{R}^{\mathfrak{m}}}\left(\operatorname{Hom}_{\widehat{S}^{\mathfrak{n}}}\left(M,E_{\widehat{S}^{\mathfrak{n}}}\left(\widehat{S}^{\mathfrak{n}}/\mathfrak{n}\widehat{S}^{\mathfrak{n}}\right)\right)\right)\\\
&=\operatorname{Gid}_{\widehat{R}^{\mathfrak{m}}}\left(\operatorname{RHom}_{R}\left(\widehat{R}^{\mathfrak{m}},M^{\vee}\right)\right)\\\
&=\operatorname{Gid}_{R}\left(M^{\vee}\right)\end{split}$

This proves the result in the second case. ∎

######  Theorem 3.11.

Let $f:(R,\mathfrak{m},k)\rightarrow(S,\mathfrak{n},l)$ be a local
homomorphism of noetherian local rings with $\operatorname{fd}_{R}(S)<\infty$
in which $R$ has a dualizing complex and $S$ is $\mathfrak{n}$-adically
complete, and $M$ a non-zero artinian $S$-module with
$\operatorname{Gid}_{R}(M)\leq\operatorname{edim}(R)-\operatorname{edim}(S)$.
Then the following assertions hold:

  1. (i)

$M$ is a Gorenstein injective $S$-module.

  2. (ii)

$f$ is an exceptional complete intersection map.

  3. (iii)

$\operatorname{Gid}_{R}(M)=\operatorname{edim}(R)-\operatorname{edim}(S)$.

In particular, if $M$ is Gorenstein injective as an $R$-module, then
$\operatorname{edim}(R)=\operatorname{edim}(S)$.

###### Proof.

Write $(-)^{\vee}=\operatorname{Hom}_{S}\left(-,E_{S}(l)\right)$. Matlis
duality theory implies that $M$ is a Matlis reflexive $S$-modules, so $M\cong
M^{\vee\vee}$, and $M^{\vee}$ is a non-zero finitely generated $S$-module. By
Lemma 3.10, we have

$\operatorname{Gfd}_{R}\left(M^{\vee}\right)=\operatorname{Gid}_{R}\left(M^{\vee\vee}\right)=\operatorname{Gid}_{R}(M)\leq\operatorname{edim}(R)-\operatorname{edim}(S).$

Therefore, Theorem 3.5 implies that $M^{\vee}$ is a totally reflexive
$S$-module, $f$ is an exceptional complete intersection map, and

$\operatorname{Gid}_{R}(M)=\operatorname{Gid}_{R}\left(M^{\vee\vee}\right)=\operatorname{Gfd}_{R}\left(M^{\vee}\right)=\operatorname{edim}(R)-\operatorname{edim}(S).$

But $S$ is $\mathfrak{n}$-adically complete, so it has a dualizing complex,
thereby another application of Lemma 3.10 yields

$\operatorname{Gid}_{S}(M)=\operatorname{Gid}_{S}\left(M^{\vee\vee}\right)=\operatorname{Gfd}_{S}\left(M^{\vee}\right)=\operatorname{Gdim}_{S}\left(M^{\vee}\right)=0,$

so $M$ is a Gorenstein injective $S$-module. ∎

######  Lemma 3.12.

Let $(R,\mathfrak{m},k)$ be a noetherian local ring, and $M$ a non-zero
finitely generated or artinian $R$-module. Let $E_{R}(k)$ denote the injective
envelope of $k$, and
$\operatorname{type}_{R}(M)=\operatorname{rank}_{k}\left(\operatorname{Ext}_{R}^{0pt_{R}(\mathfrak{m},M)}(k,M)\right)$.
Then $M$ is an injective $R$-module if and only if $M\cong
E_{R}(k)^{\operatorname{type}_{R}(M)}$.

###### Proof.

See [Fa, Lemma 3.1] ∎

######  Theorem 3.13.

Let $f:(R,\mathfrak{m},k)\rightarrow(S,\mathfrak{n},l)$ be a local
homomorphism of noetherian local rings with $\operatorname{fd}_{R}(S)<\infty$
in which $R$ has a dualizing complex and $S$ is G-regular, and $M$ a non-zero
artinian $S$-module with
$\operatorname{Gid}_{R}(M)\leq\operatorname{edim}(R)-\operatorname{edim}(S)$.
Then the following assertions hold:

  1. (i)

$M$ is an injective $S$-module.

  2. (ii)

$f$ is an exceptional complete intersection map.

  3. (iii)

$\operatorname{Gid}_{R}(M)=\operatorname{edim}(R)-\operatorname{edim}(S)$.

In particular, if $M$ is Gorenstein injective as an $R$-module, then
$\operatorname{edim}(R)=\operatorname{edim}(S)$.

