\section{Introduction}
\label{introduction}

\begin{convention*}\label{notation130604a}
In this paper, rings are commutative, noetherian, and local
with identity.
A ring is ``complete'' when it is complete with respect to its maximal ideal.
In the entire paper, $(R, \ideal{m}, k)$, $(S,\ideal{n}, \ell)$, and $(T,\ideal{r}, v)$ are local rings
where $S$ and $T$ are complete. When it is convenient, $\varphi\colon R\to S$ and $\sigma\colon S\to T$ will denote local ring
homomorphisms.
\end{convention*}

Over forty years ago, Andr\'{e}~\cite{An1, An2} and Quillen~\cite{Q}
introduced a homology theory for commutative algebras that
is known as Andr\'{e}-Quillen (or cotangent)
homology. The $n$-th Andr\'{e}-Quillen homology of the
$R$-algebra $S$ with coefficients in an $S$-module $N$, denoted $\operatorname{D}_n(S\mid R,N)$,
is the $n$-th
homology module of $\mathcal{L_{\varphi}}\otimes_S N$, where
$\mathcal{L_{\varphi}}$ is the cotangent complex of $\varphi$,
uniquely defined in the derived category of the category of $S$-modules.
The vanishing of Andr\'{e}-Quillen homology characterizes
important classes of rings and ring homomorphisms; see, e.g.,
\cite{An1,An3,Av,AHS,AI2,AI1,Q}.

To study the relation between the vanishing of Andr\'{e}-Quillen homology
and the structure of a local ring homomorphism $\varphi$, Avramov and Iyengar~\cite{AI2}
introduced the notion of Andr\'{e}-Quillen dimension of the $R$-algebra $S$, denoted $\operatorname{AQ-dim}_RS$, that is
$$
\operatorname{AQ-dim}_RS:=\sup\{n\in \mathbb{N}\mid \operatorname{D}_n(S\mid R,-)\neq 0\};
$$
in particular, $\operatorname{AQ-dim}_{R} S=-\infty$ if and only if
$\operatorname{D}_n(S\mid R,-)=0$ for all integers $n$.

Andr\'{e}~\cite{An2} and Avramov~\cite{Av} described structure of algebras of Andr\'{e}-Quillen dimension at most 2. Hence, a natural question that one may ask in general is whether there exists a way to determine the structure of algebras of finite Andr\'{e}-Quillen dimension. The only known method of generating this type of algebra has been given in~\cite[Example 2.5]{AI1}, so the question is whether all of these algebras can be obtained using this method. This leads us to the following problem posed by Avramov and Iyengar~\cite[Problem 2.8]{AI1}:

\begin{problem}\label{question130612a}
Assume that $R$ is complete and $\varphi$ is surjective.
If $\operatorname{AQ-dim}_RS<\infty$, does there exist a surjective local homomorphism $\psi\colon Q\to R$ such that
$\ker(\varphi\psi)$ is generated by a $Q$-regular sequence?
\end{problem}

It is worth noting that an affirmative answer to this problem implies the validity of the following conjecture of Quillen~\cite[(5.6)]{Q}; see~\cite[Remark 2.3, Theorem 2.6]{AI1}.

\begin{conj}\label{conj130628a}
If $S$ is an $R$-algebra with $\operatorname{AQ-dim}_R S<\!\infty$, then $\operatorname{AQ-dim}_R S\leq\! 2$.
\end{conj}

In this paper, we investigate the relation between the vanishing
of Andr\'{e}-Quillen homology and complete intersection dimensions.
As the first application of this study, in Section~\ref{sec130315a} of this paper, we use a complete local ring constructed in~\cite{AGP} and show that the answer to Problem~\ref{question130612a} is negative in general; see Example~\ref{example130316a}.
To verify that this ring works, we need the following theorem which is one of our main results in this paper; see~\ref{para130611a} for the proof.

\begin{intthm}\label{thm130316b}
Assume that
$\sigma\varphi$ is complete intersection at $\ideal{r}$ and $\operatorname{D}_n(T\mid S,v)=0$ for all $n\gg 0$.
Then $\operatorname{CI*-fd}_{S} T<\infty$. In particular, $\operatorname{CI-fd}_S T<\infty$.
\end{intthm}

Section~\ref{sec130612} is devoted almost
entirely to the proof of
the next result, which extends a result of Soto~\cite[Proposition 12]{S} and of Avramov,
Henriques, and \c{S}ega~\cite[(2.1)(3)]{AHS}; see~\ref{para130611b} for the proof.

\begin{intthm}\label{thm130316c}
Assume that $\varphi$ is essentially of finite type and $\operatorname{CI-fd}_{R} S<\infty$ (e.g. $\operatorname{CI*-fd}_{R} S<\infty$). Then $\operatorname{AQ-dim}_{R} S\leq 2$ if and only if
$\operatorname{H}_1(K(\varphi))$ is a free $S$-module.
\end{intthm}

In the process of providing a solution to Problem~\ref{question130612a}, we answer part of a question posed by Foxby about the relation
between two complete intersection dimensions; see Question~\ref{question130615a}. As we explain in section~\ref{sec130612}, the proof of Theorem~\ref{thm130316c} becomes much shorter if one can give an affirmative answer to the other part of Foxby's question.

In Section~\ref{sec130314b}, we give a characterization of algebra retracts of finite Andr\'{e}-Quillen dimension with respect to complete intersection dimensions. Avramov and Iyengar~\cite[Theorem I]{AI2} showed that if $S$ is an algebra retract of $R$, then
$\operatorname{AQ-dim}_R S<\infty$ if and only if $\operatorname{AQ-dim}_R S\leq 2$, that is, $R$ and $S$ satisfy Quillen's Conjecture~\ref{conj130628a}. Theorems~\ref{thm130316b} and~\ref{thm130316c}
enable us to prove the following result that extends the result of Avramov and Iyengar; see Theorem~\ref{thm130719a} and Corollary~\ref{cor130719a} for the proof.

\begin{intthm}\label{thm130528a}
Let $S$ be an algebra retract of $R$. Then the following  are equivalent:
\begin{itemize}
\item[(i)]
$\operatorname{AQ-dim}_R S < \infty$.
\item[(ii)]
$\operatorname{AQ-dim}_R S\leq 2$.
\item[(iii)]
$\operatorname{CI*\text{-}dim}_R S<\infty$.
\item[(iv)]
$\cidim_R S<\infty$ and $\operatorname{H}_1(K(\varphi))$ is a free $S$-module.
\end{itemize}
If $\ch(k)=0$, then these conditions are equivalent to the following:
\begin{itemize}
\item[(v)]
$\operatorname{cx}_R S<\infty$.
\end{itemize}
\end{intthm}

This theorem also extends (and gives the converse of) a
result of Soto~\cite[Proposition 21]{S} and answers Soto's Problem 24 in~\cite{S} when $\ch(k)=0$.

