# A note on cohomological vanishing theorems

Mohsen Asgharzadeh<EMAIL_ADDRESS>
###### Abstract.

We study $\operatorname{cd}(M,N):=\sup\\{j:H^{j}_{\mathfrak{m}}(M,N)\neq
0\\}$, and we prove the following over $\operatorname{AB}$-rings:
$\operatorname{cd}(M,N)<\infty\Rightarrow\operatorname{cd}(M,N)\leq 2\dim R$.
For locally free over the punctured spectrum, we present the better bound,
namely $\operatorname{cd}(M,N)<\infty\Rightarrow\operatorname{cd}(M,N)\leq\dim
R,$ and show this is sharp for maximal Cohen-Macaulay, and prove that this
detects freeness of $M$. We present some explicit examples to compute
$\operatorname{cd}(M,N)$. Now, suppose $R$ is only Cohen-Macaulay and of prime
characteristic equipped with the Frobenius map $\varphi$. We show for some
$n\gg 0$ that $\operatorname{cd}({{}^{\varphi_{n}}\\!}R,M)<\infty$ iff
$\operatorname{id}_{R}(M)<\infty.$ This presents some criteria on regularity.
Also, some vanishing results on
$\operatorname{Ext}^{i}_{R}({{}^{\varphi}\\!}R,-)$ are given, where
$(-)\in\\{R,{{}^{\varphi}\\!}R\\}$. We determine conditions under which the
vanishing $\operatorname{Ext}^{i}_{R}({{}^{\varphi}\\!}R,-)$ of restricted
many $i$-th, implies the vanishing of all.

###### Key words and phrases:

Complete-intersection; $\operatorname{AB}$-rings; local cohomology; Frobenius
map; Gorenstein rings; vanishing of Ext.

###### 2010 Mathematics Subject Classification:

Primary 13D45.

##  1\. Introduction

The Grothendieck’s vanishing theorem says that $H^{i}_{\mathfrak{m}}(M)=0$ for
all $j>d:=\dim R$ where $(R,\mathfrak{m},k)$ is noetherian and local. As soon
as the ring is not regular we see $H^{\ast}_{\mathfrak{m}}(k,k)\neq 0$. Also,
it is well-known that $H^{j}_{\mathfrak{m}}(M,N)=0$ for all $j>d$ provided
$\operatorname{pd}(M)<\infty$ or $\operatorname{id}(N)<\infty$ or even some
modifications of these, see e.g. [16]. Despite a lot of works in this area,
and up to my knowledge, there is not so data on the following basic problem:

######  Problem 1.1.

What is $\operatorname{cd}(M,N):=\sup\\{j:H^{j}_{\mathfrak{m}}(M,N)\neq 0\\}$?

And one may ask even more; what is the significance of
$\operatorname{cd}(M,N)$? Towards the symmetry in the vanishing of
$\operatorname{Ext}^{i}_{R}(-,+)$, for finitely generated modules over local
Gorenstein rings, Huneke and Jorgensen [17] introduced
“$\operatorname{AB}$-rings”. This class of rings lives between complete-
intersection and Gorenstein rings. We refer the reader to see [17] for the
systematic study of $\operatorname{AB}$-rings. Now, let $R$ be an
$\operatorname{AB}$ ring. In Section 2 we prove the following:

  1. (a)

$\operatorname{cd}(M,N)<\infty\Rightarrow\operatorname{cd}(M,N)\leq 2\dim R$.

  2. (b)

For $M$ be locally free over the punctured spectrum, we have

$\operatorname{cd}(M,N)<\infty\Rightarrow\operatorname{cd}(M,N)\leq\dim R.$

  3. (c)

For $M$ be maximal Cohen-Macaulay and locally free over the punctured
spectrum, we have

$\operatorname{cd}(M,M)<\infty\Longleftrightarrow\emph{M is free},$

when $\dim R>1$. In particular, $\operatorname{cd}(M,M)=\dim(R)$.

  4. (d)

For $M,N$ be maximal Cohen-Macaulay and locally free over the punctured
spectrum, we have $\operatorname{cd}(M,N)=\operatorname{cd}(N,M)$.

The above item (c) may be considered as the higher-dimensional version of
recent work [18, Corollary 4.12] with a connection to the famous conjecture of
Huneke and Wiegand. From now on $\varphi:R\to R$ denotes the _Frobenius
endomorphism_. In the sequel, we will see how that the above computations over
$\operatorname{AB}$-rings motive us to prove the following result over the
much larger class of rings:

  1. (e)

Let $R$ be a Cohen-Macaulay, F-finite and $n\gg 0$. Then

$\operatorname{cd}({{}^{\varphi_{n}}\\!}R,M)<\infty\Longleftrightarrow\operatorname{id}_{R}(M)<\infty.$

Recall that $\operatorname{id}_{R}(-)$ denotes the injective dimension of
$(-)$. Now, let $m>0$ be any integer and $R$ be Cohen-Macaulay. Part (e)
implies that
$\operatorname{cd}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{m}}\\!}R)<\infty$ iff
$R$ is regular. We also present the dual parallel to $(e)$, where we do not
assume F-finiteness, see Proposition 2.11. Here, we do not care about $n\gg
0$, provided the ring is complete-intersection, and even in the particular
case $R$ is normal and locally $\operatorname{AB}$.

To prove these results, we reduce the local cohomology problems to the
vanishing of certain ext-modules. In Section 3 we continue with the later:

######  Problem 1.2.

Let $-\in\\{{{}^{\varphi_{n}}\\!}R,R\\}$ and $i>0$. When is
$\operatorname{Ext}^{i}_{R}({{}^{\varphi_{n}}\\!}R,-)=0$?

The history of this comes back to [15], where Herzog assumed and studied
vanishing of all Ext-modules, see [15]. It seems this is more subtle than the
classical dual results [24, 20]. So, to the end of introduction, we assume the
rings are F-finite. In particular, we deal with the higher (lower) analogue of
the recent work [8, 1.4]. This naturally leads us to study the reflexivity
problem of ${{}^{\varphi_{n}}\\!}R$. We determine conditions under which the
vanishing of few many $i$, or even a single value of $i$,
$\operatorname{Ext}^{i}_{R}({{}^{\varphi_{n}}\\!}R,-)$ implies the vanishing
of all. Here, we summarized some of them:

  1. (i)

Suppose $0pt(R)=0$. Suppose
$\operatorname{Ext}^{1}_{R}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)=0$
for $n\gg 0$, then $R$ is a field.

  2. (ii)

Suppose $R$ is complete intersection and
$\operatorname{Ext}^{1}_{R}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)=0$
for some $n>0$, then $R$ is regular.

  3. (iii)

Suppose $R$ is normal Cohen-Macaulay. Then $R$ is Gorenstein iff
${{}^{\varphi_{n}}\\!}R$ is self-dual and
$\operatorname{Ext}^{i}_{R}({{}^{\varphi_{n}}\\!}R,R)=0$ for
$0<i\in\\{d-1,d\\}$.

  4. (iv)

We study the questions asked in [11, Page 167] and [22, 2.12], we show they
are equivalent and present some results, both in positive and negative sides.

Let us close the introduction, by recalling the relation between
$\operatorname{Ext}^{i}_{R}({{}^{\varphi}\\!}R,\omega_{R})$ and the usual
local cohomology by means of Grothendieck’s local duality.

##  2\. When is $H_{m}^{+}(M,N)=0$?

In this note $(R,\mathfrak{m},k)$ is a commutative noetherian and local ring,
and all modules are finitely generated, otherwise socialized. For two
$R$-modules $M$ and $N$, the $i^{th}$ generalized local cohomology of $M$ and
$N$ is defined by
$H^{i}_{\mathfrak{m}}(M,N):=\underset{n}{\varinjlim}\operatorname{Ext}^{i}_{R}(M/\mathfrak{m}^{n}M,N)$.

######  Proposition 2.1.

Let $R$ be an $\operatorname{AB}$ ring, let $M$ and $N$ be finitely generated
$R$-modules. If $\operatorname{cd}(M,N)<\infty$ then
$\operatorname{cd}(M,N)\leq 2\dim R$.

###### Proof.

We may assume the ring is complete. We look at the short exact sequence

$0\longrightarrow N_{1}:=\operatorname{Syz}_{1}(N)\longrightarrow
F\longrightarrow N\longrightarrow 0$

where $F$ is free. This gives the exact sequence

$H^{i-1}_{\mathfrak{m}}(M,N)\longrightarrow
H^{j}_{\mathfrak{m}}(M,N_{1})\longrightarrow
H^{j}_{\mathfrak{m}}(M,F)\quad(+)$

Recall that $\operatorname{AB}$-rings are Gorenstein, so the injective
dimension of $F$ is finite. From $(+)$ we see
$\operatorname{cd}(M,N_{1})<\infty$. Let $d:=\dim(R)$ and set
$N_{d}:=\operatorname{Syz}_{1}(N)$. By repetition of this, we see
$\operatorname{cd}(M,N_{d})<\infty$. Suppose the desired claim holds with the
additional Cohen-Macaulay assumption of $N$. Since $N_{d}$ is maximal Cohen-
Macaulay we obtain that $\operatorname{cd}(M,N_{d})\leq 2d$. Let $j>2d$, and
look at

$0=H^{j}_{\mathfrak{m}}(M,F)\longrightarrow
H^{j}_{\mathfrak{m}}(M,N_{d-1})\longrightarrow
H^{j+1}_{\mathfrak{m}}(M,N_{d}),$

we obtain that $H^{j}_{\mathfrak{m}}(M,N_{d-1})=0$. By repetition of this, we
see $\operatorname{cd}(M,N)\leq 2d$. So, without loss of generality, we may
and do assume that $N$ is Cohen-Macaulay. This allows us to apply [16, Lemma
3.4] and to deduce for any $i\geq 0$ that

$H^{d+i}_{\mathfrak{m}}(M,N)^{\vee}\cong\operatorname{Tor}^{R}_{i}(M,H^{d}_{\mathfrak{m}}(R,N)^{\vee})\cong\operatorname{Tor}^{R}_{i}(M,N^{\ast}).$

Here $(-)^{\vee}$ is the Matlis functor. So,
$\operatorname{Tor}^{R}_{i}(M,N^{\ast})=0$ for all $i\gg 0$. But, tor-index of
$\operatorname{AB}$-rings is finite, and in fact is equal to dimension of $R$
(see discussion after than [17, Proposition 3.2]). Let

$\Sigma:=\\{(L_{1},L_{2}):\operatorname{Tor}^{R}_{i}(L_{1},L_{2})=0\emph{ for
all }i\gg 0\\}.$

By definition, tor-index is

$\sup\\{n:\operatorname{Tor}^{R}_{n}(L_{1},L_{2})\neq 0\emph{ and
}\operatorname{Tor}^{R}_{>n}(L_{1},L_{2})=0\\}_{(L_{1},L_{2})\in\Sigma}.$

Let $j:=\max\\{d,\operatorname{cd}(M,N)\\}$. There is nothing to prove if
$\operatorname{cd}(M,N)<d$, so we may assume that
$i:=\operatorname{cd}(M,N)-d\geq 0$. In sum, $(M,N^{\ast})\in\Sigma$ and that

$\operatorname{Tor}^{R}_{i}(M,N^{\ast})\neq 0\emph{ and
}\operatorname{Tor}^{R}_{>i}(M,N^{\ast})=0.$

By definition,

$\operatorname{cd}(M,N)-\dim R=i\leq\operatorname{Tor-index}(R)=\dim R,$

and so $\operatorname{cd}(M,N)\leq 2\dim R$. ∎

######  Proposition 2.2.

Let $R$ be an $\operatorname{AB}$ ring, let $M$ and $N$ be finitely generated
$R$-modules so that $M$ is locally free over the punctured spectrum. If
$\operatorname{cd}(M,N)<\infty$ then $\operatorname{cd}(M,N)\leq\dim R$.