###### Proof.

We reduce to the case where $S$ is $\mathfrak{n}$-adically complete. Firstly,
$S$ is G-regular if and only if $\widehat{S}^{\mathfrak{n}}$ is G-regular; see
[Ta, Corollary 4.7]. We next note that $M$ is an artinian $S$-module, so as we
observed in the proof of Lemma 3.10, it has an
$\widehat{S}^{\mathfrak{n}}$-module structure that restricts to its original
$S$-module structure via the completion map
$S\rightarrow\widehat{S}^{\mathfrak{n}}$, and $M\cong
M\otimes_{S}\widehat{S}^{\mathfrak{n}}$ both as $S$-modules and
$\widehat{S}^{\mathfrak{n}}$-modules. Also, $M$ is an artinian
$\widehat{S}^{\mathfrak{n}}$-module. We further note that
$\operatorname{edim}\left(\widehat{S}^{\mathfrak{n}}\right)=\operatorname{edim}(S)$
and
$E_{\widehat{S}^{\mathfrak{n}}}\left(\widehat{S}^{\mathfrak{n}}/\mathfrak{n}\widehat{S}^{\mathfrak{n}}\right)\cong
E_{S}(S/\mathfrak{n})\cong E_{S}(l)$. Since $M$ is an artinian $S$-module,
$\operatorname{Ass}_{S}(M)=\\{\mathfrak{n}\\}$, so
$0pt_{S}(\mathfrak{n},M)=0$, whence
$\operatorname{type}_{S}(M)=\operatorname{rank}_{l}\left(\operatorname{Soc}_{S}(M)\right)$.
The same holds when we consider $M$ as an
$\widehat{S}^{\mathfrak{n}}$-modules. But socle is the sum of simple
submodules, and the $S$-submodules and $\widehat{S}^{\mathfrak{n}}$-submodules
of $M$ coincide, so we conclude that
$\operatorname{Soc}_{\widehat{S}^{\mathfrak{n}}}(M)=\operatorname{Soc}_{S}(M)$,
thereby
$\operatorname{type}_{\widehat{S}^{\mathfrak{n}}}(M)=\operatorname{type}_{S}(M)$.
By Lemma 3.12, $M$ is an injective $S$-module if and only if $M\cong
E_{S}(l)^{\operatorname{type}_{S}(M)}$. Similarly, $M$ is an injective
$\widehat{S}^{\mathfrak{n}}$-module if and only if $M\cong
E_{\widehat{S}^{\mathfrak{n}}}\left(\widehat{S}^{\mathfrak{n}}/\mathfrak{n}\widehat{S}^{\mathfrak{n}}\right)^{\operatorname{type}_{\widehat{S}^{\mathfrak{n}}}(M)}$.
It follows that $M$ is an injective $S$-module if and only if $M$ is an
injective $\widehat{S}^{\mathfrak{n}}$-module. Finally, by [AF1, Corollary 4.2
(b)(F)], we have
$\operatorname{fd}_{R}\left(\widehat{S}^{\mathfrak{n}}\right)\leq\operatorname{fd}_{R}(S)<\infty$.
Therefore, we can replace $S$ by $\widehat{S}^{\mathfrak{n}}$ and assume that
$S$ is $\mathfrak{n}$-adically complete.

Write $(-)^{\vee}=\operatorname{Hom}_{S}\left(-,E_{S}(l)\right)$. Matlis
duality theory implies that $M$ is a Matlis reflexive $S$-modules, so $M\cong
M^{\vee\vee}$, and $M^{\vee}$ is a non-zero finitely generated $S$-module. By
Lemma 3.10, we have

$\operatorname{Gfd}_{R}\left(M^{\vee}\right)=\operatorname{Gid}_{R}\left(M^{\vee\vee}\right)=\operatorname{Gid}_{R}(M)\leq\operatorname{edim}(R)-\operatorname{edim}(S).$

Therefore, Theorem 3.5 implies that $M^{\vee}$ is a totally reflexive
$S$-module, $f$ is an exceptional complete intersection map, and

$\operatorname{Gid}_{R}(M)=\operatorname{Gid}_{R}\left(M^{\vee\vee}\right)=\operatorname{Gfd}_{R}\left(M^{\vee}\right)=\operatorname{edim}(R)-\operatorname{edim}(S).$

But $S$ is G-regular, so $M^{\vee}$ is a free $S$-module, say $M^{\vee}\cong
S^{n}$ for some $n\geq 1$. Hence $M\cong M^{\vee\vee}\cong E_{S}(l)^{n}$ is an
injective $S$-module. ∎

######  Remark 3.14.

One should note that Theorems 3.11 and 3.13 can be specialized to a complete
intersection map as in Remark 3.8.

## Acknowledgement

It is a pleasure to express my sincerest gratitude to professors Srikanth
Iyengar and Lars Christensen for their invaluable comments and suggestions on
this manuscript which is part of my Ph.D. thesis at Clemson University.

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# Swirling against the forcing: Evidence of stable counter-directed sloshing
waves in orbital-shaken reservoirs
Supplementary Material

Alice Marcotte, François Gallaire<EMAIL_ADDRESS>Alessandro
Bongarzone  Laboratory of Fluid Mechanics and Instabilities, École
Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland

###### Abstract

In this supplementary document, we provide further details about the
derivation of the asymptotic model, the estimation of the heuristic damping
coefficient and the numerical computation of the stable wave amplitudes.

††preprint: APS/123-QED

##  I Governing equations

In the moving reference frame, any planar elliptic-like shaking can be
represented by the following equations describing the motion acceleration of
the container axis intersection with the $z=0$ plane (located at the
unperturbed free surface), parametrized in polar coordinates ($r$, $\theta$),

$\frac{\text{d}^{2}\mathbf{X}_{0}}{\text{d}t^{2}}=\begin{cases}\left(-f_{x}\cos{\left(\Omega
t\right)}\cos\theta-f_{y}\sin{\left(\Omega
t\right)}\sin\theta\right)\,\mathbf{e}_{r},\\\
\left(\,\,\,\,f_{x}\cos{\left(\Omega t\right)}\sin\theta-
f_{y}\sin{\left(\Omega
t\right)}\sin\theta\right)\,\mathbf{e}_{\theta},\end{cases}$ (1)

with $f_{x}=\overline{a}_{x}\Omega^{2}/R$ and
$f_{y}=\overline{a}_{y}\Omega^{2}/R$ are the non-dimensional major- and minor-
axis driving acceleration components, respectively, and
$\Omega=\overline{\Omega}/\sqrt{g/R}$ the non-dimensional driving angular
frequency. The bar symbol refers to the dimensional quantities.
In the potential flow limit, i.e. the flow is assumed inviscid, irrotational
and incompressible, the liquid motion is governed by the Laplace equation,
subjected to the homogeneous no-penetration condition at the solid lateral
wall, $r=R$, and bottom $z=-h$,

