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\begin{document}

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\begin{textblock}{23}(0,0)
\Title{\textcolor{NDSUGreen}{\textbf{Lifting of DG modules over DG algebras and a conjecture of Vasconcelos}}}
\end{textblock}

\begin{textblock}{8}(0,1.0)
\LHead{\textcolor{NDSUGreen}{\textbf{Saeed Nasseh and Sean Sather-Wagstaff}}
\hfil\break\break
\textcolor{NDSUGreen}{\textbf{Department of mathematics}}}
\end{textblock}


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\begin{textblock}{7}(0,2.1)
\hrule\medskip

\Head{\textcolor{orange}{Introduction}}
$(R, \mathfrak m, k)$ is assumed to be a local commutative noetherian ring with unity. When we say $R$ is complete, we mean it is complete in $\mathfrak m$-adic topology.

\textcolor{NDSUGreen}{\textbf{Definition 1.}}
Let $R\to S$ be a homomorphism of rings and let $M$ be a finitely generated $S$-module. Then the finitely generated $R$-module $L$ is called a \emph{lifting} of $M$ to $R$ if\vspace{-6mm}
    \begin{enumerate}
    \item{} $M\cong S\otimes_{R}L$, and
    \item{} Tor$^{R}_i(S,L)=0$ for all integers $i>0$.
    \end{enumerate}\vspace{-6mm}
        The $S$-module $M$ is said to be \emph{liftable} to $R$, when such an $R$-module $L$ exists.

\textcolor{NDSUGreen}{\textbf{Theorem 2.}}
(Auslander, Ding and Solberg, 1993) Suppose $R$ is complete and $\textbf{x}=x_1,\cdots,x_n$ is an $R$-regular sequence in $\mathfrak m$. Let $S=R/\textbf{x} R$ and $M$ be a finitely generated $S$-module. If Ext$\ ^2_{S}(M,M)=0$, then $M$ is liftable to $R$.

\textcolor{NDSUGreen}{\textbf{Definition 3.}}
The finitely generated $R$-module $C$ is \emph{semidualizing} if it satisfies the following:\vspace{-6mm}
\begin{enumerate}
\item The homothety map $\chi^R_C\colon R\to$ Hom$_R(C,C)$ given by $\chi^R_C(r)(c)=rc$ is an isomorphism, and

\item Ext$^i_R(C,C)=0$ for all $i>0$.
\end{enumerate}\vspace{-4mm}

\textcolor{NDSUGreen}{\textbf{Conjecture 4.}} (W. V. Vasconcelos, 1974)
The number of isomorphism classes of semidualizing modules over a Cohen-Macaulay local ring is finite.

\textcolor{NDSUGreen}{\textbf{Theorem 5.}} (L.~W.~Christensen and S.~Sather-Wagstaff, 2008)
If $R$ is \emph{Cohen-Macaulay} and \emph{equicharacteristic}, then the number of isomorphism classes of semidualizing modules is finite.\vspace{-3mm}

\textcolor{NDSUGreen}{\textbf{Sketch of Proof.}}
Pass to the completion of $R$ to assume that $R$ is complete.
Let $\textbf{x}$ be a system of parameters for $R$;
this is a maximal $R$-regular sequence. Using the Theorem of Auslander, Ding and Solberg, there is a one-to-one correspondence between the set of isomorphism classes of semidualizing $R$-modules and that of $R/\textbf{x}R$-modules. So we pass to the quotient $R/\textbf{x}R$
to assume that $R$ is artinian.
Since $R$ is equicharacteristic and artinian, Cohen's structure theorem implies $R$ is a $k$-algebra.
Now the theorem follows from a result in representation theory.

\textcolor{NDSUGreen}{\textbf{General form of Vasconcelos' Conjecture.}}
The number of isomorphism classes of semidualizing modules over a local ring is finite.

To answer this question, we consider  Differential Graded algebras and Differential Graded modules (DG algebras and DG modules).
\bigskip
%\hrule
\end{textblock}

\begin{textblock}{7}(0,8)
\hrule\medskip

\Head{\textcolor{orange}{DG Algebras And DG Modules}}
Complexes of $R$-modules ($R$-complexes for short) are indexed homologically:\vspace{-3mm}
\begin{equation*}
M = \cdots \overset{\partial_{n+2}^{M}}\longrightarrow M_{n+1} \overset{\partial_{n+1}^{M}}\longrightarrow M_n
\overset{\partial_{n}^{M}}\longrightarrow M_{n-1} \overset{\partial_{n-1}^{M}}\longrightarrow \cdots.
\end{equation*}
and if $m\in M_i$ we write $|m|=i$.

