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\graduationmonth{August}
\graduationyear{2009}
\defensedate{July 17, 2009}
\author{Meri Trema Hughes}
\committee{Dr. David Jorgensen}{Dr. Gaik Ambartsoumian}{Dr. Ren-Cang Li}{Dr. Barbara Shipman}{Dr. Michaela Vancliff}
\title{The Uniqueness of Minimal Acyclic Complexes}
\begin{document}
% \signaturepagePhD Do not need if electronicly submitted
\copyrightpage
\titlepage
\begin{acknowledgements}
I am so very grateful to Dr. David Jorgensen, undoubtedly the most patient advisor on Earth. Thank you for sharing your brilliance.
I am the luckiest mom in the world. It's true. Marshall, my all-star. Ranger, my joy. Thank you for making this part of my life so much fun.
\end{acknowledgements}
\begin{abstract}
In this paper, we discuss conditions for uniqueness among minimal acyclic complexes of finitely generated free modules over a commutative local ring which share a common syzygy module. Although such uniqueness exists over Gorenstein rings, the question has been asked whether two minimal acyclic complexes in general can be isomorphic to the left and non-isomorphic to the right. We answer the question in the negative for certain cases, including periodic complexes, sesqui-acyclic complexes, and certain rings with radical cube zero. In particular, we investigate the question for graded algebras with Hilbert series $H_R(t)=1+et+(e-1)t^2,$ and such monomial algebras possessing a special generator.
\end{abstract}
\tableofcontents
\chapter{Introduction}
With $R$ a local ring and $M$ a finitely generated $R$-module, it is well known that minimal free resolutions of $M$ are unique up to isomorphism. If we consider $M$ as a syzygy module in a minimal acyclic complex, the left-hand side of the complex is precisely its unique free resolution. As a result, minimal acyclic complexes having a syzygy module $M$ in common are ``unique to the left" of that module. However, whether $M$ can exist as a syzygy module in two non-isomorphic complexes has yet to be concluded. In other words, is it possible for the following to exist, where $\cong$ refers to the fact that modules are isomorphic and $\circlearrowleft$ refers to the fact that the squares commute:
\[
\xymatrixrowsep{2pc}
\xymatrixcolsep{1pc}
\xymatrix {\cdots \ar@{->}[r]&C_{2}\ar@{->}[rr]\ar@{->}[dd]^{\cong}&&C_{1}\ar@{->}[rr]\ar@{->}[dd]^{\cong}&&C_0\ar@{->}[rr]\ar@{->}[dd]^{\cong}\ar@{->>}[dr]&&C_{-1}\ar@{->}[rr]\ar@{->}[dd]^{\not{\cong}}&&C_{-2}\ar@{->}[r]\ar@{->}[dd]^{\text{{\normalsize or}}\,\,\,\,\,\not{\cong}}&\cdots \\&&\circlearrowleft&&\circlearrowleft&&M\ar@{^{(}->}[ur]\ar@{^{(}->}[dr]&&\text{or}\,\,\,\not{\circlearrowleft}&\\
\cdots \ar@{->}[r]&D_{2}\ar@{->}[rr]&&D_{1} \ar@{->}[rr]&&D_0\ar@{->}[rr]\ar@{->>}[ur]&&D_{-1}\ar@{->}[rr]&&D_{-2}\ar@{->}[r]&\cdots .}
\]
Thus, one major focus of this paper is to determine the conditions under which a minimal acyclic complex is ``unique to the right", or ``has no branching".
\begin{wordle}
A nonzero minimal acyclic complex of free modules \[C:\quad
\cdots \to C_1 \xra{d_1^C} C_0 \xra{d_0^C} C_{-1} \to \cdots,
\] \textbf{branches} if there exists a minimal acyclic complex \[
D:\quad \cdots \to D_1 \xra{d^D_1} D_0 \xra{d^D_0} D_{-1} \to \cdots
\] such that $C_{\ge s} \cong D_{\ge s}$ some $s\in \mathbb Z$, but $C \ncong D.$ If this is the case, we say that $C$ branches at $r$ if $r$ is the minimal integer such that $C_{\geq r} \cong D_{\geq r}.$
\end{wordle}
We attempt to explore this concept for specific types of rings and complexes. After defining the question of branching and considering the necessary properties for the ring in chapter three, we examine the traits of a periodic complex in chapter four. Chapter five provides branching results that relate to periodicity. We define the concept of pushing a matrix in a minimal acyclic complex forward in chapter six, while chapters seven, eight, and nine explore the branching results of monomial algebras with Hilbert series $H_R(t)=1+et+(e-1)t^2.$ We conclude the paper with a discussion of sesqui-acyclic complexes.