$\Delta\Phi=0,\ \ \ \ \nabla\Phi\cdot\mathbf{n}=\mathbf{0},$ (2)

and by the kinematic and dynamic boundary conditions at the free surface
$z=\eta\left(r,\theta,t\right)$ [1, 2],

$\frac{\partial\eta}{\partial
t}+\nabla\Phi\cdot\nabla\eta-\frac{\partial\Phi}{\partial z}=0.$ (3a)
$\displaystyle\frac{\partial\Phi}{\partial
t}+\frac{1}{2}\nabla\Phi\cdot\nabla\Phi+\eta=$ (3b)
$\displaystyle=r\left(f_{x}\cos{\left(\Omega
t\right)}\cos{\theta}+f_{y}\sin{\left(\Omega t\right)}\sin{\theta}\right),$

made non-dimensional using the characteristic length $R$ and velocity
$\sqrt{gR}$. $\Phi\left(r,\theta,z,t\right)$ and $\eta\left(r,\theta,t\right)$
denote potential velocity field and free surface elevation, respectively. Note
that, as in [3] and [4], surface tension effects have been neglected. By
introducing the orbit aspect ratio
$\alpha=\overline{a}_{y}/\overline{a}_{x}=f_{y}/f_{x}$, so that $f_{x}=f$ and
$f_{y}=\alpha f$, equation (3b) can be conveniently rewritten as

$\displaystyle\frac{\partial\Phi}{\partial
t}+\frac{1}{2}\nabla\Phi\cdot\nabla\Phi+\eta=$ (4) $\displaystyle
r\frac{f}{2}\left(\alpha_{{}_{A}}e^{\text{i}\left(\Omega
t-\theta\right)}+\alpha_{{}_{B}}e^{\text{i}\left(\Omega
t+\theta\right)}\right)+c.c.\,,$

with $c.c.$ denoting the complex conjugate and with
$\alpha_{{}_{A}}=\left(1+\alpha\right)/2$ and
$\alpha_{{}_{B}}=\left(1-\alpha\right)/2$ two auxiliary orbit parameter (see
Appendix B of [4]).

##  II Multiple timescale weakly nonlinear model

The weakly nonlinear multiple timescale analysis formalized in §4 of [4] is
based on the following asymptotic expansion for the flow quantities,

$\mathbf{q}\left(r,\theta,z,t\right)=\left\\{\Phi,\eta\right\\}^{T}=\epsilon\mathbf{q}_{1}+\epsilon^{2}\mathbf{q}_{2}+\epsilon^{3}\mathbf{q}_{3}+\text{O}\left(\epsilon^{4}\right),$
(5)

and on the assumption of a small forcing amplitude of order $f=\epsilon^{3}F$.
We also allow for a small frequency detuning with respect to the first
system’s natural frequency, $\omega_{0}$, such that
$\Omega=\omega_{0}+\lambda$, with $\lambda=\epsilon^{2}\Lambda$ and $\epsilon$
a small parameter $\ll 1$. The new auxiliary parameters $F$ and $\Lambda$ are
assumed of order $\text{O}\left(1\right)$.
We note that the forcing amplitude could be assumed of order $\epsilon^{2}$
(as the other parameters); however, this complicates unnecessarily the second-
order problem without modifying the final amplitude equation, even if the
values of its coefficients end up being slightly (up to order $\epsilon$)
different.
As suggested by the azimuthal periodicity of the forcing term on the right-
hand-side of (3b), $m=\pm 1$ (with $m$ a so-called azimuthal wavenumber), we
assume a leading order solution as the sum of two counter-propagating
traveling waves

$\displaystyle\mathbf{q}_{1}\left(r,\theta,z,t\right)=A_{1}\left(T_{2}\right)\hat{\mathbf{q}}_{1}^{A_{1}}\left(r,z\right)e^{\text{i}\left(\omega_{0}t-\theta\right)}\
\ \ \ \ \ \ \ \ $ (6)
$\displaystyle+B_{1}\left(T_{2}\right)\hat{\mathbf{q}}_{1}^{B_{1}}\left(r,z\right)e^{\text{i}\left(\omega_{0}t+\theta\right)}+c.c.\,.$