\section{Some Background Material}
\label{background}

This section lists important foundational material for use in this paper.
We use the notations $\mu_R(M)$ and $\operatorname{edim}(R)$ for
the minimal number of generators of the $R$-module $M$ and the embedding dimension of the ring $R$, respectively.



The objects defined in the next definition are from the work of
Avramov, Foxby, and B. Herzog~\cite{AFH}.

\begin{para}

A \emph{regular  factorization of $\varphi$}
is a diagram of local ring homomorphisms
$R\xrightarrow{\dot{\varphi}} R'\xrightarrow{\varphi'} S$
such that $\varphi=\varphi'\dot{\varphi}$, the map $\dot{\varphi}$ is
flat, the closed fibre $R'/\ideal{m} R'$ is regular, and $\varphi'$ is surjective.
A \emph{Cohen factorization of $\varphi$} is a regular factorization
$R\xrightarrow{\dot{\varphi}} R'\xrightarrow{\varphi'} S$ of $\varphi$ such that $R'$ is complete.
Since we assumed that $S$ is complete, existence of a Cohen factorization of $\varphi$ is
guaranteed by~\cite[Theorem (1.1)]{AFH}.
\end{para}



\begin{para}
The local ring homomorphism $\varphi$ is called \emph{complete intersection
at $\ideal{n}$} if in a given Cohen factorization $R\to R'\xrightarrow{\varphi'} S$ of
$\varphi$ the ideal $\ker(\varphi')$ is generated by an $R'$-regular sequence.
The complete intersection property for $\varphi$ is independent of the choice of Cohen factorization
by~\cite[Remark (3.3)]{Av}.
\end{para}


We proceed with introducing different types of complete intersection dimensions. The first three of them were defined by Avramov, Gasharov, and Peeva~\cite{AGP}, Takahashi~\cite{RT}, and
Sather-Wagstaff~\cite{SSW1}, respectively.

\begin{para}\label{fact130625a}
Fix a finitely generated $R$-module $M$ and an arbitrary $R$-module $N$.
Define
\begin{align*}
\cidim_R M
&:=\inf\left\{\pd_Q(R'\otimes_R M)-\pd_Q(R')\left| \text{
\begin{tabular}{@{}c@{}}
$R\to R'\leftarrow Q$ is a \\ quasi-deformation
\end{tabular}
}\!\!\!\right. \right\}\\
\operatorname{CI*\text{-}dim}_R M
&:=\inf\left\{\pd_Q(R'\otimes_R M)-\pd_Q(R')\left| \text{
\begin{tabular}{@{}c@{}}
$R\to R'\leftarrow Q$ is an \\ upper quasi-deformation
\end{tabular}
}\!\!\!\right. \right\}\\
\operatorname{CI-fd}_R N
&:=\inf\left\{\operatorname{fd}_Q(R'\otimes_R N)-\pd_Q(R')\left| \text{
\begin{tabular}{@{}c@{}}
$R\to R'\leftarrow Q$ is a \\ quasi-deformation
\end{tabular}
}\!\!\!\right. \right\}\\
\operatorname{CI*-fd}_R N
&:=\inf\left\{\operatorname{fd}_Q(R'\otimes_R N)-\pd_Q(R')\left| \text{
\begin{tabular}{@{}c@{}}
$R\to R'\leftarrow Q$ is an \\ upper quasi-deformation
\end{tabular}
}\!\!\!\right. \right\}.
\end{align*}
Recall that a diagram $R\to R'\leftarrow Q$ of local ring homomorphisms is a \emph{quasi-deformation of $R$} if $R\to R'$ is
flat and $R'\xleftarrow{}Q$ is surjective with kernel
generated by a $Q$-regular sequence. An \emph{upper quasi-deformation of $R$} is a quasi deformation $R\to R'\leftarrow Q$ of $R$
in which $R\to R'$ has regular closed fibre.

From the definitions we easily conclude that $\cidim_R M\leq \operatorname{CI*\text{-}dim}_R M$ and $\operatorname{CI-fd}_R N\leq \operatorname{CI*-fd}_R N$.

Given an upper quasi-deformation $R\to R'\leftarrow Q$ of $R$, it is well-known that the quantities $\pd_Q(R'\otimes_R M)$ and $\operatorname{fd}_Q(R'\otimes_R M)$ are simultaneously finite. From this, it follows that $\operatorname{CI*\text{-}dim}_R M<\infty$ if and only if $\operatorname{CI*-fd}_R M<\infty$.
\end{para}






\begin{para}
Here, we recall some notions from~\cite{AI2, L}.

Let $\overline{\varphi}\colon k\to \ell$ be the induced map between the
residue fields. Then $\varphi$ is called \emph{almost
small} if the kernel of the homomorphism
$$
\Tor[\varphi]{*}{\overline{\varphi}}\ell\colon
\Tor[R]{*}k\ell\to \Tor[S]{*}\ell\ell
$$
of graded algebras is
generated by elements of degree 1.
Assuming $k=\ell$, the homomorphism $\varphi$
is called \emph{large} if $\Tor[\varphi]{*}{\overline{\varphi}}k$ is surjective.
\end{para}

\begin{para}\label{para130711a}
For details on this definition we refer the reader to~\cite[\S 7.2]{A} and~\cite{W}.

Let $R\xrightarrow{\dot{\varphi}} R'\xrightarrow{\varphi'} S$
be a Cohen factorization of
$\varphi$. Let $R'[Y]$ denote a differential graded $R'$-algebra (DG $R'$-algebra for short;
see~\cite[\S 1.3]{A} for background on this) whose underlying algebra $R'[Y]^{\natural}$
is a tensor product of symmetric algebras of free
modules with basis $Y_n$ when $n\geq 0$ is even and exterior algebras
of free modules with basis $Y_n$ when $n\geq 1$ is odd. The DG algebra $R'[Y]$ is called a \emph{minimal model of $S$ over
$R'$} if $Y=Y_{\geq 1}$ and $\operatorname{H}(R'[Y])\cong S$, and $\partial(Y)\subseteq \ideal{m}' R'[Y]+(Y)^2 R'[Y]$, where $\ideal{m}'$ is the maximal ideal of $R'$.