\textcolor{NDSUGreen}{\textbf{Definition 6.}}
A \emph{commutative differential graded algebra over} $R$ (DG $R$-algebra for short)
  is a bounded below ($A_i=0$ for $i<0$) $R$-complex $A$ equipped with a
  chain map
  $\mu^A\colon A\otimes_RA\to A$ denoted $\mu^A(a\otimes b)=ab$ (which is called the product)
  that is\vspace{-6mm}
  \begin{enumerate}
  \item \textbf{Associative:} for all $a,b,c\in A$ we have $(ab)c=a(bc)$;
  \item \textbf{Unital:}  there is an element $1\in A_0$ such that for all $a\in A$ we have $1a=a$;
  \item \textbf{Graded Commutative:} for all $a,b\in A$ we have $ab = (-1)^{|a||b|}ba$ and $a^2=0$ when
      $|a|$ is odd.
\end{enumerate}\vspace{-6mm}
The fact that the product on $A$ is a chain map
says that $\partial^A$ satisfies the \emph{Leibniz rule}:\vspace{-3mm}
$$
\partial^A_{|a|+|b|}(ab)=\partial^A_{|a|}(a)b+(-1)^{|a|}a\partial^A_{|b|}(b).
$$
\textcolor{NDSUGreen}{\textbf{Example 7.}}
For elements $a_1,\cdots,a_n\in  \mathfrak m$,
the \emph{Koszul complex} $K=K^R(a_1,\cdots,a_n)$ is the exterior algebra on $R^n$, considered as an $R$-complex\vspace{-3mm}
$$
K=0\to K_n\overset{\partial^K_n}\longrightarrow K_{n-1}\overset{\partial^K_{n-1}}\longrightarrow\cdots\overset{\partial^K_2}\longrightarrow K_1\overset{\partial^K_1}\longrightarrow K_0\to 0.
$$
Here $K_0 =K_n = R$ and $K_1 = R^n$. Given   a basis  $e_1,\cdots,e_n\in K_1$, for each $2\leq i\leq n-1$ the
\bigskip
\end{textblock}
\begin{textblock}{7}(8,1.8)
\medskip
module $K_i=\wedge^i(K_1)$  is a free $R$-module with basis\vspace{-3mm}
$$
\{e_{j_1}\wedge e_{j_2}\wedge \cdots \wedge e_{j_i}\ |\ 1\leq j_1\leq j_2\leq \cdots\leq j_i\leq n\}
$$
and the differential $\partial_i^K$ is given as follows:\vspace{-3mm}
$$
\partial_i^K(e_{j_1}\wedge e_{j_2}\wedge \cdots \wedge e_{j_i})=\sum_{z=1}^i (-1)^{z+1}a_{j_z}(e_{j_1}\wedge e_{j_2}\wedge \cdots \wedge e_{j_{z-1}}\wedge e_{j_{z+1}}\wedge \cdots \wedge e_{j_i}).
$$
Now $K$ is
a DG $R$-algebra with product given by the wedge product.

\textcolor{NDSUGreen}{\textbf{Remark 8.}}
The natural map $R\to K$ is a "morphism of DG $R$-algebras". This morphism is our substitute for the natural surjection $R\to R/\textbf{x}R$.

\textcolor{NDSUGreen}{\textbf{Facts 9.}}\vspace{-6mm}
\begin{enumerate}
\item $K$ is like a flat ring extension of $R$, and for each $i$, H$_i(K)$ is a finite dimensional vector space over the residue field $k$.

\item $K\simeq T$ where $T$ is a finite dimensional DG algebra over $k$.
\end{enumerate}\vspace{-4mm}

\textcolor{NDSUGreen}{\textbf{Definition 10.}}
Let $A$ be a DG $R$-algebra. A \emph{differential graded module over} $A$
  (DG $A$-module for short) is an $R$-complex $M$ with a
chain map
  $\mu^M\colon A\otimes_AM\to M$ with $am:=\mu^M(a\otimes m)$.
The map $\mu^M$ is the scalar product on $M$.
The fact that the scalar product on $M$ is a chain map
says that $\partial^M$ satisfies the Leibniz rule:
  $$
  \partial^M_{|a|+|m|}(am)=\partial^A_{|a|}(a)m+(-1)^{|a|}a\partial^M_{|m|}(m)
  $$
for all $a\in A$, $m\in M$. A bounded below DG $A$-module $N$ is \emph{semifree} if its underlying $A^{\natural}$-module $N^{\natural}$ has a basis $\{e_{\lambda}\}_{\lambda\in \Lambda}$.

\textcolor{NDSUGreen}{\textbf{Facts 11.}}\vspace{-6mm}
\begin{enumerate}
 \item
  If $M$ is an $R$-complex, then $K\otimes_RM$ is a DG $K$-module.