###### Proof.

We may assume the ring is complete. Without loss of generality, we may and do
assume that $N$ is Cohen-Macaulay. Suppose $\operatorname{cd}(M,N)\geq
d:=\dim(R)$, and let $i:=\operatorname{cd}(M,N)-d\geq 0$. We show that $i=0$.
Indeed, recall that

$H^{d+i}_{\mathfrak{m}}(M,N)^{\vee}\cong\operatorname{Tor}^{R}_{i}(M,N^{\ast})\cong\operatorname{Ext}_{R}^{d+i}(M,N)^{\vee}\quad(+),$

where the last isomorphism is taken from [19, Lemma 3.5(2)]. But, ext-index of
$\operatorname{AB}$-rings is equal to $\dim R$ (see [17, Proposition 3.2]).
Let

$\Sigma:=\\{(L_{1},L_{2}):\operatorname{Ext}_{R}^{i}(L_{1},L_{2})=0\emph{ for
all }i\gg 0\\}.$

By definition, ext-index is

$\sup\\{n:\operatorname{Ext}_{R}^{n}(L_{1},L_{2})\neq 0\emph{ and
}\operatorname{Ext}_{R}^{>n}(L_{1},L_{2})=0\\}_{(L_{1},L_{2})\in\Sigma}.$

In sum, $(M,N^{\ast})\in\Sigma$ and that

$\operatorname{Ext}_{R}^{d+i}(M,N^{\ast})\neq 0\emph{ and
}\operatorname{Ext}_{R}^{>d+i}(M,N^{\ast})=0.$

By definition,

$d\leq d+i\leq\operatorname{Ext-index}(R)=\dim R,$

and so $i=0$ and consequently $\operatorname{cd}(M,N)\leq\dim R$. ∎

######  Corollary 2.3.

Let $R$ be an $\operatorname{AB}$ ring, let $M$ and $N$ be maximal Cohen-
Macaulay and locally free over the punctured spectrum. Then
$\operatorname{cd}(M,N)<\infty\Longleftrightarrow\operatorname{cd}(N,M)<\infty$.

###### Proof.

Let $d:=\dim(R)$. In view of $(+)$ from Proposition 2.2 it is enough to note
that

$\operatorname{Ext}_{R}^{>d}(M,N)=0\Longleftrightarrow\operatorname{Ext}_{R}^{>d}(N,M)=0.$

This is subject of [17, 4.3]. ∎

######  Example 2.4.

The maximal Cohen-Macaulay of both modules are needed. For example, suppose
$R$ is not regular, then $\operatorname{cd}(R,k)=0<\infty$ but
$\operatorname{cd}(k,R)=\infty$.

######  Corollary 2.5.

Let $R$ be an $\operatorname{AB}$ ring of dimension bigger than one, and $M$
be locally free over the punctured spectrum and maximal Cohen-Macaulay. If
$\operatorname{cd}(M,M)<\infty$ then $M$ is free. In particular,
$\operatorname{cd}(M,M)=\dim(R)$.

###### Proof.

According to Proposition 2.2 we observe that $\operatorname{cd}(M,M)\leq\dim
R$. By the same reasoning $\operatorname{Tor}^{R}_{+}(M,M^{\ast})=0$. In the
light of depth formula over $\operatorname{AB}$-rings (see [9, Corollary
5.3(b)]) we know $0pt(M\otimes_{R}M^{\ast})=\dim R>1$. In particular,
$H^{1}_{\mathfrak{m}}(M\otimes_{R}M^{\ast})=0$. Due to [4, 8.1] we conclude
that $M$ is free, as desired. ***One may drive this from some known cases of
Auslander–Reiten conjecture by showing some $\operatorname{Ext}^{+}_{R}(M,M)$
vanishes. As, we are interested in the previous argument, we left details of
this to the reader. ∎

From now on, $R$ denotes a local ring of prime characteristic $p>0$, and
$\varphi:R\to R$ denotes the _Frobenius endomorphism_ given by
$\varphi(a)=a^{p}$ for $a\in R$. Each iteration $\varphi_{n}$ of $\varphi$
defines a new $R$-module structure on the set $R$, and this $R$-module is
denoted by ${{}^{\varphi_{n}}\\!}R$, where $a\cdot b=a^{p^{n}}b$ for $a,b\in
R$. Recall that $R$ is said to be _F-finite_ if the Frobenius endomorphism
makes $R$ into a module-finite $R$-algebra, i.e., if ${{}^{\varphi}\\!}R$ is a
finitely generated $R$-module.

######  Corollary 2.6.

Let $(R,\mathfrak{m})$ be a normal and locally $\operatorname{AB}$-ring of
prime characteristic. If $R$ is $F$-finite, then
$\operatorname{cd}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)<\infty$ iff
$R$ is regular.

Recall that $R$ is locally $\operatorname{AB}$ means $R_{\mathfrak{p}}$ is
$\operatorname{AB}$ for any prime ideal $\mathfrak{p}$. This is an open
problem [2] that $\operatorname{AB}$ rings are locally $\operatorname{AB}$.

###### Proof.

First, suppose that
$\operatorname{cd}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)<\infty$. The
proof is proceeds via induction on $d:=\dim R$. Recall that normality means
$(S_{2})+(R_{1})$, so the desired claim is clear for $d<2$. We now want to
verify the inductive hypothesis, and drive an additional property. Since
${{}^{\varphi_{n}}\\!}R$ is maximal Cohen-Macaulay,
$H^{d+i}_{\mathfrak{m}}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)^{\vee}\cong\operatorname{Tor}^{R}_{i}(({{}^{\varphi_{n}}\\!}R),({{}^{\varphi_{n}}\\!}R)^{\ast})$,
and the later module behaves nicely with respect to localization, namely,
$({{}^{\varphi_{n}}\\!}R)_{\mathfrak{p}}={{}^{\varphi_{n}}\\!}(R_{\mathfrak{p}})$
and so
$\operatorname{Tor}^{R_{\mathfrak{p}}}_{i}(({{}^{\varphi_{n}}\\!}R_{\mathfrak{p}}),({{}^{\varphi_{n}}\\!}R_{\mathfrak{p}})^{\ast})=0$.
Again, thanks to
$H^{\operatorname{ht}(\mathfrak{p})+i}_{\mathfrak{p}R_{\mathfrak{p}}}(({{}^{\varphi_{n}}\\!}R_{\mathfrak{p}}),({{}^{\varphi_{n}}\\!}R_{\mathfrak{p}}))^{\vee}\cong\operatorname{Tor}^{R_{\mathfrak{p}}}_{i}(({{}^{\varphi_{n}}\\!}R_{\mathfrak{p}}),({{}^{\varphi_{n}}\\!}R_{\mathfrak{p}})^{\ast})$,
we deduced that
$\operatorname{cd}(({{}^{\varphi_{n}}\\!}R_{\mathfrak{p}}),({{}^{\varphi_{n}}\\!}R_{\mathfrak{p}}))<\infty$.
It is easy to see that $R_{\mathfrak{p}}$ is $F$-finite, and from the
assumption it is locally $\operatorname{AB}$. In view of the inductive
hypothesis, $R$ is regular over punctured spectrum. By a result of Kunz [20]
${{}^{\varphi_{n}}\\!}R$ is locally free over the punctured spectrum, also it
is maximal Cohen-Macaulay. The claim now follows from Corollary 2.5.

The reverse part is an immediate application of Kunz theorem along with
Grothendieck’s vanishing theorem. ∎

######  Corollary 2.7.

Let $(R,\mathfrak{m})$ be an $\operatorname{AB}$-ring of isolated singularity
of prime characteristic and dimension bigger than one. If $R$ is $F$-finite,
then $\operatorname{cd}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)<\infty$
iff $R$ is regular.

######  Corollary 2.8.

Let $R$ be complete-intersection and $F$-finite. Then

$\operatorname{cd}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)<\infty\Longleftrightarrow\emph{R
is regular}.$

In particular, if the ring is not regular, then
$H^{j}_{\mathfrak{m}}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)\neq
0\Longleftrightarrow j\geq\dim R$.

###### Proof.

First, suppose
$\operatorname{cd}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)<\infty$. The
proof is by induction on $d:=\dim R$. Due to [18] we may assume $d>1$. In view
of the inductive hypothesis, we may and do assume that the ring is of isolated
singularity. Now by an argument similar to Corollary 2.6 the ring is regular.
The reverse implication is also trivial. Following the previous result, it is
enough to prov the particular case. Since (see [6, Proposition 5.5])

$\inf\\{J:H^{j}_{\mathfrak{m}}(M,N)\neq
0\\}=\operatorname{grade}(\operatorname{Ann}_{R}(\frac{{{}^{\varphi_{n}}\\!}R}{\mathfrak{m}{{}^{\varphi_{n}}\\!}R}),{{}^{\varphi_{n}}\\!}R)=0pt({{}^{\varphi_{n}}\\!}R)=d,$

we know
$H^{j}_{\mathfrak{m}}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)=0$ for
all $j<d$. Suppose the ring is not regular and on the contrary suppose that
$H^{j}_{\mathfrak{m}}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)=0$ for
some $j\geq d$. Thanks to the proof of Proposition 2.2 we know
$\operatorname{Tor}^{R}_{j-d}({{}^{\varphi_{n}}\\!}R,({{}^{\varphi_{n}}\\!}R)^{\ast})=0$.
By complete-intersection assumption (see [23]),
$\operatorname{Tor}^{R}_{>j-d}({{}^{\varphi_{n}}\\!}R,({{}^{\varphi_{n}}\\!}R)^{\ast})=0$.
This in turn implies that
$H^{>j}_{\mathfrak{m}}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)=0$. From
the first part, we conclude that $R$ is regular. This is a contradiction that
we searched. ∎

Now, we extend the previous corollaries:

######  Proposition 2.9.

Let $R$ be a Cohen-Macaulay $F$-finite ring, $M$ be finitely generated and
$n\gg 0$. The following are equivalent:

  * $(1)$

$\operatorname{cd}({{}^{\varphi_{n}}\\!}R,M)\leq\dim R$

  * $(2)$

$\operatorname{cd}({{}^{\varphi_{n}}\\!}R,M)<\infty$

  * $(3)$

$\operatorname{id}_{R}(M)<\infty.$

In particular, for some and so all $m>0$, we have
$\operatorname{cd}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{m}}\\!}R)<\infty\Longleftrightarrow\emph{R
is regular}$.

###### Proof.

$(1)\Rightarrow(2)$: Trivial, as the ring is local.

$(2)\Rightarrow(3)$: Suppose
$\operatorname{cd}({{}^{\varphi_{n}}\\!}R,M)<\infty$. Without loss of
generality we may assume the ring is complete, so it has the canonical module.
We look at Auslander-Buchweitz approximation theory (see [7, 3.3.28]). This
gives a short exact sequence

$0\longrightarrow I\longrightarrow X\longrightarrow M\longrightarrow
0\quad(\ast)$

where $X$ is maximal Cohen-Macaulay and $I$ is of finite injective dimension.
Let $i$ be bigger than
$\max\\{\operatorname{cd}({{}^{\varphi_{n}}\\!}R,M),\operatorname{id}(I)\\}$
and be finite. This gives the exact sequence

$0=H^{i}_{\mathfrak{m}}({{}^{\varphi_{n}}\\!}R,I)\longrightarrow
H^{i}_{\mathfrak{m}}({{}^{\varphi_{n}}\\!}R,X)\longrightarrow
H^{i}_{\mathfrak{m}}({{}^{\varphi_{n}}\\!}R,M)=0\quad(+)$

From $(+)$ we see $H^{i}_{\mathfrak{m}}({{}^{\varphi_{n}}\\!}R,X)=0$ and so
$\operatorname{cd}({{}^{\varphi_{n}}\\!}R,X)<\infty$. Set
$(-)^{\dagger}:=\operatorname{Hom}(-,\omega_{R})$. Recall from [7, Theorem
3.3.10] that $X^{\dagger}=\operatorname{Hom}(X,\omega_{R})$ is maximal Cohen-
Macaulay. Let $d:=\dim(R)$. According to the proof of Proposition 2.1 that
$H^{d+i}_{\mathfrak{m}}({{}^{\varphi_{n}}\\!}R,X)^{\vee}\cong\operatorname{Tor}^{R}_{i}({{}^{\varphi_{n}}\\!}R,X^{\dagger}),$
and so for all $i\gg 0$ we have
$\operatorname{Tor}^{R}_{i}({{}^{\varphi_{n}}\\!}R,X^{\dagger})=0.$ By the
work of Koh-Lee [23, Theorem 2.2.8], $\operatorname{pd}(X^{\dagger})<\infty$.
Thanks to the Auslander-Buchsbaum formula, $X^{\dagger}=\oplus_{I}R$ is free.
In the light of [7, Theorem 3.3.10] we see

$X\cong X^{\dagger\dagger}=\oplus_{I}\omega_{R}$

which is of finite injective dimension. Apply this along with $(\ast)$ we
observe that $M$ is of finite injective dimension.