The \emph{n-th deviation of $\varphi$}, denoted $\varepsilon_n(\varphi)$, is defined to be
\[\varepsilon_n(\varphi)=
\begin{cases} \text{card($Y_1$)}-\operatorname{edim}(R'/\ideal{m} R')+\operatorname{edim}(S/\ideal{m} S)\hspace{9.2mm} \text{for $n=2$};\\
\text{card($Y_{n-1}$)}\hspace{55mm}\text{for $n\geq 3$.} \end{cases}
\]
From~\cite[Proposition (3.1)]{Av} (or~\cite[Proposition 2.4]{AI2}), it follows that the
numbers $\varepsilon_n(\varphi)$ are invariants of $\varphi$.
\end{para}

The next definition originates with work of Gulliksen and Levin~\cite{GL}; see~\cite[\S 6]{A} for details.
It involves a different type of algebra extensions introduced by Tate~\cite{T};
see~\cite{A, GL} for details.

\begin{para}
let $A$ be a DG
$R$-algebra, and assume that $\varphi$ is surjective. Let $A\langle Y\rangle$ denote a DG $R$-algebra
obtained from $A$ by adjunctions of sets of exterior
variables of positive odd degrees and of divided power
variables of positive even degrees. Then a factorization
$R\longrightarrow R\langle
Y\rangle\stackrel{\simeq}\longrightarrow S$ of
$\varphi$ is called an
\emph{acyclic closure of $\varphi$} if $\partial(Y_1)$ minimally
generates $\ker(\varphi)$, and \{cls$(\partial(y))\mid y\in Y_{n+1}$\} is
a minimal generating set of $\operatorname{H}_n(R\langle Y_{\leq n}\rangle))$ for
all $n\geq 1$.
\end{para}

\begin{para}
Let $R\to R'\xrightarrow{\varphi'} S$ and $R\to R''\xrightarrow{\varphi''} S$ be two
Cohen factorizations of $\varphi$ with
$J'=\operatorname{Ker}(\varphi')$ and $J''=\operatorname{Ker}(\varphi'')$. We may assume that
$R''=R'/I'$ and $J''=J'/I'$, where $I'$ is an ideal
generated by an $R'$-regular sequence
that extends to a minimal set of generators of $J'$. Then we have
$$
\mu_{R'}(J')-\operatorname{edim} (R'/\ideal{m} R')=\mu_{R''}(J'')-\operatorname{edim} (R''/\ideal{m} R'')=
\varepsilon_2(\varphi)-\operatorname{edim}(S/\ideal{m} S).
$$


This fact and~\cite[Corollary 1.6.13.(b)]{BH} enable us to define
the Koszul homology module of a local ring homomorphism as follows:

Let $R\to R'\xrightarrow{\varphi'} S$ be an arbitrary
Cohen factorization of $\varphi$ with
$J'=\operatorname{Ker}(\varphi')$. We define
the \emph{$n$-th Koszul homology module of $\varphi$}, denoted $\operatorname{H}_n(K(\varphi))$, by
setting $\operatorname{H}_n(K(\varphi)):=\operatorname{H}_n(K^{R'}(J'))$, in which $K^{R'}(J')$ is
the Koszul complex associated to an arbitrary set of generators
of $J'$.
\end{para}

\begin{para}
Let $M$ be a finitely generated $R$-module. The \emph{complexity of $M$
over $R$}, denoted $\operatorname{cx}_R M$,
is the number
$$
\operatorname{cx}_R M:=\inf\{d\in \mathbb{N}\mid \text{there exists}\ \gamma\in \mathbb{R}\ \text{such that}\
 \beta^R_n(M)\leq \gamma n^{d-1}\ \text{for all $n\gg 0$}\}
$$
where $\beta^R_n(M):=\rank_k(\Tor[R]nMk)$ is the $n$-th \emph{Betti number}. The \emph{Poincar\'{e} series
of $M$}, denoted $P^R_M(t)$, is defined to be $P^R_M(t):=\sum_{n\geq 0}\beta^R_n(M)t^n$.
\end{para}


\section{Theorem~\ref{thm130316b} and Problem~\ref{question130612a}}
\label{sec130315a}
This section contains the proof of Theorem~\ref{thm130316b} and
a counter-example to
Problem~\ref{question130612a} from the introduction. We start by posing the following question, which was originally asked by Foxby in his personal
communication about the relation between complete intersection dimension and complete intersection flat dimension.

\begin{question}\label{question130615a}
Let $R\xrightarrow{\dot{\varphi}} R'\xrightarrow{\varphi'} S$ be a Cohen factorization of
$\varphi$, and let $N$ be a finitely generated $S$-module. Do the following inequalities hold?
$$
\operatorname{CI*-fd}_{R} N\leq \operatorname{CI*\text{-}dim}_{R'}N\leq \operatorname{CI*-fd}_{R}N+\operatorname{edim}(R'/\ideal{m} R').
$$
\end{question}

Next lemma provides an affirmative answer to the first inequality in Question~\ref{question130615a}, while the second inequality is still open. It is worth noting that by slightly modifying the proof and using the same techniques, we can get the first inequality in the original question of Foxby.
This lemma is the key to the proof of Theorem~\ref{thm130316b}.