  \item $K\simeq T$ implies that DG $K$-modules are in one-to-one correspondence with DG $T$-modules.
  \end{enumerate}\vspace{-4mm}

\textcolor{NDSUGreen}{\textbf{Definition 12.}}
Let $A$ be a ``noetherian'' DG $R$-algebra, and let $M$ be a DG $A$-module.
Then  $M$ is a \emph{semidualizing
DG $A$-module} if $M$ is homologically finite
and the homothety morphism
$\chi^M\colon A\to$ \textbf{R}Hom$_A(M,M)$ induced by $X^M\colon A\to$ Hom$_A(M,M)$ with
$X^M(a)(m)=am$ is a quasiisomorphism. A \emph{semidualizing $R$-complex} is a semidualizing DG $R$-module.

\textcolor{NDSUGreen}{\textbf{Facts 13.}}\vspace{-6mm}
\begin{enumerate}
\item
  $M$ is a semidualizing $R$-module if and only if $K\otimes_RM$ is a semidualizing DG $K$-module.

  \item
  If $M$ and $M'$ are semidualizing $R$-modules and $K\otimes_RM\simeq K\otimes_R M'$, then $M\cong M'$.
  \end{enumerate}\vspace{-4mm}
  %\textcolor{NDSUGreen}{\textbf{Corollary 15.}}
%There is a one-to-one map form the set
%$\mathfrak{S}(R)$ to the set $\mathfrak{S}(K)$.
%\bigskip
%\hrule
\end{textblock}

\begin{textblock}{7}(8,8.5)
\hrule
\medskip
\Head{\textcolor{orange}{Solution To The General Form Of The Conjecture}}
Let $\ell$ be an algebraically closed field and
$$
A=0\lora \ell^{\alpha_m}\overset{\partial_m}\lora \ell^{\alpha_{m-1}}\overset{\partial_{m-1}}\lora \cdots \overset{\partial_{1}}\lora \ell^{\alpha_0}\lora 0
$$
be a finite dimensional commutative DG algebra over $\ell$.
Also let $W=\bigoplus_{i=0}^s W_i$
be a graded module with $\dim[\ell](W_i)=r_i$ for every $0\leq i\leq s$.
Let $\mathbf{r}=(r_0,\cdots,r_s)$ and $\od^A_{\mathbf{r}}$ be the set of all DG $A$-module structures on $W$. $\od^A_{\mathbf{r}}$ is the set of rational points of an algebraic scheme $\Mod_{\mathbf{r}}^A$ over $\ell$,
which is described in the functorial point of view as follows; for any finite dimensional commutative DG algebra $A'$ over $\ell$, we have
$$
\Mod_{\mathbf{r}}^A(A')=\Big\{\text{DG $A'\otimes_{\ell} A$-module structures on $A'\otimes_{\ell} W$}\Big\}.
$$
\textcolor{NDSUGreen}{\textbf{Theorem 14.}}
There are only finitely many isomorphism classes of semidualizing DG $A$-modules with the fixed
dimension vector $\mathbf{r}=(r_0,\cdots,r_s)$.

\textcolor{NDSUGreen}{\textbf{Sketch of Proof.}}
It can be seen that the orbit of every semidualizing DG $A$-module in $\od^A_{\mathbf{r}}$ under $\gl(W)$ is an open subscheme of
$\Mod^A_{\mathbf{r}}$. Let $\{u_{\lambda}\}_{\lambda\in \Lambda}$ be the set of these open sets, which are disjoint.
$\{u_{\lambda}\}_{\lambda\in \Lambda}$ is an open cover for the open subset $\bigcup_{\lambda\in \Lambda}u_{\lambda}$ of $\Mod^A_{\mathbf{r}}$.
Thus it has a finite subcover. Since the open sets $u_{\lambda}$'s are disjoint, $\Lambda$ is a finite set.
\bigskip
%\hrule
\end{textblock}

\begin{textblock}{7}(16,2)
%\Head{Conclusions}


\textcolor{NDSUGreen}{\textbf{Theorem 15.}}
Let $B$ be  a DG $R$-algebra such that each $B_i$ is free over $R$ of finite rank. For an element $t\in \mathfrak m$, let
$K^R(t)$  be the Koszul complex $0\to K_1\xra{t} K_0\to 0$
and let $D$  denote the DG $R$-algebra $K^R(t)\otimes_RB$. If $R$ is complete and $N$ is a semifree
DG $D$-module such that Ext$_D^2(N,N)=0$, then $N$ is
liftable to $B$.

\textcolor{NDSUGreen}{\textbf{Theorem 16.}}
Let $R$ be complete and assume that $B$ and $D$ are the DG $R$-algebras considered in Theorem 15. Let $N$ be a semifree
DG $D$-module such that Ext$_D^{d}(N,N)=0$ for some even integer $d$.
If $M$ is a lifting of $N$ to $B$, then
Ext$^d_B(M,M)=0$.