As typical of multiple timescale analyses [5], the complex amplitudes $A_{1}$
and $B_{1}$, functions of the slow time scale $T_{2}=\epsilon^{2}t$ and still
undetermined at this stage of the expansion, describe the slow time amplitude
modulation of the two oscillating waves and must be determined at a higher
order of the asymptotic expansion.
The natural frequency $\omega_{0}$ and structure
$\hat{\mathbf{q}}_{1}^{A_{1}}$ (and $\hat{\mathbf{q}}_{1}^{B_{1}}$) assume the
meaning of eigenvalue and associated eigenmode of the leading order linearized
sloshing operator, whose matrix compact form can be written as
$\left(\text{i}\omega_{0}\mathcal{B}-\mathcal{A}_{m=\pm
1}\right)\hat{\mathbf{q}}_{1}^{A_{1},\,B_{1}}=0$ (see [4]). Despite the fact
that an exact analytical solution to this generalized eigenvalue problem can
be obtained via a Bessel-function representation, in this work, as in [6], we
have opted for a numerical scheme based on a discretization technique, where
operators $B$ and $A_{m}$ are discretized in space by means of a Gauss-
Lobatto-Chebyshev pseudo-spectral collocation method with a two-dimensional
mapping implemented in Matlab, which is analogous to that described in [7] and
[8].
By pursuing the expansion to the second order in $\epsilon$, one obtains a
linear system forced by second order non-linear terms produced by combinations
of the two leading order waves through, e.g.,
$\nabla\Phi_{1}\cdot\nabla\Phi_{1}/2$ in the dynamic condition and
$\nabla\Phi_{1}\cdot\nabla\eta_{1}$ in the kinematic equation. These forcing
terms, $\bm{\mathcal{F}}_{2}$, are proportional to $A_{1}^{2}$ and $B_{1}^{2}$
(second harmonics), to $|A_{1}|^{2}$ and $|B_{1}|^{2}$ (steady and
axisymmetric mean flow corrections) and to $A_{1}B_{1}$ and
$A_{1}\overline{B}_{1}$ (cross-quadratic interactions), and therefore they
call for a second-order solution in the form

$\displaystyle\mathbf{q}_{2}=|A_{1}|^{2}\hat{\mathbf{q}}_{2}^{A_{1}\bar{A}_{1}}+|B_{1}|^{2}\hat{\mathbf{q}}_{2}^{B_{1}\bar{B}_{1}}$
(7)
$\displaystyle+\left(A_{1}^{2}\hat{\mathbf{q}}_{2}^{A_{1}A_{1}}e^{\text{i}2\left(\omega_{0}t-\theta\right)}+B_{1}^{2}\hat{\mathbf{q}}_{2}^{B_{1}B_{1}}e^{\text{i}2\left(\omega_{0}t+\theta\right)}+c.c.\right)$
$\displaystyle+\left(A_{1}B_{1}\hat{\mathbf{q}}_{2}^{A_{1}B_{1}}e^{\text{i}2\omega_{0}t}+A_{1}\overline{B}_{1}\hat{\mathbf{q}}_{2}^{A_{1}\overline{B}_{1}}e^{-\text{i}2\theta}+c.c.\right).\
$

None of the associated forcing terms being resonant, each spatial structure,
function of the $\left(r,z\right)$-space, can be computed numerically as
described in [6] by simply inverting a linear operator. The resulting
structures are shown in Fig. 1 in terms of free surface deformations.

Figure 1: (a) First-order and (b)-(e) second-order free surface deformations.
Right, interface as a function of the radial coordinate only and at
$\theta=0$, e.g. $\hat{\eta}_{1}^{A1,\,B_{1}}\left(r\right)$. Left, top-view
of the full surface deformations, rebuilt according to the corresponding
azimuthal periodicity and shown for $t=0$, i.e.
$\hat{\eta}_{1}^{A_{1},\,B_{1}}\left(r\right)\cos{m\theta}$. The first-order
solution is normalized with the amplitude and phase of the contact line
elevation (at $r=1$), such that the free surface $\eta_{1}^{A_{1},\,B_{1}}$ is
purely real, whereas the potential velocity field
$\hat{\Phi}_{1}^{A_{1},\,B_{1}}$ is purely imaginary. Note that, owing to the
symmetries of the problem, the system admits the following invariant
transformation:
$\left(\hat{\mathbf{q}},+m,\text{i}\omega_{0}\right)\longrightarrow\left(\hat{\mathbf{q}},-m,\text{i}\omega_{0}\right)$,
so that $\hat{\eta}_{1}^{A_{1}}=\hat{\eta}_{1}^{B_{1}}$,
$\hat{\eta}_{2}^{|A_{1}|^{2}}=\hat{\eta}_{2}^{|B_{1}|^{2}}$ and
$\hat{\eta}_{2}^{A_{1}^{2}}=\hat{\eta}_{2}^{B_{1}^{2}}$. In other words, only
part of the first and second order responses need to be computed explicitly.

We now move forward to the $\epsilon^{3}$–order problem, which is once again a
linear problem forced by combinations of the first (6) and second order (7)
solutions, produced by third order non-linearities through, e.g.,
$\left(\nabla\Phi_{1}\cdot\nabla\Phi_{2}+\nabla\Phi_{2}\cdot\nabla\Phi_{1}\right)/2$
in the dynamic condition or
$\nabla\Phi_{1}\cdot\nabla\eta_{2}+\nabla\Phi_{2}\cdot\nabla\eta_{1}$ in the
kinematic equation, as well as by the slow time-$T_{2}$ derivative of the
leading order solution and by the external forcing, which was assumed of order
$\epsilon^{3}$:

$\displaystyle\left(\partial_{t}\mathbf{B}-{\color[rgb]{0,0,0}\mathbf{A}_{m}}\right)\mathbf{q}_{3}=\bm{\mathcal{F}}_{3}=\
\ \ \ $ (8) $\displaystyle=-\frac{\partial A_{1}}{\partial
T_{2}}\mathbf{B}\hat{\mathbf{q}}_{1}^{A_{1}}e^{\text{i}\left(\omega_{0}t-\theta\right)}-\frac{\partial
B_{1}}{\partial
T_{2}}\mathbf{B}\hat{\mathbf{q}}_{1}^{B_{1}}e^{\text{i}\left(\omega_{0}t+\theta\right)}$
$\displaystyle+|A_{1}|^{2}A_{1}\bm{\hat{\mathcal{F}}}_{3}^{|A_{1}|^{2}A_{1}}e^{\text{i}\left(\omega_{0}t-\theta\right)}+|B_{1}|^{2}B_{1}\bm{\hat{\mathcal{F}}}_{3}^{|B_{1}|^{2}B_{1}}e^{\text{i}\left(\omega_{0}t+\theta\right)}$
$\displaystyle+|B_{1}|^{2}A_{1}\bm{\hat{\mathcal{F}}}_{3}^{|B_{1}|^{2}A_{1}}e^{\text{i}\left(\omega_{0}t-\theta\right)}+|A_{1}|^{2}B_{1}\bm{\hat{\mathcal{F}}}_{3}^{|A_{1}|^{2}B_{1}}e^{\text{i}\left(\omega_{0}t+\theta\right)}$
$\displaystyle+\alpha_{{}_{A}}F\bm{\hat{\mathcal{F}}}_{3}^{F}e^{\text{i}\left(\omega_{0}t-\theta\right)}e^{\text{i}\Lambda
T_{2}}+\alpha_{{}_{B}}F\bm{\hat{\mathcal{F}}}_{3}^{F}e^{\text{i}\left(\omega_{0}t+\theta\right)}e^{\text{i}\Lambda
T_{2}}$ $\displaystyle+\text{N.R.T.}+c.c.,$

with $\bm{\hat{\mathcal{F}}}_{3}^{F}=\left\\{0,r/2\right\\}^{T}$ and where
N.R.T. stands for non-resonating terms. The latter terms are not strictly
relevant for further analysis and can therefore be neglected. Amplitudes
equations for $A_{1}$ and $B_{1}$ are obtained by requiring that secular terms
do not appear in the solution to equation (8), where secularity results from
all resonant forcing terms in $\bm{\mathcal{F}}_{3}$ (see Appendix D of [6]
for its explicit expression), i.e. all terms sharing the same frequency and
wavenumber of $\mathbf{q}_{1}$, e.g. $\left(\omega_{0},m=\pm 1\right)$, and in
effect all terms explicitly written in (8). It follows that a compatibility
condition must be enforced through the Fredholm alternative [9], which imposes
the amplitudes $A=\epsilon A_{1}e^{-\text{i}\lambda t}$ and $B=\epsilon
B_{1}e^{-\text{i}\lambda t}$ to obey the following normal form

$\frac{\text{d}A}{\text{d}t}=-\text{i}\lambda A+\text{i}\,\alpha_{{}_{A}}\mu
f+\text{i}\,\nu|A|^{2}A+\text{i}\,\xi|B|^{2}A,$ (9a)
$\frac{\text{d}B}{\text{d}t}=-\text{i}\lambda B+\text{i}\,\alpha_{{}_{B}}\mu
f+\text{i}\,\nu|B|^{2}B+\text{i}\,\xi|A|^{2}B,$ (9b)

where the physical time $t=T_{2}/\epsilon^{2}$ has been reintroduced and where
forcing amplitude and detuning parameter are recast in terms of their
corresponding physical values, $f=\epsilon^{3}F$ and
$\lambda=\epsilon^{2}\Lambda=\Omega-\omega_{0}$, so as to eliminate the small
implicit parameter $\epsilon$ [10, 11]. The various normal form coefficients,
which turn out to be real-valued quantities due to the absence of dissipation,
are computed as scalar products between the adjoint mode,
$\left(\hat{\mathbf{q}}_{1}^{A_{1}\dagger},\hat{\mathbf{q}}_{1}^{B_{1}\dagger}\right)$,
associated with
$\left(\hat{\mathbf{q}}_{1}^{A_{1}},\hat{\mathbf{q}}_{1}^{B_{1}}\right)$, and
the third order resonant forcing terms:

$\displaystyle\text{i}\,\mathcal{I}_{{}_{SC}}\,\mu=<\hat{\mathbf{q}}_{1}^{A_{1}\dagger},\bm{\hat{\mathcal{F}}}_{3}^{F}>=\int_{0}^{1}\left(r/2\right)\overline{\hat{\eta}}_{1}^{A_{1}\dagger}\,r\text{d}r,$
(10a)
$\displaystyle\text{i}\,\mathcal{I}_{{}_{SC}}\,\nu=<\hat{\mathbf{q}}_{1}^{A_{1}\dagger},\bm{\hat{\mathcal{F}}}_{3}^{|A_{1}|^{2}A_{1}}>=$
(10b)
$\displaystyle=\int_{0}^{1}\left(\overline{\hat{\eta}}_{1}^{A_{1}\dagger}\hat{\mathcal{F}}_{3_{\text{dyn}}}^{|A_{1}|^{2}A_{1}}+\overline{\hat{\Phi}}_{1}^{A_{1}\dagger}\hat{\mathcal{F}}_{3_{\text{kin}}}^{|A_{1}|^{2}A_{1}}\right)\,r\text{d}r,$
$\displaystyle\text{i}\,\mathcal{I}_{{}_{SC}}\,\xi=<\hat{\mathbf{q}}_{1}^{A_{1}\dagger},\bm{\hat{\mathcal{F}}}_{3}^{|B_{1}|^{2}A_{1}}>=$
(10c)
$\displaystyle=\int_{0}^{1}\left(\overline{\hat{\eta}}_{1}^{A_{1}\dagger}\hat{\mathcal{F}}_{3_{\text{dyn}}}^{|B_{1}|^{2}A_{1}}+\overline{\hat{\Phi}}_{1}^{A_{1}\dagger}\hat{\mathcal{F}}_{3_{\text{kin}}}^{|B_{1}|^{2}A_{1}}\right)\,r\text{d}r.$

where

$\displaystyle\mathcal{I}_{{}_{SC}}=<\hat{\mathbf{q}}_{1}^{A_{1}\dagger},\mathbf{B}\hat{\mathbf{q}}_{1}^{A_{1}}>=$
(11)
$\displaystyle=\int_{0}^{1}\left(\overline{\hat{\eta}}_{1}^{A_{1}\dagger}\hat{\Phi}_{1}^{A_{1}}+\overline{\hat{\Phi}}_{1}^{A_{1}\dagger}\hat{\eta}_{1}^{A_{1}}\right)\,r\text{d}r.$