$(3)\Rightarrow(1)$: If $\operatorname{id}(M)<\infty$, then
$\operatorname{id}(M)=0pt(R)$. As the ring is Cohen-Macaulay***Following Bass’
conjecture we do not assume this., we see $\operatorname{Ext}^{i}_{R}(-,M)=0$
for all $i>\dim R$, and so
$\operatorname{cd}({{}^{\varphi_{n}}\\!}R,M)\leq\dim R$.

To see the particular case it is enough to apply Avramov-Iyengar-Miller [5,
Theorem 13.3]. ∎

The dual version is easy:

######  Observation 2.10.

i) Let $R$ be a Cohen-Macaulay $F$-finite ring with canonical module, $M$ be
finitely generated and $n\gg 0$. Then
$\operatorname{cd}(M,({{}^{\varphi_{n}}\\!}R)^{\dagger})<\infty$ iff
$\operatorname{pd}_{R}(M)<\infty.$

ii) If in addition $R$ is Gorenstein, then
$\operatorname{cd}(M,{{}^{\varphi_{n}}\\!}R)<\infty$ iff
$\operatorname{pd}_{R}(M)<\infty.$

###### Proof.

i) This is included in the proof of Proposition 2.9, where we left the
straightforward modification to the reader.

ii) Recall that
$({{}^{\varphi_{n}}\\!}R)^{\dagger}=({{}^{\varphi_{n}}\\!}R)^{\ast}$ and that
over Gorenstein rings $({{}^{\varphi_{n}}\\!}R)$ is self-dual, see [12, 1.1].
Now, it is enough to apply item i). ∎

As a sample, we present a method to drop F-finiteness:

######  Proposition 2.11.

i) Let $R$ be a Cohen-Macaulay ring of prime characteristic with canonical
module, $M$ be finitely generated and $n\gg 0$. Then
$\operatorname{cd}(M,\omega_{{{}^{\varphi_{n}}\\!}R})<\infty$ iff
$\operatorname{pd}_{R}(M)<\infty.$

ii) If in addition $R$ is Gorenstein, then
$\operatorname{cd}(M,{{}^{\varphi_{n}}\\!}R)<\infty$ iff
$\operatorname{pd}_{R}(M)<\infty.$

###### Proof.

We prove both claims at the same times. Suppose
$\operatorname{cd}(M,\omega_{{{}^{\varphi_{n}}\\!}R})<\infty$. Recall that $M$
is finitely generated. It turns out there is a Grothendieck’s spectral
sequence
$\operatorname{Ext}_{R}^{p}(M,H^{q}_{\mathfrak{m}}(\omega_{{{}^{\varphi_{n}}\\!}R}))\Rightarrow
H^{p+q}_{\mathfrak{m}}(M,\omega_{{{}^{\varphi_{n}}\\!}R}).$ Recall that
$\omega_{{{}^{\varphi_{n}}\\!}R}$ is maximal Cohen-Macaulay over the algebra
${{}^{\varphi_{n}}\\!}R$. In particular, it is big Cohen-Macaulay as an
$R$-module. Following definition,
$H^{q}_{\mathfrak{m}}(\omega_{{{}^{\varphi_{n}}\\!}R})=0$ for all $q\neq
d:=\dim R$, i.e., the spectral sequence collapses and we have

$H^{d+q}_{\mathfrak{m}}(M,\omega_{{{}^{\varphi_{n}}\\!}R})\cong\operatorname{Ext}_{R}^{p}\big{(}M,H^{d}_{\mathfrak{m}}(\omega_{{{}^{\varphi_{n}}\\!}R})\big{)}.$

There is a local ring extension $(S,\mathfrak{n})$ of $(R,\mathfrak{m})$ such
that $\mathfrak{m}S=\mathfrak{n}$, $S$ is $F$-finite, $S$ is faithfully flat
over $R$, and $S$ has infinite residue field. For example, letting
$\widehat{R}\cong k[[x_{1},\cdots,x_{m}]]/I$ for some ideal $I$, we can pick
$S=\bar{k}[[x_{1},\cdots,x_{m}]]/I\bar{k}[[x_{1},\cdots,x_{m}]]$, where
$\bar{k}$ denotes the algebraic closure of $k$. Note, if $L$ is an $R$-module,
then it follows that

$(L\otimes_{R}{{}^{\varphi_{n}}\\!}R)\otimes_{R}S\cong(L\otimes_{R}S)\otimes_{S}{{}^{\varphi_{n}}\\!}S\quad(\flat)$

By base change theorem, the map

${{}^{\varphi_{n}}\\!}R\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}R\otimes_{R}{{}^{\varphi_{n}}\\!}R\longrightarrow
S\otimes_{R}{{}^{\varphi_{n}}\\!}R\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}{{}^{\varphi_{n}}\\!}S\quad(+)$

is flat. Also, recall that
$\operatorname{Hom}_{A}(B,\omega_{A})\stackrel{{\scriptstyle(\ast)}}{{=}}\omega_{B}$,
where $A\to B$ is a morphism of local rings of same dimension and equipped
with canonical modules so that $B$ is finitely generated as an $A$-module.
This gives:

$\omega_{{{}^{\varphi_{n}}\\!}R}\otimes_{R}S\cong\omega_{{{}^{\varphi_{n}}\\!}R}\otimes_{{{}^{\varphi_{n}}\\!}R}({{}^{\varphi_{n}}\\!}R\otimes_{R}S)\stackrel{{\scriptstyle(\flat)}}{{\cong}}\omega_{{{}^{\varphi_{n}}\\!}R}\otimes_{{{}^{\varphi_{n}}\\!}R}{{}^{\varphi_{n}}\\!}S\stackrel{{\scriptstyle(+)}}{{\cong}}\omega_{{{}^{\varphi_{n}}\\!}S}\stackrel{{\scriptstyle(\ast)}}{{\cong}}\operatorname{Hom}_{S}({{}^{\varphi_{n}}\\!}S,\omega_{S})\quad(\natural)$

Due to the assumptions, we know
$\operatorname{Ext}_{R}^{p}\big{(}M,H^{d}_{\mathfrak{m}}(\omega_{{{}^{\varphi_{n}}\\!}R})\big{)}=0$
for all $p\gg 0$. Let
$(-)^{\dagger}:=(-)^{\dagger_{S}}:=\operatorname{Hom}_{S}(-,\omega_{S})$, and
recall that $M$ is finitely generated. This in turn implies the second
implication of the following:

$\displaystyle\operatorname{Ext}_{R}^{i}\big{(}M,H^{d}_{\mathfrak{m}}(\omega_{{{}^{\varphi_{n}}\\!}R})\big{)}=0$
$\displaystyle\Longleftrightarrow$
$\displaystyle\operatorname{Ext}_{R}^{i}\big{(}M,H^{d}_{\mathfrak{m}}(\omega_{{{}^{\varphi_{n}}\\!}R})\big{)}\otimes_{R}S=0$
$\displaystyle\Longleftrightarrow$
$\displaystyle\operatorname{Ext}_{S}^{i}\big{(}M\otimes_{R}S,H^{d}_{\mathfrak{m}}(\omega_{{{}^{\varphi_{n}}\\!}R})\otimes_{R}S\big{)}=0$
$\displaystyle\Longleftrightarrow$
$\displaystyle\operatorname{Ext}_{S}^{i}\big{(}M\otimes_{R}S,H^{d}_{\mathfrak{m}}(\omega_{{{}^{\varphi_{n}}\\!}R}\otimes_{R}S)\big{)}=0$
$\displaystyle\stackrel{{\scriptstyle(\natural)}}{{\Longleftrightarrow}}$
$\displaystyle\operatorname{Ext}_{S}^{i}\big{(}M\otimes_{R}S,H^{d}_{\mathfrak{m}}(({{}^{\varphi_{n}}\\!}S)^{\dagger_{S}})\big{)}=0$
$\displaystyle\Longleftrightarrow$
$\displaystyle\operatorname{Ext}_{S}^{i}\big{(}M\otimes_{R}S,H^{d}_{\mathfrak{n}}(({{}^{\varphi_{n}}\\!}S)^{\dagger_{S}})\big{)}=0,$

where in the third (resp. five) implication we used the flat base change
(resp. independence) theorem. Also, there is an spectral sequence

$\operatorname{Ext}_{S}^{p}(M\otimes_{R}S,H^{q}_{\mathfrak{n}}({{}^{\varphi_{n}}\\!}S)^{\dagger})\Rightarrow
H^{p+q}_{\mathfrak{n}}(M\otimes_{R}S,({{}^{\varphi_{n}}\\!}S)^{\dagger}).$

Recall that $({{}^{\varphi_{n}}\\!}S)^{\dagger}$ is Cohen-Macaulay. This shows

$H^{d+q}_{\mathfrak{n}}(M\otimes_{R}S,({{}^{\varphi_{n}}\\!}S)^{\dagger})\cong\operatorname{Ext}_{R}^{p}\big{(}M\otimes_{R}S,H^{d}_{\mathfrak{n}}({{}^{\varphi_{n}}\\!}S)^{\dagger})\big{)}.$

So, we conclude that
$\operatorname{cd}(M\otimes_{R}S,({{}^{\varphi_{n}}\\!}S)^{\dagger_{S}})<\infty$.
One may deduce from $\operatorname{pd}_{S}(M\otimes_{R}S)<\infty$ that
$\operatorname{pd}(M)<\infty$. Also, it follows that ascent through the flat
map $R\to S$ implies that $S$ is Cohen-Macaulay (Gorenstein). From these, and
by replacing $R$ with $S$, we may and do assume that $R$ is $F$-finite. Now,
the desired claim is in Observation 2.10. ∎

Concerning the next item, it slightly extends [22, 2.9+2.11], as we do not
assume $R$ is generically Gorenstein:

######  Observation 2.12.

Suppose $R$ is Cohen-Macaulay, $\underline{x}$ is a system of parameter and
$M$ is maximal Cohen-Macaulay. If
$\operatorname{Tor}_{i}^{R}(M/\underline{x}M,{{}^{\varphi_{n}}\\!}R)=0$ for
some $n\gg 0$ then $M$ is free. In particular, if $R$ has canonical module and
$\operatorname{Tor}_{i}^{R}(\omega/\underline{x}\omega,{{}^{\varphi_{n}}\\!}R)=0$
then $R$ is Gorenstein.