\begin{lem}\label{prop130607a}
Let $R\xrightarrow{\dot{\varphi}} R'\xrightarrow{\varphi'} S$ be a Cohen factorization of
$\varphi$, and let $N$ be a finitely generated $S$-module. Then $\operatorname{CI*-fd}_{R} N\leq \operatorname{CI*\text{-}dim}_{R'}N$.
\end{lem}

\begin{proof}
Assume without loss of generality that $\operatorname{CI*\text{-}dim}_{R'} N<\infty$. Then, by definition, there exists an upper quasi-deformation $R'\xrightarrow{\alpha}
R''\xleftarrow{} Q$ for which we have $\operatorname{CI*\text{-}dim}_{R'} N=\pd_{Q}(R''\otimes_{R'} N)-\pd_Q(R'')$.
Now, the third step in the next display comes from~\cite[Theorem 4.1 (F)]{AF} and the fact that $\dot{\varphi}$ and $\alpha$ are flat:
\begin{align*}
\operatorname{CI*\text{-}dim}_{R'} N&=\pd_{Q}(R''\otimes_{R'} N)-\pd_Q(R'')\\
&\geq \operatorname{fd}_{Q}(R''\otimes_{R'} N)-\pd_Q(R'')\\
&\geq\operatorname{fd}_{Q}((R''\otimes_{R'} N)\otimes_R R')-\pd_Q(R'')\\
&= \operatorname{fd}_{Q}(R''\otimes_{R} N)-\pd_Q(R'')\\
&\geq \operatorname{CI*-fd}_{R} N.
\end{align*}
Associativity and tensor cancellation give the forth step. The last step follows from the definition of $\operatorname{CI*-fd}$ and the fact that $R\xrightarrow{\alpha\dot{\varphi}} R''\xleftarrow{} Q$ is an upper quasi-deformation; see~\cite[Fact 2.2]{NS}.
\end{proof}


Next, we recall a construction used by Iyengar and
Sather-Wagstaff~\cite{IW}, essentially given by Avramov, Foxby, and B. Herzog~\cite{AFH}. We use this construction in the proof of Theorem~\ref{thm130316b}.


\begin{construction}\label{fact130607c}
Let $R\xrightarrow{\dot{\varphi}}R'\xrightarrow{\varphi'} S$ and $R'\xrightarrow{\dot{\rho}}
R''\xrightarrow{\rho'} T$
be Cohen factorizations of $\varphi$ and $\sigma\varphi'$,
respectively. Then $\rho'$ factors through the tensor
product $S'=R''\otimes_{R'}S$ and gives the following commutative
diagram of local ring homomorphisms
\[
\xymatrixrowsep{2.5pc}
\xymatrixcolsep{2.5pc}
\xymatrix{
&& R''
\ar@{->}[dr]^{\varphi''}
\ar@/^2.5pc/[ddrr]^{\rho'=\sigma'\varphi''}
\\
& R'
\ar@{->}[dr]^{\varphi'}
\ar@{->}[ur]^{\dot\rho}
&& S'
\ar@{->}[dr]^{\sigma'}
\\
R
\ar@{->}[rr]^{\varphi}
\ar@/^2.5pc/[uurr]^{\dot\rho \dot\varphi}
\ar@{->}[ur]^{\dot\varphi}
&& S
\ar@{->}[rr]^{\sigma}
\ar@{->}[ur]^{\dot\sigma}
&& T
}
\]
in which $\dot{\sigma}$ and $\varphi''$ are the natural maps to
the tensor product and the diagrams
$S\to S'\to T$,
$R\to R''\to T$, and
$R'\to R''\to S'$ are Cohen factorizations.
\end{construction}


Here is the proof of Theorem~\ref{thm130316b}.

\begin{para}[Proof of Theorem~\ref{thm130316b}]\label{para130611a}
Using the notation from Construction~\ref{fact130607c}, we get the diagram
\begin{equation}\label{eq130607a}
R''\xrightarrow{\varphi''} S'\xrightarrow{\sigma'}T
\end{equation}
of surjective local ring homomorphisms.
Since $\sigma\varphi$ is complete intersection at $\ideal{r}$, from~\cite[(1.7), (1.8)]{Av} we get that
$\sigma'\varphi''$ is complete intersection at
$\ideal{r}$. Hence, by~\cite[Corollary (4.9)]{AI2} we conclude that
$\varphi''$ is an almost small ring homomorphism. Again,
using~\cite[Lemma (1.7)]{Av} and our assumption, we have $\operatorname{D}_n(T\mid S',v)=0$
for all $n\gg 0$. From the Jacobi-Zariski exact sequence arising
from diagram~\eqref{eq130607a}, we find that $\operatorname{D}_n(S'\mid R'',v)=0$ for all $n\gg 0$.
Since $\varphi''$ is almost small, it follows from~\cite[(6.4)]{AI2}
that $\varphi''$ is complete intersection at $\ideal{r}$. Thus, $\operatorname{Ker}(\varphi'')$ is generated by an
$R''$-regular sequence.

To prove $\operatorname{CI*-fd}_{S} T<\infty$, by Lemma~\ref{prop130607a}, it
suffices to show $\operatorname{CI*\text{-}dim}_{S'} T<\infty$. For this, consider
the diagram $S'\xrightarrow{\id} S'\xleftarrow{\varphi''} R''$ of local ring homomorphisms, that is an upper quasi-deformation
by Construction~\ref{fact130607c}.
Note that $\ker(\sigma'\varphi'')$ is generated by an
$R''$-regular sequence because $\sigma'\varphi''$ is surjective and complete intersection at $\ideal{r}$.
Hence, we conclude that $\pd_{R''}(T\otimes_{S'} S')=\pd_{R''} T<\infty$. Therefore,
$\operatorname{CI*\text{-}dim}_{S'} T<\infty$, as desired. \qed
\end{para}

As our first application to Theorem~\ref{thm130316b}, we show that the answer to Problem~\ref{question130612a}
from the introduction is negative in general.
For this we use a short ring constructed in~\cite{AGP}, used also in~\cite[Theorem 4.5]{AHS}.

\begin{ex}\label{example130316a}
Let $k$ be a field with $\ch(k)\neq 2$. Consider the complete local ring
$R:=k[w,x,y,z]/I$
where the ideal $I$ is generated by the elements
$$
w^2, wx-y^2, wy-xz, wz, x^2+yz, xy, z^2.
$$
Let $a$ be the image of $x$ in
$R$, and let $S:=R/aR$. By~\cite[Theorem 4.5]{AHS} and~\cite{BMR} we have $\operatorname{D}_n(S\mid R,\ell)=0$ for all $n\geq 3$.

Consider the natural surjection
$\varphi\colon R\to S$. Suppose that there exists a surjective local ring homomorphism $\psi\colon Q\to R$
such that $\ker(\varphi\psi)$ is generated by a $Q$-regular sequence.
Then from Theorem~\ref{thm130316b} and~\ref{fact130625a},
we must have
$\operatorname{CI\text{-}dim}_{R} S<\infty$, which is a contradiction; see~\cite[Theorem 4.5]{AHS}.
\end{ex}


\section{Theorem~\ref{thm130316c}}\label{sec130612}

This section is devoted almost
entirely to the proof of Theorem~\ref{thm130316c} that extends a result of Soto~\cite[Proposition 12]{S} and of Avramov,
Henriques, and \c{S}ega~\cite[(2.1)(3)]{AHS}.