Here
$\left(\hat{\mathbf{q}}_{1}^{A_{1}\dagger},\hat{\mathbf{q}}_{1}^{B_{1}\dagger}\right)=\left(\overline{\hat{\mathbf{q}}}_{1}^{A_{1}},\overline{\hat{\mathbf{q}}}_{1}^{B_{1}}\right)$,
since the inviscid problem is self–adjoint with respect to the Hermitian
scalar product
$<\mathbf{a},\mathbf{b}>=\int_{V}\overline{\mathbf{a}}\cdot\mathbf{b}\,\text{d}V$,
with $\mathbf{a}$ and $\mathbf{b}$ two generic vector (see [12] for a thorough
discussion and derivation of the adjoint problem).
For the sake of brevity, we do not report the expression of the various
forcing terms. As an example, the full expression of
$\hat{\mathcal{F}}_{3_{kin}}^{|A_{1}|^{2}A_{1}}$ is given in Appendix D of
[6]. The other forcing terms are calculated analogously.

##  III Phenomenological damping coefficient

In contradistinction with the original system proposed in Appendix B
(B$\,5a$)-(B$\,5b$) of [4], we account here for a heuristic damping
coefficient, $\sigma$, whose value, by analogy with [13], is estimated
according to the well-known expression

$\displaystyle\sigma=\underbrace{\frac{2k^{2}}{Re}}_{\text{bulk}}+\sqrt{\frac{\omega_{0}}{2Re}}\left(\underbrace{\frac{k\cosh^{2}{kH}}{\sinh{2kH}}}_{\text{surf.
contamination}}\right)\ \ \ \ \ \ $ (12)
$\displaystyle+\sqrt{\frac{\omega_{0}}{2Re}}\left(\underbrace{\frac{k}{\sinh{2kH}}}_{\text{bottom}}+\underbrace{\frac{1}{2}\frac{1+\left(1/k\right)^{2}}{1-\left(1/k\right)^{2}}-\frac{kH}{\sinh{2kH}}}_{\textit{sidewall}}\right).$

The damping associated with lowest natural frequency,
$\omega_{0}=\overline{\omega}_{0}/\sqrt{g/R}=\sqrt{k\tanh{\left(kH\right)}}=1.3547$
(with wavenumber $k=1.8412$) [14], in a container of diameter
$D=2R=0.172\,\text{m}$ filled to a depth $H=h/R=1.744$ with distilled water,
i.e. $\rho=1000\,\text{kg/m${}^{3}$}$, $\mu=0.001\,\text{kg/ms}$ and
$\gamma=0.072\,\text{N/m}$, for which $Re=\rho\sqrt{gR^{3}}/\mu=78\,952$
(Reynolds number), amounts to $\sigma=0.0055$.
Typically the viscous damping rate can be interpreter as a slow damping
process [15, 7], i.e. $1/\sigma\approx 180$, over a faster time scale
represented by the wave oscillation, i.e. $1/\omega_{0}\approx 0.5$. When this
hypothesis holds, as in the present experimental study, the damping
coefficient is assumed to be small of order $\epsilon^{2}$, such that damping
terms as $\sigma A$ and $\sigma B$, both of order
$\sim\text{O}\left(\epsilon^{3}\right)$
($A,B\sim\text{O}\left(\epsilon\right)$), can be phenomenologically added a
posteriori to the final inviscid amplitude equations.
Expression (12) englobes different effects, i.e. viscous dissipation occurring
in the Stokes boundary layers (at the solid lateral and bottom wall), bulk
dissipation and possible sources of dissipation associated with free surface
contamination effects [16, 17]. However, it does not account for any form of
dissipation induced by contact angle dynamics [18, 19, 20, 21, 8], by wave
breaking and overtuning [22] or by Prandtl mass-transport phenomena [23, 24].
Moreover, as pointed out in Appendix A of [6], prediction (12) is only valid
for free capillary-gravity waves, whereas the dissipation rates of forced wave
motions is generally more complex, i.e. it is typically a function of the wave
amplitude [22]. A rigorous viscous analysis would indeed produce complex
eigenfunctions and, therefore, complex-valued normal form coefficients [11],
e.g.
$\nu_{{}_{SC}}=\text{Re}\left[\nu_{{}_{SC}}\right]+\text{i}\,\text{Im}\left[\nu_{{}_{SC}}\right]$
(same for $\xi_{{}_{SC}}$), so that the effective damping will read
asymptotically, e.g. for amplitude $A$ (similar for $B$),
$\left(\sigma+\text{Im}\left[\nu_{{}_{SC}}\right]|A|^{2}+\text{Im}\left[\xi_{{}_{SC}}\right]|B|^{2}\right)$.
Hence, we do not expect the heuristic damping model to provide an accurate
estimation of the actual amplitude-dependent dissipation of the system,
crucial for a correct prediction of the phase-lag between forcing and the
system response [25]. However, accounting for a damping coefficient $\sigma$
in equations (13a)-(13b) is essential in order to regularize the WNL
prediction as the orbit aspect ratio $\alpha$ approaches $1$, i.g. for
circular orbits, which otherwise would unrealistically predict counter-waves
even for $\alpha\rightarrow 1$ [3, 22], thus implying that the response curves
branching is not a continuous function of $\alpha$, which is in contradiction
with experimental evidences [26].