###### Proof.

We may assume that $R$ is of positive dimension. Since
${{}^{\varphi_{n}}\\!}R$ is big Cohen-Macaulay,
$\operatorname{Tor}_{+}^{R}(R/\underline{x}R,{{}^{\varphi_{n}}\\!}R)=0$. Then
there is a first quadrant spectral sequence

$\operatorname{Tor}_{i}^{R}(\operatorname{Tor}_{j}^{R}(M,R/\underline{x}R),{{}^{\varphi_{n}}\\!}R)\Rightarrow\operatorname{Tor}_{i+j}^{R}(M,{{}^{\varphi_{n}}\\!}R/\underline{x}{{}^{\varphi_{n}}\\!}R).$

Since $M$ is maximal Cohen-Macaulay,
$\operatorname{Tor}_{j}^{R}(M,R/\underline{x}R)=0$. This means the spectral
sequence collapses. From this,
$\operatorname{Tor}_{i}^{R}(M,{{}^{\varphi_{n}}\\!}R/\underline{x}{{}^{\varphi_{n}}\\!}R)=\operatorname{Tor}_{i}^{R}(M/\underline{x}M,{{}^{\varphi_{n}}\\!}R)=0$.
But we know from [26, Corollary 3.3] that $k$ is a direct summand of
${{}^{\varphi_{n}}\\!}R/\underline{x}{{}^{\varphi_{n}}\\!}R$. This means
$\operatorname{pd}(M)<\infty$, and so it is free. ∎

######  Corollary 2.13.

Suppose $R$ is a Cohen-Macaulay F-finite ring with canonical module,
$\underline{x}$ is a system of parameter and $M$ is maximal Cohen-Macaulay.
For $n\gg 0$, the following are equivalent:

  * $(1)$

$H^{i}_{\mathfrak{m}}(M/\underline{x}M,({{}^{\varphi_{n}}\\!}R)^{\dagger})=0$
for all $i>\dim R$

  * $(2)$

$H^{i}_{\mathfrak{m}}(M/\underline{x}M,({{}^{\varphi_{n}}\\!}R)^{\dagger})=0$
for some $i>\dim R$

  * $(3)$

$M$ is free.

###### Proof.

$(1)\Rightarrow(2)$: Trivial.

$(2)\Rightarrow(3)$: Set $j:=\dim R-i$. Recall from the proof of Proposition
2.1 that

$\operatorname{Tor}_{j}^{R}(M/\underline{x}M,{{}^{\varphi_{n}}\\!}R)=\operatorname{Tor}_{j}^{R}(M/\underline{x}M,({{}^{\varphi_{n}}\\!}R)^{\dagger\dagger})=H^{i}_{\mathfrak{m}}(M/\underline{x}M,({{}^{\varphi_{n}}\\!}R)^{\dagger})^{\vee}=0.$

Now, we apply the previous observation to deduce $M$ is free.

$(3)\Rightarrow(1)$: Since $M$ is free, thanks to Koszul complex with respect
to $\underline{x}$ over $M$, we know
$\operatorname{pd}(M/\underline{x}M)<\infty$. This implies that
$\operatorname{cd}(M/\underline{x}M,({{}^{\varphi_{n}}\\!}R)^{\dagger})<\infty$,
and so $(1)$ easily follows, for instance see [16, Theorem 3.2]. ∎

##  3\. Vanishing of $\operatorname{Ext}^{\ast}({{}^{\varphi_{n}}\\!}R,-)$

We say $M$ is (strongly) tor-rigid, if $\operatorname{Tor}_{i}^{R}(M,N)=0$
then $\operatorname{Tor}_{>i}^{R}(M,N)=0$ when $N$ is (not necessarily)
finitely generated. Also, $M$ is called e-rigid if
$\operatorname{Ext}_{R}^{1}(M,M)=0$. Recall that if $0pt(R)=1$ then
${{}^{\varphi_{n}}\\!}R$ is tor-rigid (see [23]), so the following may be
considered as higher analogue of [8, 1.4].

######  Proposition 3.1.

Let $R$ be $(S_{2})$, F-finite and reduced. If ${{}^{\varphi_{n}}\\!}R$ is
e-rigid and strongly tor-rigid, then $R$ is regular.

Instead of the e-rigidity property of ${{}^{\varphi_{n}}\\!}R$, and throughout
this section, one may assume only that
$\operatorname{Ext}^{1}_{R}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{m}}\\!}R)=0$
for some $m$.

###### Proof.

Let $d:=\dim R$. There is nothing to prove if $d=0$, as the ring is assumed to
be reduced. If $\dim R=1$ the claim is in [8, 1.4] that we do not need the
tor-rigidity assumption. Recall that
$({{}^{\varphi_{n}}\\!}R)_{\mathfrak{p}}={{}^{\varphi}\\!}(R_{\mathfrak{p}})$
and that ${{}^{\varphi}\\!}(R_{\mathfrak{p}})$ is e-rigid. Apply this for any
height one prime ideal, we deduce that $R$ satisfies $(R_{1})$. Following
Serre’s characterization of normality, we observe that $R$ is normal. By
$\operatorname{Tr}(-)$ we mean the Auslander’s transpose. Set
$T:=\operatorname{Tr}(\operatorname{Syz}_{1}({{}^{\varphi_{n}}\\!}R))$, and
look at the following exact sequence taken from [2]

(3.1)
$\operatorname{Tor}_{2}^{R}(T,{{}^{\varphi_{n}}\\!}R)\rightarrow\operatorname{Ext}^{1}_{R}({{}^{\varphi_{n}}\\!}R,R)\otimes_{R}{{}^{\varphi_{n}}\\!}R\rightarrow\operatorname{Ext}^{1}_{R}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)\rightarrow\operatorname{Tor}_{1}^{R}(T,{{}^{\varphi_{n}}\\!}R)\rightarrow
0.$

Recall that the ext-rigidity means
$\operatorname{Ext}^{1}_{R}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)=0$.
So, $\operatorname{Tor}_{1}^{R}(T,{{}^{\varphi_{n}}\\!}R)=0$. Recall that the
tor-rigidity implies that
$\operatorname{Tor}_{2}^{R}(T,{{}^{\varphi_{n}}\\!}R)=0$. Due to the displayed
exact sequence
$\operatorname{Ext}^{1}_{R}({{}^{\varphi_{n}}\\!}R,R)\otimes_{R}{{}^{\varphi_{n}}\\!}R=0$.
By Nakayama’s lemma, $\operatorname{Ext}^{1}_{R}({{}^{\varphi_{n}}\\!}R,R)=0$.
In the light of Matlis duality, we observe

$0=\operatorname{Ext}^{1}_{R}({{}^{\varphi_{n}}\\!}R,R)^{\vee}=\operatorname{Ext}^{1}_{R}({{}^{\varphi_{n}}\\!}R,E(k)^{\vee})^{\vee}=\operatorname{Tor}^{R}_{1}({{}^{\varphi_{n}}\\!}R,E(k)).$

Thanks to tor-rigidity, one may deduce that
$\operatorname{Tor}^{R}_{+}({{}^{\varphi_{n}}\\!}R,E(k))=0$. By Matlis
duality, $\operatorname{Ext}^{+}_{R}({{}^{\varphi_{n}}\\!}R,R)=0$. Similarly,
$\operatorname{Ext}^{+}_{R}({{}^{\varphi_{n}}\\!}R,{{}^{\varphi_{n}}\\!}R)=0$.
According to Auslander–Reiten over normal rings, see [19, 3.13(3)], we
conclude that ${{}^{\varphi_{n}}\\!}R$ is free. Recall from Kunz [20] that $R$
is regular if and only if ${{}^{\varphi^{r}}\\!}R$ is a flat $R$-module. The
proof is now completed. ∎

Strongly tor-rigid modules with dimension $<\dim R-1$ are so special:

######  Remark 3.2.

Suppose $R$ is Cohen-Macaulay and let $M$ be a finitely generated module such
that $\dim M<\dim R-1$. Then $M$ is not strongly tor-rigid.

###### Proof.

Let $d:=\dim R$, $d_{0}:=d-\dim M$ and $d_{1}:=d-\dim M-1$. The Check complex
provides a flat resolution of $H:=H^{d}_{\mathfrak{m}}(R)$. So,
$\operatorname{Tor}^{R}_{d_{1}}(M,H)=H^{\dim M+1}_{\mathfrak{m}}(M)=0$, but
$\operatorname{Tor}^{R}_{d_{0}}(M,H)=H^{\dim M}_{\mathfrak{m}}(M)\neq 0$ and
$d_{0}>d_{1}$. ∎

######  Corollary 3.3.

Suppose $0pt(R)=0$, it is $F$-finite and ${{}^{\varphi_{n}}\\!}R$ is e-rigid
for $n\gg 0$. Then $R$ is a field.

###### Proof.

It follows from the previous argument that
$\operatorname{Tor}_{1}^{R}(\operatorname{Tr}(\operatorname{Syz}_{1}({{}^{\varphi_{n}}\\!}R)),{{}^{\varphi_{n}}\\!}R)=0$.
By the work of Koh-Lee [23, Theorem 2.2.8],
$\operatorname{pd}(\operatorname{Tr}(\operatorname{Syz}_{1}({{}^{\varphi_{n}}\\!}R)))<\infty$.
As $(\operatorname{Syz}_{1}({{}^{\varphi_{n}}\\!}R))^{\ast}$ is the second
syzygy of $\operatorname{Tr}(\operatorname{Syz}_{1}({{}^{\varphi_{n}}\\!}R))$
we deduce that
$\operatorname{pd}((\operatorname{Syz}_{1}({{}^{\varphi_{n}}\\!}R))^{\ast})<\infty$.
Thanks to Auslander-Buchsbaum formula, we know
$\operatorname{Syz}_{1}({{}^{\varphi_{n}}\\!}R)^{\ast}$ is free. Again, we are
going to use the depth-zero assumption. Namely, following Ramras [25, 2.6], we
know $\operatorname{Syz}_{1}({{}^{\varphi_{n}}\\!}R)$ is free. So,
$\operatorname{pd}({{}^{\varphi_{n}}\\!}R)<\infty$. In the light of Kunz [20]
we observe that $R$ is regular, and so it is field. ∎

The ring $R$ is called $(G_{1})$, if $R$ is Gorenstein in codimension less
than or equal to one. Recall that $R$ quasi-normal means $(S_{2})+(G_{1})$.
Concerning the next result, part $(c)$ is well-known (see [10]):

######  Fact 3.4.

Assume one of the following:

  1. (a)

$M$ is reflexive and $\operatorname{Ext}^{1}_{R}(M,R)=0$.

  2. (b)

$R$ is Gorenstein and $M$ is almost Cohen-Macaulay.

  3. (c)

Let $M$ be torsionless and $R$ be quasi-normal.

Then $\operatorname{Syz}_{1}(M)$ is reflexive.

###### Proof.

$a)$ There is a free module $F$ and an exact sequence $0\to M^{\ast}\to
F^{\ast}\to\operatorname{Syz}_{1}(M)^{\ast}\to\operatorname{Ext}^{1}_{R}(M,R)=0$.
This yields

$\begin{CD}0@>{}>{}>\operatorname{Syz}_{1}(M)^{\ast\ast}@>{}>{}>F^{\ast\ast}@>{}>{}>M^{\ast\ast}\\\
@A{}A{}A\cong @A{}A{}A=@A{}A{}A\\\
0@>{}>{}>\operatorname{Syz}_{1}(M)@>{}>{}>F@>{}>{}>M@>{}>{}>0,\\\ \end{CD}$

which shows $\operatorname{Syz}_{1}(M)$ is reflexive.

b): As $\operatorname{Syz}_{1}(M)$ is maximal Cohen-Macaulay, and of finite
$G$-dimension, by Auslander-Bridger should be totally reflexive.

c): As $M$ is first syzygy, $\operatorname{Syz}_{1}(M)$ is second syzygy and
so reflexive, see [10, Theorem 3.6]. ∎

######  Corollary 3.5.

Suppose $R$ is complete intersection, $F$-finite and ${{}^{\varphi_{n}}\\!}R$
is e-rigid for some $n>0$. Then $R$ is regular.

###### Proof.

Let $d:=\dim R$. Thanks to Corollary 3.3 we may assume $d>0$. The case $d=1$
is subject of [8, 1.4]. So, we may assume $d>1$. This is in the above argument
that
$\operatorname{pd}((\operatorname{Syz}_{1}({{}^{\varphi_{n}}\\!}R))^{\ast})<\infty$.
As ${{}^{\varphi_{n}}\\!}R$ is maximal Cohen-Macaulay, its syzygy, and the
dual of its syzygy is maximal Cohen-Macaulay. By Auslander-Buchsbaum,
$(\operatorname{Syz}_{1}({{}^{\varphi_{n}}\\!}R))^{\ast}$ is free. So,
$(\operatorname{Syz}_{1}({{}^{\varphi_{n}}\\!}R))^{\ast\ast}$ is free. But,
recall that the claim is clear in dimension one, so:

  1. (a)

$({{}^{\varphi_{n}}\\!}R)_{\mathfrak{p}}$ is reflexive for all $\mathfrak{p}$
with $0ptR_{\mathfrak{p}}\leq 1$, and

  2. (b)

$0pt({{}^{\varphi_{n}}\\!}R)_{\mathfrak{p}}\geq 2$ for all $\mathfrak{p}$ with
$0ptR_{\mathfrak{p}}\geq 2$.