Recall that there exists a sequence $\{V_n(R)\}_{n\geq 0}$ of $k$-vector spaces associated to the local ring $R$, where $V_n(R)$ is the $n$-th component of the graded $k$-space of the indecomposable elements of the Hopf algebra with divided powers $\Tor[R]{}kk$; see~\cite{av} for details. Andr\'{e} and Quillen~\cite[19.21, 20.26, and 20.27]{An2} show that there exist isomorphisms $V_n(R)\cong \operatorname{D}_n(k\mid R, k)$ for all $n\leq \pi(R)$, where $\pi(R)=\infty$ if $k$ has characteristic 0 and $\pi(R)=2p$ if $k$ has characteristic $p$; see~\cite[(1.3) and (1.4)]{av}.

The following lemma is an
extension of~\cite[Lemma 1(2)]{R1}.

\begin{lem}\label{lem130316a}
If $\operatorname{fd}_{R}S <\infty$, then the homomorphism
$\operatorname{D}_m(S\mid R,\ell)\xrightarrow{\alpha} \operatorname{D}_m(\ell\mid R,\ell)$ is
trivial for $m=2, 4$.
\end{lem}

\begin{proof}
For each $n\geq 0$, the local ring homomorphism $\varphi$ induces the homomorphism $V_n(R)\otimes_k \ell\to V_n(S)$ as $\ell$-vector spaces; see~\cite{av}. Since $\operatorname{fd}_RS<\infty$, by~\cite[Th\'{e}or\`{e}me]{Avramov} we have $V_{2n}(R)\otimes_k \ell\to V_{2n}(S)$ is injective for all $n\geq 0$. This implies that the map
$\operatorname{D}_m(k\mid R, k)\otimes_k \ell\to \operatorname{D}_m(\ell\mid S, \ell)$ is injective for $m=2, 4$. It follows from~\cite[Lemme 3.20]{An2} that the map $\operatorname{D}_m(k\mid R, \ell)\to \operatorname{D}_m(\ell\mid S, \ell)$ is injective for $m=2, 4$. From the Jacobi-Zariski exact sequence of $R\to k\to \ell$ and the fact that $\operatorname{D}_n(\ell\mid k, \ell)=0$ for all $n\geq 2$ we conclude that $\operatorname{D}_m(k\mid R, \ell)\cong \operatorname{D}_m(\ell\mid R, \ell)$ for $m=2, 4$. Therefore, the map $\beta$ in the Jacobi-Zariski exact sequence
$$
\operatorname{D}_m(S\mid R, \ell)\xrightarrow{\alpha} \operatorname{D}_m(\ell\mid R, \ell)\xrightarrow{\beta} \operatorname{D}_m(\ell\mid S, \ell)
$$
of $R\to S\to \ell$ is injective for $m=2, 4$. This implies that $\alpha$ is trivial, as desired.
\end{proof}





In the proof of Theorem~\ref{thm130316c}, we will use the above lemma when $m=2$. The case where $m=4$ gets used in the proof of Theorem~\ref{thm130528a} in the next section.

\begin{lem}\label{prop130720a}
If $\operatorname{AQ-dim}_{R} S\leq 2$, then
$\operatorname{H}_1(K(\varphi))$ is a free $S$-module.
\end{lem}

\begin{proof}
Let $R\xrightarrow{\dot{\varphi}} R'\xrightarrow{\varphi'} S$ be a Cohen factorization of $\varphi$.
By assumption we get $\operatorname{D}_n(S\mid R,\ell)=0$ for all $n\geq 3$. Hence, $\operatorname{D}_n(S\mid R',\ell)=0$ for all $n\geq 3$ by~\cite[Lemma (1.7)]{Av}. This implies that $\operatorname{D}_n(S\mid R',-)=0$ for all $n\geq 3$. Now~\cite[Corollary 3]{BMR} implies that $\operatorname{H}_1(K(\varphi'))$ is a free $S$-module. The assertion now follows from 2.7.
\end{proof}

Here is the proof of Theorem~\ref{thm130316c}.

\begin{para}[Proof of Theorem~\ref{thm130316c}]\label{para130611b}
Assume that
$\operatorname{H}_1(K(\varphi))$ is a free $S$-module.
Since $\varphi$ is essentially of finite type, it admits a regular
factorization $R\to R[X]_{\frak M}\to S$ in which
$R[X]:=R[x_1,x_2,...,x_n]$ for some $n\geq 1$, and
$\frak M$ is
a prime ideal of $R[X]$ lying over $\ideal{m}$.

Since $\operatorname{CI-fd}_{R} S<\infty$, there is a quasi-deformation
$R\to R''\xleftarrow{\eta} Q$ of $R$ such
that $\operatorname{fd}_{Q}(S\otimes_{R} R'')<\infty$. Set $R':=R[X]_{\frak M}$,
$P:=R'\otimes_{R} R''$, $U:=S\otimes_{R} R''$, and consider the diagram
$Q\xrightarrow{\alpha} P
\xrightarrow{\beta} U$ of natural ring
homomorphisms in which $\beta$ is surjective.
Let $\frak u\in \operatorname{Spec}(U)$, and set
$\ideal{p}:=\beta^{-1}(\frak u)$ and
$\ideal{q}:=\alpha^{-1}(\ideal{p})$. Then we have the diagram
\begin{equation}\label{eq130610a}
Q_{\ideal{q}}\xrightarrow{\alpha_{\ideal{q}}}
P_{\ideal{p}}\xrightarrow{\beta_{\ideal{p}}}
U_{\frak u}
\end{equation}
of local ring homomorphisms
that gives us the
commutative diagram
$$\xymatrix{
Q_{\ideal{q}}\ar[r]^{\alpha_{\ideal{q}}} \ar[d]_{\beta_{\ideal{p}}\alpha_{\ideal{q}}} & P_{\ideal{p}}\ar[r]^{\beta_{\ideal{p}}} \ar[d]^{\beta_{\ideal{p}}} & U_{\frak u}\ar[d] \\
U_{\frak u}\ar[r]^{\id}\ar[d] & U_{\frak u}\ar[r]\ar[d] & k(\frak u)\\
k(\frak u)\ar[r]^{\id} & k(\frak u)}
$$
of local ring homomorphisms.
This diagram induces the following commutative
diagram of $k(\frak u)$-vector spaces
\begin{equation}\label{eq130610b}
\begin{split}
\xymatrix{
\operatorname{D}_3(k(\frak u)\mid U_{\frak u},k(\frak u))\ar[r]^{f} & \operatorname{D}_2(U_{\frak u}\mid P_{\ideal{p}},k(\frak u))\ar[r] & \operatorname{D}_2(k(\frak u)\mid P_{\ideal{p}},k(\frak u))\\
\operatorname{D}_3(k(\frak u)\mid U_{\frak u},k(\frak u))\ar[r]^{g}\ar@{=}[u]& \operatorname{D}_2(U_{\frak u}\mid Q_{\ideal{q}},k(\frak u))\ar[r]^{\theta}\ar[u]^{h} & \operatorname{D}_2(k(\frak u)\mid Q_{\ideal{q}},k(\frak u))\\
& \operatorname{D}_2(P_{\ideal{p}}\mid Q_{\ideal{q}},k(\frak u))\ar[u]^{\lambda}}
\end{split}\end{equation}