\chapter{Preliminaries}
Throughout this paper, assume $R$ is a commutative ring with unity. This chapter contains preliminary concepts which will be used in the later chapters. Most of these definitions can be found in [7] and [12].
\section{Properties and Classes of Rings}
The types of rings we study in this paper are noetherian local rings that are also graded. To define this idea, we start by defining certain useful properties of rings, and use these definitions to expand our classification of rings.
\subsection{Properties of General Commutative Rings}
\begin{itemize}
\item A ring is {\it noetherian} if every ideal is finitely generated.
\item The {\it Krull dimension} of a ring $R,$ $\dim (R)=d,$ also called simply the {\it dimension}, is the length of the longest chain of prime ideals in $R,$ $$p_0\subset p_1 \subset \cdots \subset p_d.$$ We always assume that $\dim R$ is finite.
\end{itemize}
\subsection{Local Rings}
\begin{itemize}
\item A {\it maximal ideal} of a ring is a proper ideal that is not contained in any other proper ideal.
\item A Noetherian ring that has a unique maximal ideal $\m$ is called a {\it local ring.} In this case, we let $k$ denote the residue field $R/\m.$
\item The {\it embedding dimension} of a local ring $R$ is the finite number $\edim(R)=\dim_k(\m/\m^2).$ In other words, the minimal number of generators of the maximal ideal of $R.$
\item The {\it codimension } of a local ring $R$ is the number $\edim(R) - \dim (R).$
\item If $M$ is a finitely generated $R$-module, a {\it regular sequence} on $M$ is a sequence of elements $x_1,\dots,x_n$ in $R$ such that $x_1$ is a nonzerodivisor on $M$ and each $x_i$ is a nonzerodivisor on $M/{(x_1,\dots,x_{i-1})M},$ where $(x_1,\dots,x_n)M\neq M.$
\item Let $M$ be a module over a local ring $(R,\m,k).$ Then $\soc M=(0:\m)_M \cong \Hom_R(k,M)$ is called the {\it socle} of $M.$ In other words, it is the annihilator in $M$ of the maximal ideal.
\item The {\it depth} of a local ring $R$ is the length of the longest regular sequence on $R$ contained in $\m.$
\end{itemize}
\subsection{Classes of Local Rings}
\begin{itemize}
\item For a local ring $(R,\m)$ of dimension $d,$ $R$ is a {\it regular ring} if $\m$ can be generated by exactly $d$ elements. When $\dim_k(R)=\edim(R),$ $R$ is regular.
\item A local ring is a {\it complete intersection} if it is the quotient of a regular local ring by an ideal generated by a regular sequence.
\item A local ring $R$ is called a {\it Gorenstein ring} if it has finite injective dimension as a module over itself.
\item When $\depth (R) = \dim (R),$ the ring $R$ is a {\it Cohen-Macaulay ring,} which is the class of rings studied in this paper.
\end{itemize}
\subsection{Graded Rings}
In addition, we focus on noetherian local rings that are also graded. We only consider non-negatively graded rings.
\begin{itemize}
\item A {\it graded ring} is a ring $R$ together with a direct sum decomposition $R=R_0\oplus R_1\oplus R_2\oplus \cdots,$ where $R_i$ is called the $i$th homogeneous component of $R,$ such that $R_iR_j \subset R_{i+j},$ for $i,j \geq 0.$ The ring of polynomials $R=R_0[x_1, \cdots, x_e]$ is a graded ring where $R_i$ consists of homogeneous polynomials of degree $i$ and $R_0$ is a ring. The examples used in this paper are primarily quotients of polynomial rings.