From this, or even without this, ${{}^{\varphi_{n}}\\!}R$ is reflexive. It
turns out that $(\operatorname{Syz}_{1}({{}^{\varphi_{n}}\\!}R))$ is
reflexive. Consequently, $\operatorname{Syz}_{1}({{}^{\varphi_{n}}\\!}R)$ is
free. Therefore, $\operatorname{pd}({{}^{\varphi_{n}}\\!}R)<\infty$. By Kunz
[20], $R$ is regular. ∎

The self-dual property of ${{}^{\varphi_{n}}\\!}R$ was asked in [15] and
studied in [12]. One may ask the reflexivity problem of
${{}^{\varphi_{n}}\\!}R$. In the positive side:

######  Corollary 3.6.

Suppose $R$ is quasi-normal and F-finite. Then ${{}^{\varphi_{n}}\\!}R$ is
reflexive.

######  Example 3.7.

There are several situations, e.g. when betti numbers are strictly increasing,
for which ${{}^{\varphi_{n}}\\!}R$ is not reflexive.

###### Proof.

We are going to use [25], where Ramras introduced a class of rings coined with
$BNSI$. Over this class of rings any reflexive module is free. Now recall this
class included in rings of depth zero. Thanks to Kunz [20], it is enough to
take any $BNSI$ ring of prime characteristic, which is not a field. To be more
explicit, just let $R:=\bar{\mathbb{F}_{p}}[[X,Y]]/(X,Y)^{2}$, and follow some
lines from [3]. ∎

Directed limit of Matlis-reflexive modules is every thing, but not inverse
limit:

######  Lemma 3.8.

Suppose that $(R,\mathfrak{m})$ is local and complete. Let $\\{E_{i}\\}$ be an
inverse system of Matlis-reflexive modules. The following assertions are
valid:

  * $(i)$

$\operatornamewithlimits{\varinjlim}(E_{i}^{\vee})\cong[\operatornamewithlimits{\varinjlim}(E_{i}^{\vee})]^{\vee\vee}$
iff
$\operatornamewithlimits{\varinjlim}(E_{i}^{\vee})\cong(\operatornamewithlimits{\varprojlim}E_{i})^{\vee}$.

  * $(ii)$

One has
$\operatornamewithlimits{\varprojlim}E_{i}\cong(\operatornamewithlimits{\varprojlim}E_{i})^{\vee\vee}$.

###### Proof.

i) First, suppose $\operatornamewithlimits{\varinjlim}(E_{i}^{\vee})$ is
Matlis-reflexive. Let $A:=\operatornamewithlimits{\varinjlim}(E_{i}^{\vee})$.
Recall that:

$A=A^{\vee\vee}=(\operatornamewithlimits{\varinjlim}(E_{i}^{\vee}))^{\vee\vee}=[\operatorname{Hom}(\operatornamewithlimits{\varinjlim}E_{i}^{\vee},E)]^{\vee}=[\operatornamewithlimits{\varprojlim}\operatorname{Hom}(E_{i}^{\vee},E)]^{\vee}=[\operatornamewithlimits{\varprojlim}E_{i}]^{\vee}.$

The reverse implication also is easy.

ii) Set $A:=\operatornamewithlimits{\varprojlim}E_{i}$, let
$M_{i}:=E_{i}^{\vee}$, and $B:=\operatornamewithlimits{\varinjlim}M_{i}$. Then
$E_{i}=M_{i}^{\vee}$ and

$A=\operatornamewithlimits{\varprojlim}E_{i}=\operatornamewithlimits{\varprojlim}\operatorname{Hom}(M_{i},E)=\operatorname{Hom}(\operatornamewithlimits{\varinjlim}M_{i},E)=B^{\vee},$

and so
$A^{\vee\vee}=B^{\vee\vee\vee}=[\operatorname{Hom}(\operatorname{Hom}(B,E),E)]^{\vee}=[B\otimes\operatorname{Hom}(E,E)]^{\vee}=(B\otimes\hat{R})^{\vee}=B^{\vee}=A.$
∎

######  Lemma 3.9.

Let $R$ be a Cohen-Macaulay ring and of prime characteristic with canonical
module. Let $N$ be maximal Cohen-Macaulay and locally free over the punctured
spectrum. Then
$\operatorname{Tor}^{R}_{i}(\mathcal{M},N)\cong\operatorname{Ext}^{d+i}_{R}(\mathcal{M},N^{\dagger})^{\vee}$
for all Matlis-reflexive module $\mathcal{M}$.

###### Proof.

Step I) First, assume $\mathcal{M}:=M$ is finitely generated. A slight
modification of the proof of [19, Lemma 3.5(2)] yields the desired claim.

Step II) Write $\mathcal{M}=\operatornamewithlimits{\varinjlim}_{j\in J}M_{j}$
where $M_{j}$ is finitely generated. Recall that
$\operatorname{Tor}^{R}_{i}(\mathcal{M},N)$ is Matlis-reflexive. We use this
(resp. step I) along with 3.8 in the third (resp. second) identification of:

$\displaystyle\operatorname{Tor}^{R}_{i}(\mathcal{M},N)$ $\displaystyle\cong$
$\displaystyle\operatornamewithlimits{\varinjlim}\operatorname{Tor}^{R}_{i}(M_{j},N)$
$\displaystyle\cong$
$\displaystyle\operatornamewithlimits{\varinjlim}[\operatorname{Ext}^{d+i}_{R}(M_{j},N^{\dagger})^{\vee}]$
$\displaystyle\cong$
$\displaystyle[\operatornamewithlimits{\varprojlim}\operatorname{Ext}^{d+i}_{R}(M_{j},N^{\dagger})]^{\vee}$
$\displaystyle\cong$
$\displaystyle\operatorname{Ext}^{d+i}_{R}(\operatornamewithlimits{\varinjlim}M_{j},N^{\dagger})^{\vee}$
$\displaystyle=$
$\displaystyle\operatorname{Ext}^{d+i}_{R}(\mathcal{M},N^{\dagger})^{\vee},$

yields the desired claim. ∎

######  Question 3.10.

Let $R$ be Cohen-Macaulay and of prime characteristic with canonical module
and $n\gg 0$.

  * $(1)$

(See [11, Page 167]) Suppose
$\operatorname{Ext}^{j}_{R}({{}^{\varphi_{n}}\\!}R,R)=0$ for some $j>\dim R$.
Is $R$ Gorenstein?

  * $(2)$

(See [22, 2.12]) Suppose
$\operatorname{Tor}^{R}_{i}({{}^{\varphi_{n}}\\!}R,\omega_{R})=0$ for some
$i>0$. Is $R$ Gorenstein?

When is $({{}^{\varphi_{n}}\\!}R)^{\vee\vee}={{}^{\varphi_{n}}\\!}R$? For
instantiate, $R$ is F-finite.

######  Proposition 3.11.

Suppose $R$ is F-finite. Then Question 3.10(1) $\equiv$ Question 3.10(2).

###### Proof.

The proof is by induction on $d:=\dim R$. First, assume $d\leq 1$. We claim
both questions are true. Let us deal with Question 3.10(1). Indeed, as depth
of the ring is at most one, ${{}^{\varphi_{n}}\\!}R$ is tor-rigid. From this,
$\operatorname{Ext}^{>d}_{R}({{}^{\varphi_{n}}\\!}R,R)=0$. It follows from
[11, Theorem 1.2] that $R$ is Gorenstein. We should show that Question 3.10(2)
is true in this case, as well. Indeed, as ${{}^{\varphi_{n}}\\!}R$ is tor-
rigid, $\operatorname{Tor}^{R}_{>i}({{}^{\varphi_{n}}\\!}R,\omega_{R})=0$. It
follows from Koh-Lee that $\operatorname{pd}(\omega_{R})<\infty$. Following
Auslander-Buchsbaum formula, $\omega_{R}$ is free, and consequently, the ring
is Gorenstein. Suppose the claim is true in rings of dimension less that $d$.
We now want to verify the inductive hypothesis, and drive an additional
property. The assumptions behave well with respect to the localization. In
view of the inductive hypothesis, $R$ is Gorenstein over punctured spectrum.
So, $\omega_{R}$ is locally free over the punctured spectrum. This allows us
to apply Lemma 3.9 to see
$\operatorname{Tor}^{R}_{i}({{}^{\varphi_{n}}\\!}R,\omega_{R})\cong\operatorname{Ext}^{d+i}_{R}({{}^{\varphi_{n}}\\!}R,R)^{\vee}.$
The proof is now completed. ∎

It may be worth to note that the ring in [11, Page 167] was not assumed Cohen-
Macaulay. Also, the restriction $j>\dim R$ is superfical at least the ring is
$F$-finite, see [11, Corollary 3.5]. So, one may reformulate Question 3.10(1)
as follows:

######  Question 3.12.

Suppose $\operatorname{Ext}^{j}_{R}({{}^{\varphi}\\!}R,R)=0$ for some $j>0$.
Is $R$ Gorenstein?

It seems Question 3.12 is not true, and so negative answer to [23, Conjecture
7.1].

######  Fact 3.13.

(See [10, Theorem 3.8]) i ) Let $k>2$. Let $R$ be a local ring which satisfies
$(S_{k})$, and be $(G_{1})$. The module $M$ satisfies $(S_{k})$ iff $M$ is
reflexive and $\operatorname{Ext}^{i}_{R}(M^{*},R)=0$ for $1\leq i\leq k-2$.

ii) (See [15, Kollar 5] and [12]) Let $R$ be UFD and of prime characteristic.
Then ${{}^{\varphi}\\!}R$ is self-dual.

######  Discussion 3.14.

There is a UFD local ring $R$ of prime characteristic with prefect residue
field so that it is $(S_{3})$ but not Cohen-Macaulay.

###### Proof.

Let $A$ be any complete local ring of prime characteristic with prefect
residue field so that it $(S_{3})$ but not Cohen-Macaulay. Following the work
of Heitmann [14] there is a UFD local ring $R$ such that $\hat{R}=A$.
Following [7, Proposition 2.1.16] $R$ satisfies $(S_{3})$ but it is not Cohen-
Macaulay. ∎

######  Proposition 3.15.

Suppose in addition to Discussion 3.14, the ring is F-finite. Then Question
3.12 is not true.

###### Proof.

Since Frobenius behaves well with respect to localization,
${{}^{\varphi}\\!}R$ satisfies $(S_{3})$. We apply Fact 3.13(i) for $k:=3$
along with Fact 3.13(ii) to observe that
$\operatorname{Ext}^{1}_{R}({{}^{\varphi}\\!}R,R)=0$ but $R$ is not
Gorenstein.∎

The following reduces checking of d-spots vanishing of
$\operatorname{Ext}^{i}_{R}({{}^{\varphi}\\!}R,R)$ into at most 3-spots
vanishing:

######  Observation 3.16.

Suppose $R$ is Cohen-Macaulay, $F$-finite and of dimension $d$. The following
are equivalent:

  * $(1)$

$R$ is Gorenstein.

  * $(2)$

${{}^{\varphi}\\!}R$ is self-dual,
$\operatorname{Ext}^{i}_{R}({{}^{\varphi}\\!}R,R)=0$ for
$0<i\in\\{1,d-1,d\\}$.

Suppose in addition, $R$ is quasi-normal. Then these are equivalent with:

  * $(3)$

${{}^{\varphi}\\!}R$ is self-dual and
$\operatorname{Ext}^{i}_{R}({{}^{\varphi}\\!}R,R)=0$ for $0<i\in\\{d-1,d\\}$.