##  IV Lowest order asymptotic solution

In conclusion, the lowest order asymptotic solution governing the close-to-
resonance interaction of the two $m=\pm 1$ counter-propagating waves is ruled
by the following system of complex amplitude equations

$\frac{\text{d}A}{\text{d}t}=-\left(\sigma+\text{i}\lambda\right)A+\text{i}\,\mu\alpha_{{}_{A}}f+\text{i}\,\nu|A|^{2}A+\text{i}\,\xi|B|^{2}A,$
(13a)
$\frac{\text{d}B}{\text{d}t}=-\left(\sigma+\text{i}\lambda\right)B+\text{i}\,\mu\alpha_{{}_{B}}f+\text{i}\,\nu|B|^{2}B+\text{i}\,\xi|A|^{2}B.$
(13b)

It follows that the leading order free surface deformation writes

$\displaystyle\eta\left(r,\theta,t\right)=\hat{\eta}_{1}^{A_{1},\,B_{1}}\left(r\right)\left(Ae^{\text{i}\left(\Omega
t-\theta+\Phi_{A}\right)}+Be^{\text{i}\left(\Omega
t+\theta+\Phi_{B}\right)}\right)$ $\displaystyle+c.c.\,.\ \ \ \ $ (14)

Given the mode normalization, i.e.
$\hat{\eta}_{1}^{A_{1},\,B_{1}}\left(r=1\right)=1$, we can express the
dimensionless contact line elevation, $\delta\left(\theta,t\right)/R$, at any
azimuthal coordinate, e.g. at $\theta=0$, as

$\delta\left(0,t\right)/R=\left(A+B\right)e^{\text{i}\Omega t}+c.c.\,$ (15)

(with $R$ the container’s radius), which is the quantity that has been used
for comparison with our experimental measurements.

##  V Stationary wave amplitude solutions and their stability

$H=h/R$ | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 | 1.7 | 1.744 | 1.8 | 1.9 | 2.0
---|---|---|---|---|---|---|---|---|---|---|---
$\mu$ | -0.279 | -0.280 | -0.281 | -0.282 | -0.283 | -0.283 | -0.283 | -0.283 | -0.284 | -0.284 | -0.284
$\nu$ | 1.414 | 1.407 | 1.406 | 1.407 | 1.409 | 1.410 | 1.411 | 1.412 | 1.412 | 1.412 | 1.413
$\xi$ | -7.487 | -7.914 | -8.101 | -8.211 | -8.281 | -8.328 | -8.359 | -8.369 | -8.381 | -8.395 | -8.405
$\omega_{0}$ | 1.334 | 1.341 | 1.346 | 1.349 | 1.352 | 1.353 | 1.354 | 1.355 | 1.355 | 1.356 | 1.356
Table 1: Value of the normal form coefficients appearing in (13a)-(13b)
computed at different non-dimensional fluid depths $H=h/R$ (as reported in
table 1 of [4]) and associated with the lowest natural frequency mode. The
subscript $SC$ was used in [6] and [4] to indicate the shape of the associated
free surface response close to harmonic resonance, initially denominated
single-crest (SC) by [26]. Here the subscript SC has been omitted but in
practice, $\mu$, $\nu$ and $\xi$ coincide with $\mu_{{}_{SC}}$,
$\nu_{{}_{SC}}$ and $\xi_{{}_{SC}}$ in [4]. For completeness we also report
the value of the system’s lowest natural frequency $\omega_{0}$. The bold
values correspond to those used in the main document for comparison with
experiments.

By turning (13a)-(13b) into polar coordinates, i.e.
$A=|A|e^{\text{i}\Phi_{A}}$ and $B=|B|e^{\text{i}\Phi_{B}}$, we can split real
and imaginary parts, hence obtaining

$\frac{\text{d}|A|}{\text{d}t}=-\sigma|A|+\alpha_{{}_{A}}\mu f\sin{\Phi_{A}},$
(16a) $|A|\frac{\text{d}\Phi_{A}}{\text{d}t}=-\lambda|A|+\alpha_{{}_{A}}\mu
f\cos{\Phi_{A}}+\nu|A|^{3}+\xi|B|^{2}|A|,$ (16b)
$\frac{\text{d}|B|}{\text{d}t}=-\sigma|B|+\alpha_{{}_{B}}\mu f\sin{\Phi_{B}},$
(16c) $|B|\frac{\text{d}\Phi_{B}}{\text{d}t}=-\lambda|B|+\alpha_{{}_{B}}\mu
f\cos{\Phi_{B}}+\nu|B|^{3}+\xi|A|^{2}|B|.$ (16d)

Let us then decompose amplitudes and phases as the sum of stationary values
plus time-dependent small perturbations of order $\epsilon\ll 1$.