\item If $R$ is a graded ring, then a {\it graded module} over $R$ is an $R$-module $M$ that can be decomposed as $M=\oplus_{i=0}^\infty M_i,$ where $M_i$ is called the $i$th homogeneous component of $M,$ such that $R_iM_j \subset M_{i+j}$ for all $i,j.$
\item Such a module is {\it free} if it has a linearly independent generating set over the graded ring consisting of homogeneous elements.
\item If $R$ is graded, $R_0=k$ is a field, and $M$ is a graded $R$-module, then the formal power series $H_M(t)=\Sigma_{i=0}^\infty \dim_k(M_i)t^i$ is called the {\it Hilbert Series of $M$.}
\item The module $R^n:=\oplus^n R$ is a graded free $R$-module with {\it standard basis} consisting of $e_i,1 \leq i \leq n,$ where $e_i=(0, \cdots,0,1,0,\cdots,0)$ has $1$ in the $i$th component and zero elsewhere.
\item A homomorphism $\phi: M\xra{} N$ of graded $R$-modules is {\it homogeneous of degree d} if $\phi(M_i) \subseteq N_{i+d}$ for every $i.$ An $R$-module homomorphism $\phi:R^n \xra{}R^m$ of free $R$-modules is represented with respect to the standard bases of $R^n$ and $R^m$ by a matrix $[f_1 \cdots f_n]$ where the $f_i \in R^m$ are columns. If $R$ is graded, then $\phi$ being homogeneous implies that the entries of $[f_1 \cdots f_n]$ are homogeneous elements of $R.$
\item Moreover, if $R_0=k,$ then $\phi|_{R_i^n}:R_i^n \xra{} R^m_{i+d}$ is also a linear transformation of vector spaces, and thus fixing vector space bases of $R_i^n$ and $R_{i+d}^m$ one may represent the linear transformation $\phi|_{R_i^n}$ by a matrix $T,$ the associated linear transformation of $\phi$ of degree $i.$
\end{itemize}
\section{Homological Algebra}
We now give some general definitions from homological algebra, which is the study of chain complexes of algebraic structures, in our case, $R$-modules.
\subsection{Complexes}
\begin{itemize}
\item A {\it complex} is a sequence of $R$-modules and $R$-linear maps
\[
C:\quad \cdots \to C_{i+1}\xra{d_{i+1}^{C}} C_i\xra{d_i^{C}}
C_{i-1}\to\cdots
\]
with $d_i^C\circ d_{i+1}^C=0$ for all $i$, alternately, the image of $d_{i+1}^C$ is contained in the kernel of $d_i^C.$ The maps $d^C_i$ are called the {\it differentials}.
\item Let $C$ and $D$ be complexes. A {\it homomorphism of complexes} $f: C \to D$ is a set of homomorphisms $f_n: C_n \to D_n$ such that for every $n$ the following diagram commutes:
\[
\xymatrixrowsep{2pc}
\xymatrixcolsep{2pc}
\xymatrix{\cdots \ar@{->}[r] &
C_{n} \ar@{->}[r]^{d^C_{n}} \ar@{->}[d]^{f_{n}}&
C_{n-1} \ar@{->}[r] \ar@{->}[d]^{f_{n-1}}&\cdots\\
\cdots \ar@{->}[r]&
D_{n} \ar@{->}[r]^{d^D_n}&
D_{n-1} \ar@{->}[r]&\cdots,}\]
more formally, $f_{n-1}d^C_n=d^D_nf_n.$
\item A complex is {\it exact at} $C_i$ if $\Ima d^C_{i+1} = \ker d^C_i.$ If $C_i$ is exact for each $i,$ then the complex is said to be {\it exact.}
% \item Suppose $L \xra{\phi} M$ and $M \xra{\psi} N$ are homomorphisms of $R$-modules. The sequence $L \xra{\phi} M \xra{\psi}N$ is called {\it exact} at $M$ if $\ker (\psi) = \Ima (\phi).$ If a sequence of homomorphisms $F_n \xra{\sigma_n}F_{n-1}\xra{\sigma_{n-1}} F_{n-2}\xra{\sigma_{n-2}} \cdots \xra{\sigma_2} F_1\xra{\sigma_1} F_0$ is exact at $F_1, F_2, \cdots, F_{n-1},$ then the sequence is said to be {\it exact.}