\textcolor{NDSUGreen}{\textbf{Theorem 17.}}
Suppose that $R$ is complete and $\underline{t}=t_1,\cdots,t_n$ is a sequence of elements of $\mathfrak m$.
If $N$ is a semidualizing DG module over $K:=K^R(\underline{t})$, then there
exists a semidualizing $R$-complex $M$ which is a lifting of $N$ to $R$.

\textcolor{NDSUGreen}{\textbf{Sketch of Proof.}}
We can replace $N$ by a semifree DG resolution over $K$ and assume that $N$ is semifree DG $K$-module.
The existence of an $R$-complex $M$ which is a lifting of $N$ to $R$ follows
by induction on $n$ from the previous two theorems. But note that
\begin{eqnarray*}
K\ \ \simeq\ \ \DHom[K]NN&=&{\DHom[K]{K\otimes_RM}{K\otimes_RM}}\\
&\simeq& {\DHom[R]{M}{K\otimes_RM}}\\
&\simeq& {K\otimes_R \DHom[R]{M}{M}},
\end{eqnarray*}
where the last isomorphism is a special case of tensor-evaluation. Therefore,
$R\simeq \DHom[R]{M}{M}$.

\textcolor{NDSUGreen}{\textbf{Notation 18.}}
Let $A$ be a commutative local DG algebra. $\mathfrak{S}(A)$ (resp. $\mathfrak{S}(R)$) denotes the set of shift-isomorphism classes of semidualizing
DG $A$-modules (resp. semidualizing $R$-complexes).

\textcolor{NDSUGreen}{\textbf{Corollary 19.}}
There is a one-to-one and onto map from the set
$\mathfrak{S}(R)$ to the set $\mathfrak{S}(K)$.

\textcolor{NDSUGreen}{\textbf{Theorem 20.}}
There are only finitely many
semidualizing $R$-complexes up to shift isomorphism.\vspace{-3mm}

\textcolor{NDSUGreen}{\textbf{Sketch of Proof.}}
Without loss of generality we may assume that $R$ is complete and $k$ is algebraically closed.

Let $\mathbf{a}=a_1,\cdots,a_d$ be a minimal set of generators for $m$ and set $K=K^R(\mathbf{a})$.
We know that $K\simeq T$ where $T$ is of the form
$$
T=0\lora k^{\alpha_m}\overset{\partial_m}\lora k^{\alpha_{m-1}}\overset{\partial_{m-1}}\lora \cdots\lora k^{\alpha_1}\overset{\partial_1}\lora k\lora 0
$$
where each $\alpha_i$ is a nonnegative integer. In fact $T$ is a finite dimensional DG algebra over $k$. Now by Corollary 19 and Fact 11(2) we get
the following equivalences
$$
\mathfrak{S}(R)\overset{\sim}\longleftrightarrow\mathfrak{S}(K)\overset{\sim}\longleftrightarrow \mathfrak{S}(T).
$$
%In other words, finiteness of each one of $\mathfrak{S}(R)$, $\mathfrak{S}(K)$ or $\mathfrak{S}(T)$ implies finiteness of the rest.
Now suppose that $C$ is a semidualizing $R$-complex and denote by $\widetilde{C}$ the semidualizing DG $T$-module corresponding to $C$.
Without loss of generality we can assume that $\inf(C)=0$.
By using a minimal semifree DG resolution, without loss of generality we can suppose that $\widetilde{C}$ is a semifree DG $T$-module.
We also can replace $\widetilde{C}$ by its left truncation $\tau(\widetilde{C})_{\leq n}$  where $n=\cmd R+\edimR$ and suppose that
$$
\widetilde{C}=0\to \widetilde{C}_n\to \widetilde{C}_{n-1}\to \cdots\to \widetilde{C}_1\to \widetilde{C}_0\to 0
$$
where for every $i$, $\widetilde{C}_i=\bigoplus_{j=0}^i T^{\beta'_{i-j}}_{j}$.
Now it can be seen that
$$
\sum_{i=0}^n\dim[k]{(\widetilde{C_i})}\leq \lgt[]{(T)}\Big(\sum_{i=0}^n \mu_R^{i+\dep R}(R)\Big)
$$
where the right-hand side is a constant. It is obvious that there are only finitely many $(r_0,\cdots,r_n)\in \mathbb{N}^{n+1}$ with
$\sum_{i=0}^nr_i\leq \lgt[]{(T)}\Big(\sum_{i=0}^n \mu_R^{i+\dep R}(R)\Big)$. Therefore, Theorem
14 implies that $\mathfrak{S}(T)$ and hence $\mathfrak{S}(R)$ is finite.
\end{textblock}


\end{document}