###### Proof.

The case $d\leq 2$ follows by a result of Goto, see [12, Theorem 1.1]. This in
turn implies that $R$ is Gorenstein in codimension less than or equal to two.
So, we may assume $d\geq 3$. Since Frobenius behaves well with respect to
localization, ${{}^{\varphi}\\!}R$ satisfies $(S_{d})$. Thanks to Fact 3.13(i)
$\operatorname{Ext}^{i}_{R}({{}^{\varphi}\\!}R,R)=0$ for all $i<d-1$. It
remains to apply [12, Theorem 1.1]. ∎

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[1]Mark Hannam These authors contributed equally to this work These authors
contributed equally to this work These authors contributed equally to this
work

1]Gravity Exploration Institute, Cardiff University, The Parade, Cardiff CF24
3AA , UK

2]Departament de Física, Universitat de les Illes Balears, Crta. Valldemossa
km 7.5, E-07122 Palma, Spain

3]LIGO Laboratory, California Institute of Technology, Pasadena, California,
USA

4]LIGO Laboratory, Massachusetts Institute of Technology, Cambirdge, MA, USA

5]University of Oregon, Eugene, OR 97403, USA

6]Institut de Fìsica d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Barcelona, Spain

7]Missouri University of Science and Technology, Rolla, MO, USA

8]Max Planck Institute for Gravitational Physics, Potsdam, Germany

9]Institute for Gravitational Wave Astronomy & School of Physics and
Astronomy, University of Birmingham, Birmingham B15 2TT, UK

10]School of Physics and Astronomy, Monash University, Clayton, VIC 3800,
Australia

11]OzGrav: The ARC Centre of Excellence for Gravitational Wave Discovery,
Clayton, VIC 3800, Australia

12]Department of Applied Mathematics and Theoretical Physics, Cambridge CB3
0WA, UK

13]Instituto Galego de Fisica de Altas Enerxias, Universidade de Santiago de
Compostela, Santiago de Compostela, Galicia, Spain

14]SUPA, University of Glasgow, Glasgow G12 8QQ, UK

15]Rochester Institute of Technology, Rochester, NY, USA

16]University of Texas at Austin, Austin, TX, USA

# General-relativistic precession in a black-hole binary

<EMAIL_ADDRESS>Charlie Hoy  Jonathan E. Thompson  Stephen Fairhurst
Vivien Raymond  Marta Colleoni  Derek Davis  Héctor Estellés  Carl-Johan
Haster  Adrian Helmling-Cornell  Sascha Husa  David Keitel  T. J. Massinger
Alexis Menéndez-Vázquez  Kentaro Mogushi  Serguei Ossokine  Ethan Payne
Geraint Pratten  Isobel Romero-Shaw  Jam Sadiq  Patricia Schmidt  Rodrigo
Tenorio  Richard Udall  John Veitch  Daniel Williams  Anjali Balasaheb Yelikar
Aaron Zimmerman  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [

###### Abstract

The general-relativistic phenomenon of spin-induced orbital precession has not
yet been observed in strong-field gravity. Gravitational-wave observations of
binary black holes (BBHs) are prime candidates, since we expect the
astrophysical binary population to contain precessing binaries (Farr:2017uvj,
; LIGOScientific:2021psn, ). Imprints of precession have been investigated in
several signals (LIGOScientific:2020stg, ; LIGOScientific:2020ufj, ;
LIGOScientific:2021djp, ), but no definitive identification of orbital
precession has been reported in any one of the 84 BBH observations to date
(LIGOScientific:2018mvr, ; LIGOScientific:2020ibl, ; LIGOScientific:2021djp, )
by the Advanced LIGO and Virgo detectors (LIGOScientific:2014pky, ;
VIRGO:2014yos, ). Here we report the measurement of strong-field precession in
the LIGO-Virgo-Kagra (LVK) gravitational-wave signal GW200129. The binary’s
orbit precesses at a rate ten orders of magnitude faster than previous weak-
field measurements from binary pulsars (Taylor:1982zz, ; Kramer:1998id, ;
Burgay:2003jj, ; Breton:2008xy, ). We also find that the primary black hole is
likely highly spinning. According to current binary population estimates a
GW200129-like signal is extremely unlikely, and therefore presents a direct
challenge to many current binary formation models.

###### keywords:

black hole physics — gravitational waves — orbital precession

Spin-induced general-relativistic orbital precession is exhibited in one of
the events in the most recent LVK data release (LIGOScientific:2021djp, ),
GW200129_065458, which we refer to throughout this paper as GW200129. We find
that, given our astrophysical priors, observational biases, and an assumption
of Gaussian noise, the precessing hypothesis is favoured to the non-precessing
hypothesis by a factor of at least 30:1. Considering noise effects alone,
there is only a 1 in 25,000 chance that the imprint of precession on this
signal is entirely due to noise. Regarding priors, we use the most agnostic
statistical priors available, i.e., flat in the component masses and spin
magnitude and the cosine of the spin misalignment angle, and these are the
same as those used in all LVK detections LIGOScientific:2021djp . We also
verify that the main features of our results are (a) not explained by noise,
by repeating our analysis on a theoretical signal injected into detector data,
and on a theoretical signal as it would appear in a noise-free detector
network and (b) the most accurate and reliable available, by comparing our
model to the raw output from solving Einstein’s equations directly.

GW200129 was reported with a signal-to-noise ratio (SNR) of 26.5 across the
three-detector LIGO-Virgo network (LIGOScientific:2021djp, ). This makes it
the loudest BBH signal yet observed, slightly louder than the first detection,
GW150914, initially reported with a network SNR of 24 (LIGOScientific:2016aoc,
). The LVK source properties for GW200129 were based on two analyses. One
reported high precession broadly consistent with the results we present here,
and the other did not. The LVK analysis gave equal weight to both analyses,
leaving the properties of GW200129 unclear (LIGOScientific:2021djp, ). Our
work shows that both analyses rely on theoretical gravitational waveform
models that may not be sufficiently accurate to measure GW200129. However, a
third theoretical model _is_ sufficiently accurate and with that model we are
able to identify that GW200129 is indeed highly precessing. Our measurements
of the binary’s properties, with 90% credible intervals, are given in Tab. 1.
The details of our parameter-estimation and systematics analysis are given in
the Methods section.

Primary mass, $m_{1}$ ($M_{\odot}$) | $39_{-7}^{+6}$
---|---
Secondary mass, $m_{2}$ ($M_{\odot}$) | $22_{-4}^{+8}$
Total mass, $M=m_{1}+m_{2}$ ($M_{\odot}$) | $62_{-3}^{+3}$
Mass ratio, $q=m_{2}/m_{1}$ | $0.6_{-0.2}^{+0.4}$
Primary spin, $a_{1}/m_{1}$ | $0.9_{-0.5}^{+0.1}$
Primary spin tilt angle, $\theta_{\rm LS_{1}}$ (rad) | $1.4_{-0.5}^{+0.4}$
Secondary spin, $a_{2}/m_{2}$ | (undetermined)
Binary inclination, $\theta_{\rm JN}$ (rad) | $0.5_{-0.3}^{+0.3}$
Luminosity distance, $D_{L}$ (Mpc) | $1000_{-200}^{+200}$
Redshift, $z$ | $0.21_{-0.04}^{+0.03}$
Table 1: Properties of the binary-black-hole source of GW200129. Source
parameter measurements from our analysis of GW200129, with uncertainties at
the 90% credible interval. The posterior distributions for both the secondary
spin magnitude and tilt angle $\theta_{LS_{2}}$ are essentially flat, and for
this reason we have indicated the secondary spin as “undetermined”. This is
expected for a signal of GW200129’s strength, as explained in the main text. A
naive application of our 90% credible-interval calculation gives
$a_{2}/m_{2}\in[0.05,0.95]$ and $\cos(\theta_{LS_{2}})\in[-0.93,0.85]$, i.e.,
an arbitrary 90% stretch of the range of possible values.

The key feature of GW200129 that we focus on in our analysis is the
measurement of strong-field general-relativistic orbital and spin precession.
Precession was previously measured in binary pulsars, both from the precession
of one of the pulsars’ spins (Kramer:1998id, ; Breton:2008xy, ;
Manchester:2010dh, ; Fonseca:2014qla, ; Desvignes:2019uxs, ), or precession of
the binary’s orbital plane Krishnan:2020txo . In the limit of a point particle
orbiting a much larger body, the precession of the test body’s spin due to the
presence of the larger body is known as de Sitter precession, while the
precession of a test body’s orbit due to the spin of a larger body is Lense-
Thirring precession. In the general case, where the two bodies can be of
comparable mass, the two effects can be seen to counteract each other, such
that the direction of the total angular momentum of the binary remains
constant and the total spin precesses at the same rate as the orbital plane
(Barker:1975ae, ; Apostolatos:1994mx, ; Kidder:1995zr, ). Black-hole mergers
are in the strong-field nonlinear regime of general relativity well beyond
leading-order effects, but one quantity that can be compared across all
regimes is the system’s precession frequency.

For example, the pulsar PSR B1913+16 was found to have a precession rate of
approximately 1.2 degrees per year (Kramer:1998id, ), or $1.1\times 10^{-10}$
Hz. Later, the double pulsar PSR J0737-3039 was found to have a precession
rate of about 4.8 degrees per year (Breton:2008xy, ), or $4.4\times 10^{-10}$
Hz. In contrast, we find GW200129 to have an average precession rate in the
LIGO-Virgo band of $\sim$3 Hz, i.e., _ten orders of magnitude higher than
previously measured._ Fig. 1 shows the precession frequency $\Omega_{\rm p}$
as a function of time. We see that the precession frequency is at least 1 Hz
when the signal enters the detector sensitivity band, and rises to over 10 Hz
at merger, which is indicated here by $t=0$, the time of maximum signal
amplitude. The binary’s orbital plane is inclined to the total angular
momentum by $\sim$0.5 rad; although this angle increases through inspiral
Apostolatos:1994mx , it does not change significantly over the short duration
of this observation. We discuss further the subtleties of this measurement in
the Methods section.

Figure 1:  Precession frequency of GW200129. The precession frequency
$\Omega_{\rm p}$ of GW200129 as a function of time, from 0.2s before merger,
along with the 90% credible interval.

We now turn to the black hole’s spin angular momentum, $S$. The spin of a
black hole with mass $m$ is usually represented in geometric units by the
dimensionless quantity $a/m=S/m^{2}$, which ranges from zero (non-spinning) to
one (extremal spin). Fig. 2 shows the posterior distribution of our
measurement of the spin of the primary black hole $a_{1}/m_{1}$, and its angle
of misalignment (“tilt”) with respect to the orbital angular momentum. The
misalignment is close to 90∘, and therefore the spin lies almost entirely in
the orbital plane; this is the cause of the significant orbital precession. We
also find that the primary black hole’s spin is larger than $\sim$0.4, with a
strong preference for much higher values, with
$a_{1}/m_{1}=0.9_{-0.5}^{+0.1}$. The signal is not strong enough for the
secondary spin to be measured; this is the case in all BBH observations to
date, and in general reliable measurements of both spins are not expected for
SNRs less than $\sim$100 (Purrer:2015nkh, ).

Figure 2:  The magnitude and tilt of the spin of the larger black hole. Two-
dimensional posterior probability for our measurement of the dimensionless
spin $\vec{a}_{1}/m_{1}$ of the primary black hole, and its mis-alignment
(tilt) with the direction of the orbital angular momentum. Tilt angles of
$90^{\circ}$ means that the spin vector lies within the plane of the binary.
The colour indicates the posterior probability per pixel. This plot is
produced by using histogram bins that are constructed linearly in spin
magnitude and the cosine of the tilt angles such that each bin contains
identical prior probability. The probabilities are marginalized over the
azimuthal angles. Contours represent the 50% and 90% credible intervals.