Since $\operatorname{fd}_{Q_{\ideal{q}}} U_{\frak u}<\infty$, it follows from Lemma~\ref{lem130316a} that
$\theta$ is trivial.

It is also straightforward to see that
$\operatorname{H}_1(K^{P_{\ideal{p}}}(\operatorname{Ker}(\beta_{\ideal{p}})))$ is a free $U_{\frak u}$-module. Thus,
from~\cite[Theorem 1]{R2} we conclude that $f$ is trivial.













Since $\operatorname{D}_n(R[X]\mid R,-)=0$ for all $n\geq 1$, we get
$\operatorname{D}_n(R'\mid R,-)=0$ for all $n\geq 1$. The
faithful flatness of $R''$ over $R$ also yields $\operatorname{D}_n(P\mid R'',-)=0$
for all $n\geq 1$.
Hence, setting $\ideal{p}'':=\ideal{p}\cap R''$, we get $\operatorname{D}_n(P_{\ideal{p}}\mid R''_{\ideal{p}''}, k(\ideal{p}))=0$ for all $n\geq 1$.

Consider the diagram $Q_{\ideal{q}}\xrightarrow{\eta_{\ideal{q}}} R''_{\ideal{p}''}\xrightarrow{}
P_{\ideal{p}}$ of local ring homomorphisms. Since $\operatorname{Ker}(\eta_{\ideal{q}})$ is generated by a $Q_{\ideal{q}}$-regular
sequence, from the Jacobi-Zariski exact sequence we get
$\operatorname{D}_n(P_{\ideal{p}}\mid Q_{\ideal{q}},k(\ideal{p}))=0$ for all $n\geq 2$. Thus,
$\lambda$ in diagram~\eqref{eq130610b} is injective. Commutativity of
diagram~\eqref{eq130610b} implies that $\operatorname{D}_2(U_{\frak u}\mid Q_{\ideal{q}},k(\frak u))=0$. Hence,
by~\cite[Proposition (1.8)]{Av} we get $\operatorname{D}_n(U_{\frak u}\mid Q_{\ideal{q}},k(\frak u))=0$
for all $n\geq 2$. Now, from the Jacobi-Zariski exact sequence of
diagram~\eqref{eq130610a} we conclude that $\operatorname{D}_n(U_{\frak u}\mid P_{\ideal{p}},k(\frak u))=0$
for all $n\geq 3$.
Since $\frak u$ was an arbitrary prime ideal of $U$,
we conclude that $\operatorname{D}_n(U\mid P,-)=0$ for all
$n\geq 3$; see~\cite{An2}. Now, faithful flatness of $R''$ over $R$ implies that
$\operatorname{D}_n(S\mid R',-)=0$ for all $n\geq 3$. Thus, from~\cite[Lemma (1.7)]{Av} we have
$\operatorname{D}_n(S\mid R,\ell)=0$ for all $n\geq 3$. Since
$\varphi$ is essentially of finite type, from~\cite[Lemma 8.7]{I} we
conclude that $\operatorname{AQ-dim}_{R} S\leq 2$.

Converse follows immediately from Lemma~\ref{prop130720a}. \qed
\end{para}

\begin{disc}
Let $R\xrightarrow{\dot{\varphi}} R'\xrightarrow{\varphi'} S$ be a Cohen factorization of $\varphi$. If either the Foxby's inequality $\cidim_{R'} S\leq \operatorname{CI-fd}_R S+\operatorname{edim}(R'/\ideal{m} R')$, or the inequality
$\operatorname{CI*\text{-}dim}_{R'} S\leq \operatorname{CI*-fd}_R S+\operatorname{edim}(R'/\ideal{m} R')$ in Question~\ref{question130615a} from Section~\ref{sec130315a} holds, then we can give a short proof to Theorem~\ref{thm130316c}. In fact, the assumption
$\operatorname{CI-fd}_R S<\infty$ (or $\operatorname{CI*-fd}_R S<\infty$) implies that $\cidim_{R'} S<\infty$. Since $\operatorname{H}_1(K^{R'}(\ker (\varphi')))$ is a free $S$-module, from \cite[Proposition 12]{S} we conclude that
$\operatorname{D}_n(S\mid R',\ell)=0$ for all $n\geq 3$. Now, the assertion follows from~\cite[Lemma (1.7)]{Av} and~\cite[Lemma 8.7]{I}.
\end{disc}

\section{Algebra retracts and Theorem~\ref{thm130528a}}\label{sec130314b}

In this section, we prove Theorem~\ref{thm130528a} as an application of Theorems~\ref{thm130316b} and~\ref{thm130316c}.


Recall that $S$ is called an \emph{algebra retract of
$R$} if there exists a local ring homomorphism $\psi\colon S\to R$ such that
$\varphi\psi=\id_S$.