$\displaystyle\mathbf{y}\left(t\right)=\left(\begin{matrix}|A|\left(t\right)\\\
\Phi_{A}\left(t\right)\\\ |B|\left(t\right)\\\
\Phi_{B}\left(t\right)\end{matrix}\right)=\left(\begin{matrix}a_{0}\\\
\phi_{A,0}\\\ b_{0}\\\
\phi_{B,0}\end{matrix}\right)+\epsilon\left(\begin{matrix}a_{1}\left(t\right)\\\
\phi_{A,1}\left(t\right)\\\ b_{1}\left(t\right)\\\
\phi_{B,1}\left(t\right)\end{matrix}\right)=$
$\displaystyle=\mathbf{y}_{0}+\epsilon\mathbf{y}_{1}\left(t\right)=\mathbf{y}_{0}+\epsilon\left(\hat{\mathbf{y}}_{1}e^{st}+\text{c.c.}\right),\
\ \ $ (17)

with $s=s_{{}_{R}}+\text{i}s_{{}_{I}}\in\mathbb{C}$ an eigenvalue and c.c.
denoting the complex conjugate part of the small linear perturbation. The
ubstitution of (V) in (16a)-(16d) and the linearization around
$\mathbf{y}_{0}$, lead to two problems at order $\epsilon^{0}$ and $\epsilon$,
respectively. As the nonlinear system of equations at order $\epsilon^{0}$
does not admit an analytical solution, we apply a numerical procedure after
rewriting the problem in the form:

$\mathcal{F}=0=\begin{cases}\alpha_{{}_{A}}\mu f\sin{\phi_{A,0}}-\sigma
a_{0},\\\ \alpha_{{}_{A}}\mu f\cos{\phi_{A,0}}-a_{0}\left(\lambda-\nu
a_{0}^{2}-\xi b_{0}^{2}\right),\\\ \alpha_{{}_{B}}\mu f\sin{\phi_{B,0}}-\sigma
b_{0},\\\ \alpha_{{}_{B}}\mu f\cos{\phi_{B,0}}-b_{0}\left(\lambda-\nu
b_{0}^{2}-\xi a_{0}^{2}\right).\end{cases}$ (18)

System (18) is then solved in Matlab function using the built-in function
fsolve and prescribing some initial guesses (ig) for
$\left(a_{0}^{ig},\phi_{A,0}^{ig},b_{0}^{ig},\phi_{B,0}^{ig}\right)$. In
practice, we provide in input the external control parameters, $\Omega$,
$a_{x}=\overline{a}_{x}/R$ and $\alpha$, whereas the associated combination of
stationary amplitudes and phases,
$\left(a_{0},\phi_{A,0},b_{0},\phi_{B,0}\right)$ are computed as outputs.
In the following we study the stability properties of these steady-state
amplitude and phase solutions. Given the ansatz
$\mathbf{y}_{1}\left(t\right)=\hat{\mathbf{y}}_{1}e^{st}+c.c.$, at order
$\epsilon$ the linearized and unsteady system, describing the evolution of
small amplitude perturbations around the stationary states, can be written in
a matrix form as

$s\mathbf{M}\hat{\mathbf{y}}_{1}=\mathbf{K}\hat{\mathbf{y}}_{1},$ (19)

with matrices $\mathbf{M}$ and $\mathbf{K}$ reading

$\mathbf{M}=\left[\begin{matrix}1&0&0&0\\\ 0&a_{0}&0&0\\\ 0&0&1&0\\\
0&0&0&b_{0}\end{matrix}\right],\ \ \
\mathbf{K}=\left[\begin{matrix}K_{11}&K_{12}&0&0\\\ K_{21}&K_{22}&K_{24}&0\\\
0&0&K_{33}&K_{34}\\\ K_{41}&0&K_{43}&K_{44}\end{matrix}\right],$ (20)

$\hat{\mathbf{y}}_{1}=\left(\hat{a}_{1},\hat{\phi}_{A,1},\hat{b}_{1},\hat{\phi}_{B,1}\right)^{T}$
and

$K_{11}=-\sigma,\ K_{33}=-\sigma,$ (21a) $K_{12}=\alpha_{{}_{A}}\mu
f\cos{\phi_{A,0}},\ K_{34}=\alpha_{{}_{B}}\mu f\cos{\phi_{B,0}},$ (21b)
$K_{21}=-\lambda+3\nu a_{0}^{2}+\xi b_{0}^{2},\ K_{43}=-\lambda+3\nu
b_{0}^{2}+\xi a_{0}^{2},$ (21c) $K_{22}=-\alpha_{{}_{A}}\mu
f\sin{\phi_{A,0}},\ K_{44}=-\alpha_{{}_{B}}\mu f\sin{\phi_{B,0}},$ (21d)
$K_{24}=2\xi a_{0}b_{0},\ K_{41}=2\xi a_{0}b_{0},$ (21e)

We proceed as follows. For each
$\left(a_{0},\phi_{A,0},b_{0},\phi_{B,0}\right)$, solution of (18), we obtain
four eigenvalues $s$. If the real part of at least one of these eigenvalues is
positive, then that configuration, associated with the set of external
parameters $\left(\Omega,a_{x},\alpha\right)$, is labeled as unstable.

##  VI Values of the normal form coefficients

In table 1 we report the values of the normal form coefficients, $\mu$, $\nu$
and $\xi$ appearing in (13a)-(13b) as a function of the non-dimensional fluid
depth $H=h/R$. Experiments reported in the main document have been performed
at a fluid depth $H=1.744$.
For completeness we also report the value of the system’s lowest natural
frequency $\omega_{0}$, which satisfies the well-known dispersion relation for
gravity waves
$\omega_{0}=\overline{\omega}_{0}/\sqrt{g/R}=\sqrt{k\tanh{\left(kH\right)}}$
(with $k=1.8412$) [14]. We do not report the value of the damping coefficient
$\sigma$ as a function of $H$, since for $H\geq 1$ the fluid depth does not
affect significantly its value estimated according to (12).

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