Black-hole spin and orbital precession leave only subtle imprints on the
waveform. This is why a high SNR is in general necessary for precession to be
measured. Fig. 3 shows the plus polarisation of the theoretical signal
preferred by our analysis, over a $\sim$0.25 s window that corresponds to
roughly half of the duration of data that we analyse. It is possible to make
an approximate decomposition of the signal into two non-precessing harmonics,
and to recover the mild precession modulations as the beating between these
two harmonics (Fairhurst:2019vut, ). The leading harmonic (“harmonic 0”) makes
up the dominant part of the signal, while the next harmonic (“harmonic 1”)
provides the power in precession. Fig. 3 also shows these two harmonics. The
harmonics are in phase at merger and roughly ninety degrees out of phase 0.2 s
earlier, illustrating that the system undergoes roughly one precession cycle
while in the detector’s sensitivity band from 0.65 s before meger.

Figure 3:  Anatomy of GW200129. The figure shows the theoretical maximum
likelihood waveform from our parameter-estimation analysis (see Methods
section), and its approximate decomposition into the two strongest (out of
five) non-precessing harmonics (Fairhurst:2019vut, ). It is the second
harmonic that provides the power in precession.

Having argued that the signal GW200129 was likely produced by a precessing
high-spin binary, we address the question of how it may have formed. A single
observation is not informative about the overall binary population. However,
GW200129 does tell us that black holes can form with high spins, and can end
up in binaries with a large spin misalignment. The 84 LVK BBH observations
suggest that black-hole spins are typically low, with half of spin magnitudes
less than 0.26 (LIGOScientific:2021psn, ), and this is consistent with the
expectation that black holes that form through stellar collapse will not be
highly spinning (Fuller:2019sxi, ).

Although formation of a binary like GW200129 would be rare from the formation
mechanisms that are consistent with the currently observed astrophysical
population, there are a number of viable routes. One is a hierarchical merger,
where successive black-hole mergers could ultimately lead to a binary where
one component is highly spinning, and misaligned with the secondary.
Hierarchical mergers may have already been identified through gravitational
wave observations; see, e.g., Refs. Kimball:2020qyd ; Tiwari:2020otp . Another
is a field binary where the binary black hole is formed from isolated stellar
progenitors. Although various aspects of this field binary tend to suppress
spin misalignment Kalogera:1999tq , there are regions of the parameter space
that can produce binaries with high spins and significant precession
Steinle:2020xej . Other options to produce high spins include chemically
homogeneous evolution (Mandel:2015qlu, ) (although this would _not_ lead to
spin misalignment), formation in AGN (Tagawa:2020dxe, ) (including accretion),
and a binary that has formed from a triple-black-hole system
(Antonini:2017tgo, ).

In some of these scenarios an event such as GW200129 would be extremely rare,
and if that is the case then we do not expect to observe another signal of
this type for some time. Indeed, following the techniques described in Ref.
Fairhurst:2019srr  and assuming the best estimate for the spin distribution of
black holes based on current observations LIGOScientific:2021psn , we
calculate that a GW200129-like system will be observed only once in every 1200
GW observations. This means that a similar system is unlikely to be observed
again in the next two LVK observing runs LIGOObservingPlans:2019aaa  and
therefore GW200129 will likely be the only event with significant precession
and high spin in the first decade of GW astronomy. Given the exceptional
nature of GW200129 within current binary population estimates
LIGOScientific:2021psn , it is more likely that GW200129 presents a
significant challenge to the two favoured BBH formation mechanisms: field
binaries Kalogera:1999tq , and dynamic binaries (formed when two black holes
become gravitationally bound in dense stellar environments)
rodriguez2016illuminating . In fact, we note that in the majority of observed
BBH signals the SNR has not been high enough for precession to be measured,
even if the black-hole spins are large and mis-aligned, and so these may be
more common than previously thought. If so, more binaries such as this will be
observed during the upcoming LVK observing runs and they will provide clues as
to the specific mechanism that produces high-spin, high-misalignment binaries,
and how frequently they form.

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##  1 Methods

We analyse public LVK data from the second half of the third observing run
(O3b) (LIGOScientific:2021djp, ; LIGOScientific:2019lzm, ), which ran from
November 1, 2019 until March 27, 2020. The signal GW200129 was observed on
January 29, 2020, and enters the detector sensitivity band at 20 Hz
approximately 0.65s before merger. During that time the binary completes
roughly nine orbits. GW200129 was reported by the LVK with SNRs in each of the
three operational detectors of 21.2 in Livingston, 14.6 in Hanford and 6.3 in
Virgo. The total network SNR is 26.5.

We determine the properties of the signal source by comparing the data against
theoretical predictions from general relativity. We calculated predicted
signals using the most accurate theoretical waveform model available for
configurations consistent with this signal, NRSur7dq4 (Varma:2019csw, ): when
we analyse the data starting at 20 Hz, the model is appropriate for sources at
a redshift of 0.2 with total binary masses above 56.7 $M_{\odot}$ and where
the primary black hole is no more than four times more massive than the
secondary. We find that the binary’s mass is above 58 $M_{\odot}$ with 99%
confidence, and the mass ratio is less than 1:3 with 99% confidence.

We perform our analysis using the Bayesian Markov-Chain Monte-Carlo (MCMC)
code LALInference (Veitch:2014wba, ) that has been used in LVK analyses since
the first GW detection in 2015 (LIGOScientific:2016vlm, ), and has been
rigorously tested and refined since that time. We used the same prior, sampler
settings, power spectral densities, and calibration envelopes as those used in
LVK GWTC-3 analyses, except for a cut in the prior parameter space to
accommodate the NRSur7dq4 model, i.e., limiting the mass ratio to be less than
1:4, the chirp mass to be between $14.5\,M_{\odot}$ and $49\,M_{\odot}$ and
the total mass to be above $68\,M_{\odot}$ in the detector frame, which
corresponds to $56.7\,M_{\odot}$ at redshift 0.2. With respect to the
component spins, our prior was flat in spin magnitude and the cosine of the
misalignment angle, and flat in the azimuthal angle of each spin.

Our analysis is restricted to the $\ell\leq 3$ multipoles of the signal. The
signal power in the $\ell>3$ multipoles has an SNR of less than 1.0, and does
not significantly change the results. There is also some power lost in the
$(3,\pm 3)$ multipoles between 20 Hz and 30 Hz for the lowest-mass
configurations that are sampled, due to these multipoles starting at a
frequency $3/2$ times higher than the dominant $(2,\pm 2)$ multipoles. We have
also verified that this missing power is negligible and also does not affect
the results. Finally, we performed analyses on both the raw public detector
data and the public data that had undergone glitch removal. The “de-glitched”
data were used in the official LVK results, and we do the same here. The raw
data gave very similar results, with slightly increased support for
precession.

We find that the signal power due to precession has an SNR of
$\rho_{p}=4_{-2}^{+3}$. If precession effects were due to noise, we expect
$\rho_{p}$ to be no higher than around 2 (Green:2020ptm, ). The $\rho_{p}$
estimate of power in precession makes use of an approximate two-harmonic
decomposition of the signal (Fairhurst:2019vut, ), which becomes less
applicable for high-mass signals; however, in these cases it is most likely an
_underestimate_ of the total power due to precession. We also find, following
Ref. (Hoy:2021dqg, ), that there is only a 1 in 25,000 chance that noise alone
would produce the inferred precession SNR that we measure.

Despite the clear measurement of a high in-plane primary spin in Tab. 1 and
Fig. 2, we now quantify our statistical confidence that GW200129 was produced
by a precessing binary. We compare the marginal likelihood from our parameter-
estimation analysis with that from a second analysis, which restricts the in-
plane spin components to be zero, i.e., a non-precessing binary. We calculate
that the precessing-binary hypothesis is favoured over the non-precessing
hypothesis by a factor of 30:1, or a $\log_{e}$ Bayes factor of 3.4
($\log_{10}$ Bayes factor of 1.5). This Bayes factor is consis- tent with the
output from independent nested samplers Veitch:2014wba ; Smith:2019ucc ;
Speagle:2019ivv .

We estimate the time evolution of the precession frequency, and the opening
angle between the orbital and total angular momenta (i.e., the inclination of
the orbital plane), from a subset of theoretical waveforms within the 90%
credible region of our parameter-estimation results. It is possible to
estimate these dynamical properties from the multipole structure of the
waveforms (Schmidt:2010it, ). The precession frequency and opening angle
calculated from the signal are not identical to those in the binary dynamics
(Hamilton:2021pkf, ), but the differences are smaller than the overall
uncertainty in our measurements. The signal-based precession frequency as
measured from the theoretical model also contains oscillations that we know
are not present in the orbital dynamics (by comparison with numerical-
relativity results), and we filter those out of the results shown in Fig. 1.

There are four potential sources of error in our results. (1) uncertainties in
the waveform model have biassed the results, (2) the parameter-estimation code
has settled on the wrong source-parameter values, (3) prior effects, and (4)
the results are due to noise artifacts. We now discuss each of these in
detail.

(1) Waveform model uncertainties. The waveform model was tuned to numerical-
relativity simulations between mass ratios of 1:1 and 1:4, and black-hole
spins up to 0.8, and is well extrapolated up to extreme spins (Varma:2019csw,
; Islam:2020reh, ). All tests reported in the literature, and our own, suggest
that the extrapolation to extreme spins is well-behaved, i.e., the properties
of the waveforms (changes in phasing, amplitude and precession angles) extend
smoothly from zero spin to extreme spin.

The usual measure of the accuracy of a model waveform is the mismatch between
the model and a fiducial “true” waveform. The mismatch between two waveforms
is calculated from a noise-weighted inner product between the normalised
waveforms; the mismatch is the deviation of this quantity from unity, and a
value of 0 indicates that the waveforms agree perfectly (up to an overall
amplitude rescaling), and a value of 1 that they are completely orthogonal.
The mismatch also allows us to estimate the SNR at which two signals would be
indistinguishable (Baird:2012cu, ). If the mismatch between two signals is
$\mathcal{M}$, then they will be indistinguishable when,

$\mathcal{M}\leq\frac{\chi_{k}^{2}(1-p)}{2\rho^{2}},$ (1)

where $\chi_{k}^{2}(1-p)$ is calculated from the cumulative distribution
function of the $\chi^{2}$ distribution with $k$ degrees of freedom at
probability $p$. A binary black hole system undergoing non-eccentric inspiral
has eight physical parameters (the two masses, and the components of each spin
vector); with eight degrees of freedom, two waveforms will be
indistinguishable at 90% confidence if their mismatch satisfies
$\mathcal{M}\leq 6.68/\rho^{2}$. Given that we do not measure most of the spin
components, we might instead consider four degrees of freedom (e.g., the two
masses, and the in-plane and aligned spin contributions), which provides a
stronger mismatch criterion of $\mathcal{M}\leq 3.89/\rho^{2}$. For an SNR of
27, two signals will be indistinguishable if their mismatch is less than
$9.2\times 10^{-3}$ if we assume eight degrees of freedom. If we apply the
more stringent criterion of four degrees of freedom, the mismatch must be
below $5.3\times 10^{-3}$ for them to be indistinguishable.

Within the model’s calibration parameter space, the mismatch error between the
model and fully general relativistic numerical relativity simulations was less
than $4\times 10^{-3}$ for 95% of configurations considered in Ref.
(Varma:2019csw, ). This is well within both criteria proposed above, and
indicates that the model is well within the accuracy requirements to measure
GW200129.