We start with the next result that contains a part of Theorem~\ref{thm130528a}.

\begin{thm}\label{thm130719a}
Let $S$ be an algebra retract of $R$. Then the following  are equivalent:
\begin{itemize}
\item[(i)]
$\operatorname{AQ-dim}_R S < \infty$.
\item[(ii)]
$\operatorname{AQ-dim}_R S\leq 2$.
\item[(iii)]
$\operatorname{CI*\text{-}dim}_R S<\infty$.
\end{itemize}
If $\ch(k)=0$, then these conditions are equivalent to the following:
\begin{itemize}
\item[(iv)]
$\operatorname{cx}_R S<\infty$.
\end{itemize}
\end{thm}

\begin{proof}
(i)$\implies$(ii) follows from~\cite[Theorem I]{AI2}.
(ii)$\implies$(iii)
follows from Theorem~\ref{thm130316b} and~\ref{fact130625a}.
(iii)$\implies$(iv)
follows from~\ref{fact130625a}, and~\cite[Theorem (5.3)]{AGP1}.

(iii)$\implies$(i)
Assume that $\operatorname{CI*\text{-}dim}_R S<\infty$. Let $R\to R'\leftarrow Q$ be an upper quasi-deformation such that $\pd_Q(S\otimes_R R')<\infty$. Let $S':=S\otimes_R R'$, and let $\ell'$ be the residue field of $S'$. The vanishing of $\operatorname{D}_n(Q\mid R',-)$ for all $n\geq 2$ implies that $\operatorname{D}_n(S'\mid Q,\ell')\cong \operatorname{D}_n(S'\mid R',\ell')$ for all $n\geq 3$. Therefore, we get a commutative diagram
$$
\xymatrix{
&&\operatorname{D}_4(S'\mid Q,\ell')\ar[r]^{\alpha}& \operatorname{D}_4(\ell'\mid Q,\ell')\ar[d]\\
&&\operatorname{D}_4(S'\mid R',\ell')\ar[r]\ar[u]^{\cong}& \operatorname{D}_4(\ell'\mid R',\ell')\ar[d]^{g}\\
\operatorname{D}_5(\ell'\mid R,\ell')\ar[r]^{\gamma}&\operatorname{D}_5(\ell'\mid S,\ell')\ar[r]^{\delta}&\operatorname{D}_4(S\mid R,\ell')\ar[r]^{\beta}\ar[u]^{f}&
\operatorname{D}_4(\ell'\mid R,\ell')
}
$$
in which the bottom row is the Jacobi-Zariski exact sequence of $R\to S\to \ell'$. It follows from Lemma~\ref{lem130316a} that $\alpha=0$. Also, by flat base change $f$ is an isomorphism, and $g$ is an isomorphism by~\cite[Lemma 18]{JM}. Therefore, $\beta=0$. On the other hand, since $S$ is a retract of $R$ and $\operatorname{D}_5(\ell'\mid -,\ell')$ is a functor, $\gamma$ is surjective. This implies that $\delta=0$, and hence, $\operatorname{D}_4(S\mid R,\ell')=0$. Therefore by~\cite[Thorem I]{AI2} (see also \cite[Remark 25]{JM}) we get $\operatorname{AQ-dim}_R S<\infty$, as desired.

(iv)$\implies$(i)
Let $\psi\colon S\to R$ be local homomorphisms such
that $\varphi\psi=\id_S$.
Assume that $\ch(k)=0$ and $\operatorname{AQ-dim}_R S=\infty$. Then using the
Jacobi-Zariski exact sequence of $S\xrightarrow{\psi} R\xrightarrow{\varphi} S$,
it follows that $\psi$ is not
complete intersection at $\ideal{m}$. However, it follows from~\cite[Corollary 4.9]{AI2}
that $\psi$ is an almost small homomorphism. Hence, by~\cite[Theorem 5.6]{AI2} it
has finite weak category;
see, e.g., ~\cite[\S 5]{AI2}
for background on this. Thus, from~\cite[Theorem 5.4 (iv)]{AI2} there is an increasing sequence of
natural numbers $n_j$ such that
the sequence $\{\varepsilon_{n_j}(\psi)\}$ has exponential growth. Since $\ch(k)=0$,
it follows from the Jacobi-Zariski exact sequence of $S\to R\to S$ and~\cite[Corollary (4.3)]{Av} that
the sequence $\{\varepsilon_{n_j+1}(\varphi)\}$ also has exponential growth.

Now let
$R\langle Y\rangle\to S$ be the acyclic closure of $S$ over
$R$. Since $\varphi$ is large,
it follows from~\ref{para130711a} and~\cite[3.3.1]{AI3} that the sequence $\{\operatorname{card}(Y_{n_j})\}$
has exponential growth. By~\cite[Corollary 2.7]{AI3}, we find that
$R\langle Y\rangle$ is the minimal free resolution of $S$ over $R$.
Therefore,
$$
\beta_{n_j}^R(S)=\dim_{k}\operatorname{H}_{n_j}(R\langle
Y\rangle\otimes_{R}k)=\dim_{k}(k\langle
Y\rangle_{n_j}).
$$
It is clear that $\operatorname{card}(Y_n)\leq \dim_k(k\langle
Y\rangle_n)$. Thus the sequence $\{\beta_{n_j}^R(S)\}$ has exponential growth, which is
a contradiction. This shows that $\operatorname{AQ-dim}_R S<\infty$.
\end{proof}

The next result contains the rest of Theorem~\ref{thm130528a}.\footnote{This result has already been obtained by Soto~\cite[Proposition 23]{S}. However, we state it as a corollary of our main results. It is worth mentioning that as we see in Example~\ref{example130316a} this result does not hold in general.}

\begin{cor}\label{cor130719a}
Let $S$ be an algebra retract of $R$. Then $\operatorname{AQ-dim}_R S\leq 2$ if and only if $\cidim_R S<\infty$ and $\operatorname{H}_1(K(\varphi))$ is a free $S$-module.
\end{cor}

\begin{proof}
Proof follows from~\ref{fact130625a}, Theorem~\ref{thm130719a}, and Theorem~\ref{thm130316c}.
\end{proof}

\begin{disc}\label{disc130711a}
At this time, we do not know if the assumption $\ch(k)=0$
is necessary in Theorem~\ref{thm130528a}. However, as we will see in Proposition~\ref{cor130528b}, when we work with $R$ as an algebra retract of its trivial extension, we are able to remove this assumption. This can be done using a result of Herzog.
\end{disc}






In the rest of this section, we work with a specific algebra retract.
Recall that the \emph{trivial extension of $R$  by an $R$-module $M$}, denoted $R\ltimes M$, is the underlying  $R$-module  $R\oplus M$ equipped with a ring structure given by the multiplication rule
$(r,m)\cdot (r',m')=(rr',rm'+r'm)$.
There are ring homomorphisms $\rho\colon R\to R\ltimes M$ with $\rho(r)=(r,0)$ and $\pi\colon R\ltimes M\to R$ with $\pi(r,m)=r$.
Note that the composition $\pi\rho$ is the identity on $R$, that is, $R$ is an algebra retract of $R\ltimes M$.







\begin{lem}\label{lem130711a}
Let $M$ be a finitely generated $R$-module. Then for each non-negative integer $i$ we have $(\mu_R(M))^i\leq\beta^{R\ltimes M}_i(R)$.
\end{lem}

\begin{proof}
By~\cite[Corollary 1]{He} we have $P^{R\ltimes M}_R(t)=1+tP_R^{R\ltimes M}(t)P^R_M(t)$. Hence, by induction on $i$, there exists a sequence $\{\gamma_i\}$ of natural numbers such that $\beta^{R\ltimes M}_i(R)=\gamma_i+(\mu_R(M))^i$. Therefore, $(\mu_R(M))^i\leq\beta^{R\ltimes M}_i(R)$.
\end{proof}

The following result gives a characterization of local
rings admitting non-trivial semidualizing modules. Recall that a finitely generated $R$-module $M$ is called \emph{semidualizing} if $R\cong \Hom[R]MM$ and $\Ext[R]{i}MM=0$ for all $i>0$. For instance, every finitely generated projective $R$-module of rank 1 is
semidualizing. Also, every dualizing module is semidualizing with finite injective dimension.

\begin{prop}\label{cor130528b}
Let $M$ be a finitely generated $R$-module with $\operatorname{Ann}_R(M)=0$ (e.g., $M$ is semidualizing). Then the following conditions are equivalent:
\begin{itemize}
\item[(i)]
$M\cong R$.
\item[(ii)]
$\operatorname{AQ-dim}_{R\ltimes M}R=2$.
\item[(iii)]
$\operatorname{CI*\text{-}dim}_{R\ltimes M}R<\infty$.
\item[(iv)]
$\cidim_{R\ltimes M}R<\infty$.
\item[(v)]
$\operatorname{cx}_{R\ltimes M}R<\infty$.
\end{itemize}
If one of the above conditions holds, then $\operatorname{CI*\text{-}dim}_{R\ltimes M}R=\cidim_{R\ltimes M}R=0$ and $\operatorname{cx}_{R\ltimes M}R=1$.
\end{prop}

\begin{proof}
We  prove the implications (i)$\implies$(ii) and (v)$\implies$(i). The other implications
follow as we wrote in the proof of Theorem~\ref{thm130528a}.

(i)$\implies$(ii)
Assume that $M\cong R$.
It is straightforward to see that $\operatorname{Ker}(\pi)=(x)$, where $x$
is an exact zero divisor in the sense of~\cite{HS}, that is, $\operatorname{Ann}_{R\ltimes M}(x)\cong (R\ltimes M)/x(R\ltimes M)$. Hence,
$\operatorname{AQ-dim}_{R\ltimes M} R=2$ by~\cite[1.7, 1.9]{AHS}.

(v)$\implies$(i)
Assume that $\operatorname{cx}_{R\ltimes M} R<\infty$. Hence, the sequence of the Betti numbers of $R$ over $R\ltimes M$ has polynomial growth. Since, by Lemma~\ref{lem130711a}, for each non-negative integer $i$ we have $(\mu_R(M))^i\leq\beta^{R\ltimes M}_i(R)$,
we conclude that $\mu_R(M)=1$, which implies that $M\cong R$.

Note that when $\operatorname{CI*\text{-}dim}_{R\ltimes M}R<\infty$, then $\operatorname{CI*\text{-}dim}_{R\ltimes M}R=\cidim_{R\ltimes M}R=0$ by~\cite[Lemma 3.14]{SSW} and~\cite{RT}. Note also that when $M\cong R$, $\ker(\pi)$ is generated by an exact zero divisor $x$. Therefore, $$\cdots\xrightarrow{x} R\ltimes R\xrightarrow{x}R\ltimes R\xrightarrow{\pi} R\to 0$$
is a free resolution of $R$ over $R\ltimes R$, that is, $\operatorname{cx}_{R\ltimes R}R=1$.
\end{proof}

\begin{disc}\label{disc130707a}
Let $M$ be a semidualizing $R$-module. If $\operatorname{cx}_{R\ltimes M}R<\infty$, then by Proposition~\ref{cor130528b} we have $\operatorname{cx}_RM=0$. So, a natural question to ask is:\vspace{2mm}

\emph{Does
$\operatorname{cx}_RM<\infty$ imply that $\operatorname{cx}_{R\ltimes M}R<\infty$? In other words, does
$\operatorname{cx}_RM<\infty$ imply that $M\cong R$?\vspace{2mm}}

We do not know the answer of this question at this moment. However, if we remove the semidualizing assumption, then the answer is obviously negative; let $M$ be a free $R$-module of rank $>1$.

When $R$ is Cohen-Macaulay with dualizing module $M$, this question has been asked by Jorgensen and Leuschke~\cite{JL}. In this case, an affirmative answer to this question provides an affirmative answer to the following question of C. Huneke.\vspace{2mm}\footnote{This question has not been posed in any of Huneke's papers and is still open. One can find a different approach to this question in~\cite{BSW}.}

\emph{If $R$ is not Gorenstein, must the sequence $\{\dim_k(\Ext[R]ikR)\}$ of the Bass numbers of $R$ be unbounded?}\vspace{2mm}

In fact, if $R$ is not Gorenstein, then $\operatorname{cx}_R M=\infty$, that is, the sequence of the Betti numbers of $M$ is not bounded. Therefore, our claim follows from the fact that the $\depth(R)+i$-th Bass number of $R$ equals to the $i$-th Betti number of $M$.
\end{disc}

\section*{Acknowledgments}
We are deeply grateful to Javier Majadas for sharing implication (iii)$\implies$(i) in Theorem~\ref{thm130528a}, and for his valuable comments on the first version of this paper, and to Sean Sather-Wagstaff
for for his comments on some parts of this paper.



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