We explicitly test these accuracy statements for signals in the region of
parameter space of GW200129, for NRSur7dq4 and for the two alternative
theoretical models that were used in the LVK analysis. These were selected
because the total mass and mass-ratio limitations of the NRSur7dq4 model mean
that it cannot be used to measure the properties of binaries with low total
mass. Since most of the LVK detections to date were indeed at masses below the
model’s low-mass limit, the LVK analysis of the O3b data made use of two other
independent models, PhenomXPHM (Pratten:2020fqn, ; Garcia-Quiros:2020qpx, ;
Pratten:2020ceb, ) and SEOBNRv4PHM (Ossokine:2020kjp, ). Both models are on
average less accurate than NRSur7dq4 over its calibration region, but have the
advantage of being applicable at both higher mass ratios, and lower masses,
and their accuracy is sufficient at the SNRs and configurations (in
particular, low spins) of most LVK observations. However, they may not meet
the accuracy requirements of the high-SNR high-spin merger GW200129. Extended
Fig. 1 shows mismatches for the three different models against a set of
publicly available numerical-relativity waveforms (SXS:catalog, ). All five
simulations are at mass-ratio 1:2, and have different magnitudes of in-plane
spin on the larger black hole. The simluations used are SXS:BBH:1128,
SXS:BBH:1096, SXS:BBH:0800, SXS:BBH:1097, SXS:BBH:1215, listed in order of
increasing in-plane spin. The mismatches are calculated at a binary
inclination of $\pi/6$ (close to that recovered for GW200129), a total mass of
73 $M_{\odot}$ (the maximum likelihood total mass in the detector frame), and
are optimised over the template phase, polarization and in-plane spin
direction, while SNR-weighted averaging over signal polarization and phase.
(More details on the mismatch calculation that we use are given in Ref.
(Hamilton:2021pkf, ).) The figure also shows the mismatch accuracy threshold
for a signal with SNR 27 with eight and four degrees of freedom. We see that
NRSur7dq4 meets the eight-degrees-of-freedom requirement for high spins, and
indeed is the only model that also meets the requirement with four degrees of
freedom. For the simulation with the highest in-plane spin of
$a_{1\perp}/m_{1}=0.85$, SXS:BBH:1215, we find a mismatch of $2.23\times
10^{-3}$, corrsponding to a distinguisable SNR of $\sim$42 (with four degrees
of freedom). We note that this configuration is _outside_ the calibration
region of the model, since $a_{1}/m_{1}>0.8$.

In Extended Fig. 2 we show an alternative mismatch-accuracy check. Here we
assume that the NRSur7dq4 model accurately represents binary signals, and
calculate the mismatch against that model evaluated at the parameters that
yield the maximum likelihood in our parameter-estimation analysis, and
instances of the PhenomXPHM and SEOBNRv4PHM models evaluated at the same
parameters, and optimised as in Extended Fig. 1. We also calculate mismatches
for model evaluations with the same parameters, but with a range of values of
the primary spin magnitude $a_{1}/m_{1}$. We see again that both models do not
meet the indistinguishability requirement when the spins are high. We also see
that the high-spin mismatches are better for the PhenomXPHM model, which is
consistent with that model recovering signs of precession in GW200129, while
SEOBNRv4PHM does not. Note, however, that systematics errors are likely still
a problem for PhenomXPHM, as discussed in Ref. (Biscoveanu:2021nvg, ).

A final source of systematic uncertainty is that we assume non-eccentric
inspiral. As an example of how this can affect a measurement, the earlier GW
observation GW190521 also appears to have a large in-plane spin when analysed
with a non-eccentric-binary model LIGOScientific:2020ufj , but the apparent
precession signature may be due to orbital eccentricity (Romero-Shaw:2020thy,
; Gayathri:2020coq, ) or even a head-on collision (CalderonBustillo:2020odh,
). The key difference with GW200129 is that the binary has lower mass, and so
more GW cycles are detectable, and over this number of cycles it should be
possible to distinguish eccentricity from precession; GW200129 undergoes
roughly one precession cycle in the detectors’ sensitivity bands, while
eccentricity appears at the orbital timescale. Therefore orbital eccentricity
would produce $\sim$10 modulations in the amplitude and phase of the observed
signal, while precession would only produce one modulation.

Extended Figure 1:  Mismatches between waveform models and numerical-
relativity waveforms. Mismatches between three waveform models, and five
waveforms from numerical-relativity simulations. The simulations were of
binaries with mass-ratio 1:2, and varying values of the in-plane spin
magnitude on the primary black hole, $a_{1\perp}/m_{1}$, which is what drives
precession. The dotted line shows the accuracy threshold assuming eight
degrees of freedom, and the dashed-dotted line the threshold assuming only
four degrees of freedom (see text for discussion). Only the NRSur7dq4 is well
within the accuracy thresholds for GW200129.  Extended Figure 2:  Mismatches
between the most accurate waveform model and those used in the LVK analysis.
Mismatches between theoretical signals (calculated using the NRSur7dq4 model)
against the PhenomXPHM and SEOBNRv4PHM models. The model parameters are those
recovered from our analysis of GW200129, but with a range of values of the
primary spin magnitude $a_{1}/m_{1}$. We see that neither model meets the
accuracy thresholds for GW200129 at high spins, but the better agreement of
PhenomXPHM is consistent with it recovering results closer to those reported
in this work.

(2) Parameter-estimation uncertainties. The results have several features that
at first sight raise concerns. In preliminary parameter-estimation runs the
one-dimensional posterior distribution functions for the masses were bi-modal,
which is often a sign of a noise artefact, or that the MCMC method has not
converged, or some other issue. However, we found that the results became much
cleaner when the sampler was run for longer, and the final production run
produced $1.93\times 10^{5}$ samples. (By comparison with standard GW
applications of the LALinference sampler, we would normally consider
$\sim$$10^{4}$ samples to be sufficient.) In the final results, the
probability distribution for the binary’s mass ratio has a tail that extends
to equal masses. However, our astrophysical priors have a preference for equal
masses, and so this result is not surprising, and, despite this prior
preference, over 80% of the samples are at mass ratios above 1:1.35.

Another concern is the apparent “railing” of the spin measurement against
extremal spin; this can also be a sign of noise issues. There have also been
studies that suggest that the astrophysical prior and observational biases
will pull the spin-magnitude measurement down to lower values, even if the
source contains a highly spinning black hole Chatziioannou:2018wqx . However,
in those studies the large spin was aligned with the orbital angular momentum,
and the prior on the aligned-spin components has a strong preference for low
spin. In our case, the priors on the spin magnitudes are flat.

Finally, we find that if we restrict the analysis to the Livingston and Virgo
detectors, then a similar precessing configuration is recovered, but if we
restrict to Hanford and Virgo, we recover a configuration closer to equal-
mass, and with minimal precession. This suggests that the data in the
Livingston and Hanford detectors may not be consistent.

To investigate these effects, we checked whether we would find similar results
if the precessing-binary signal matching our preferred parameters were
observed in a detector network with zero noise. We performed a parameter-
estimation run on an idealised example of a theoretical signal from a binary
with our best-estimate parameters (those with the maximum likelihood in our
main parameter-estimation results), as it would be observed in a network of
detectors with the same frequency-dependent sensitivity, but no noise. For
this exercise we compared with an analysis on raw data, and starting at 30 Hz.
The results are shown in Extended Fig. 3. We recovered parameters consistent
with our measurements in real data: the spin-magnitude again “rails” against
extreme spins and the mass-ratio measurement has a tail that extends towards
equal-mass systems. In addition, we again find that precession is identified
in the Livingston and Virgo detectors, but _not_ when repeated with only the
Hanford and Virgo detectors. This can be explained by the lower SNR in the
Hanford detector: the precession SNR will be only $\sim$2.4 in that detector,
and so difficult to distinguish from noise.

Extended Figure 3:  Comparison between GW200129 results and those from a
model-waveform injection. One-dimensional posterior distributions for the
primary black hole’s spin, $a_{1}/m_{1}$, and the binary’s mass ratio,
$q=m_{2}/m_{1}\leq 1$, from a parameter-estimation analysis of GW200129 in raw
detector data starting at 30 Hz, and an idealised zero-noise injection. The
main results for the three-detector network (left) are broadly consistent
between the real data and the injection. We also find in both the real data
and the injection that analysis of a Hanford-Virgo-only analysis (middle)
prefers equal-mass binaries and much weaker evidence for precession, while a
Livingston-Virgo-only analysis (right) identifies an unequal-mass precessing
binary; this can be explained by the lower SNR in Hanford (14.6, vs 21.2 in
Livingston), which reduces the measurability of precession.

(3) Prior effects. The choice of priors for the spin magnitudes and tilt
angles can strongly affect parameter estimates (see e.g. Ref. Zevin:2020gxf ).
Given the relatively small number of GW observations to date, there is no
clear motivation to use a prior based on the currently observed population or
motivated by other astrophysical considerations. Indeed, the LVK
collaborations still maintain a prior that remains agnostic about black-hole
masses and spin magnitudes and orientations LIGOScientific:2021djp . In our
analysis we used the same priors as those used by the LVK collaborations,
i.e., flat priors in the component masses and spin magnitude and the cosine of
the misalignment angle. Of course, alternative spin priors consistent with
different astrophysical binary formation scenarios can be used to perform
model selection on those scenarios (see e.g. Farr:2017uvj ; Tiwari:2018qch ;
Hoy:2021rfv ), and this would be an interesting topic for future work.

(4) Noise effects. The LVK analysis noted some noise artifacts near the time
of the observation, but mitigation procedures were applied to the data, and
these de-glitched data were used for the LVK analysis. We have analysed both
the raw and de-glitched data, and find broadly consistent results. However,
the support for precession is slightly _higher_ in the raw data, and an
interesting topic for future work would be to quantify more precisely the
effect of the de-glitching procedures on the precession measurement. As a
final consistency check, we also perform injections of the preferred waveform
into data within 2 s of GW200129, and recovery of these signals also gives
consistent results.

Data Availability. The posterior samples from the analyses performed in this
work are available on Zenodo:
https://zenodo.org/record/6672460https://zenodo.org/record/6672460. Public
documentation is available at the following URL:
https://data.cardiffgravity.org/GW200129-precession/https://data.cardiffgravity.org/GW200129-precession/.

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Acknowledgements We thank Thomas Dent, Shrobana Ghosh, Eleanor Hamilton,
Panagiota Kolitsidou, Lionel London and Frank Ohme for useful discussions. We
also thank Keith Riles for his guidance during the internal LIGO review
process. The authors were supported in part by Science and Technology
Facilities Council (STFC) grant ST/V00154X/1 and European Research Council
(ERC) Consolidator Grant 647839. Calculations were performed using the
supercomputing facilities at Cardiff University operated by Advanced Research
Computing at Cardiff (ARCCA) on behalf of the Cardiff Supercomputing Facility
and the HPC Wales and Supercomputing Wales (SCW) projects. We acknowledge the
support of the latter, which is part-funded by the European Regional
Development Fund (ERDF) via the Welsh Government. In part the computational
resources at Cardiff University were also supported by STFC grant
ST/I006285/1. We are also grateful for computational resources provided by
LIGO Laboratory and supported by National Science Foundation Grants
PHY-0757058 and PHY-0823459. This material is based upon work supported by
NSF’s LIGO Laboratory, which is a major facility fully funded by the National
Science Foundation. This research has made use of data, software and/or web
tools obtained from the Gravitational Wave Open Science Center
(https://www.gw-openscience.orghttps://www.gw-openscience.org), a service of
LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo
Collaboration. LIGO is funded by the U.S. National Science Foundation. Virgo
is funded by the French Centre National de Recherche Scientifique (CNRS), the
Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef,
with contributions by Polish and Hungarian institutes.

Plots were prepared with Matplotlib (2007CSE…..9…90H, ), GWpy (gwpy, ) and
PESummary (Hoy:2020vys, ). Parameter estimation was performed with the
LALInference (Veitch:2014wba, ) and LALSimulation libraries within LALSuite
(lalsuite, ), as well as the BILBY (Ashton:2018jfp, ; Romero-Shaw:2020owr, )
and PBILBY Libraries (Smith:2019ucc, ) and the DYNESTY nested sampling package
(Speagle:2019ivv, ). NumPy numpy , Scipy mckinney-proc-scipy-2010  and
positive London:2018nxs ; lionel_london_2020_3901856  were used during the
analysis.

Author contributions

M.H. initiated and lead the study and writing of the text. C.H. and J.T.
performed the parameter estimation calculations on the detector data, and the
synthetic injections. J.T. performed the mismatch calculations. V.R. and S.F.
provided valuable input to the analysis; V.R. verified the PE results and S.F.
produced Fig. 3. M.H., C.H., J.T., S.F. and V.R. all contributed to the
interpretation of the results and writing the paper. Although they did not
directly contribute to the results presented here, the other authors
contributed to the LVK’s initial understanding of GW200129, which was the
foundation of this work.

Competing interests

The authors declare that they have no competing financial interests.