\item The {\it homology} $H_i(C)$ at $C_i$ is the module ${\ker d^C_i}{/\Ima d^C_{i+1}}.$
\item The homology of the complex is given by $H(C)=\oplus H_i(C).$
\end{itemize}
\subsection{Projective and Free Resolutions}
\begin{itemize}
\item An $R$-module $P$ is {\it projective} if for every epimorphism of $R$-modules $\alpha:M\xra{}N$ and every map $\beta:P\xra{}N,$ there exists a map $\gamma:P\xra{}M$ such that $\beta=\alpha\gamma,$ as in the following figure:$$\xymatrixrowsep{4pc}
\xymatrixcolsep{2pc}
\xymatrix {&P\ar@{->}[d]^{\beta}\ar@{-->}[dl]^{\gamma}\\M\ar@{->>}[r]^{\alpha}&N.}$$
\item Free modules are projective. To see this, if $P$ is free on a set of generators $p_i,$ then choose elements $q_i$ of $M$ that map to $\beta(p_i) \in N,$ and let $\gamma$ send $p_i$ to $q_i.$
\item A {\it projective resolution} of an $R$-module $M$ is a complex $${F}: \cdots \xra{}F_n \xra {d^F_n} \cdots \xra{} F_1 \xra{d^F_1} F_0$$ of projective $R$-modules such that $\Coker d^F_1 \cong M$ and $F$ is an exact complex. If, in addition, each $F_i$ is free, $F$ is called a {\it free resolution of} $M.$ If $M$ is finitely generated and $R$ is noetherian, then each $F_i$ can be chosen to be finitely generated. Free resolutions serve to compare projective modules with free modules.
\item If for some $n < \infty,$ we have $F_{n+1}=0,$ but $F_i \neq 0$ for $0 \leq i \leq n,$ then $F$ is a {\it finite resolution of length $n.$}
\item Assume that $R$ is either local or graded with (homogeneous) maximal ideal $\m.$ Then $F,$ as above, is {\it minimal} if entries of each matrix representing the $d_i^F$ are in $\m.$
\item Assuming that $\m$ is finitely generated and $R$ is neotherian, the $\rank F_i=b_i$ are called the {\it Betti numbers} of $M.$
\item The {\it Poincare series} of the $R-$module $M$ is the power series in $t$, $$P_M^R(t)=\Sigma_{i\geq 0}b_it^i=b_0+b_1t+b_2t^2+\cdots,$$ where the $b_i$ are the Betti numbers.
\item An $R$-module $I$ is {\it injective} if for every monomorphism of $R$-modules $\alpha: N\xra{}M$ and every homomorphism of $R$-modules $\beta:N\xra{}I,$ there exists a homomorphism of $R$-modules $\gamma:M \xra{}I$ such that $\beta = \gamma \alpha,$ as in the following figure:
$$\xymatrixrowsep{4pc}
\xymatrixcolsep{2pc}
\xymatrix {N\hskip .1in\ar@{>->}[r]^{\alpha}\ar@{->}[d]^{\beta}&M\ar@{-->}[dl]^{\gamma}\\I&.}$$
\item If $M$ is an $R$-module, we may embed $M$ in an injective module $I_0.$ We may then embed the cokernel, $I_0/M,$ in an injective module $I_1.$ Continuing in this way, we get an {\it injective resolution} $$0 \xra{}M\xra{}I_0\xra{}I_1\xra{}I_2\xra{}\cdots$$ of $M;$ that is, an exact sequence of the given form in which all the $I_i$ are injectives.
\end{itemize}
\subsection{Minimal Acyclic Complexes}
\begin{itemize}
\item An {\it acyclic complex} of free $R$-modules is a complex
\[
C\quad \cdots \to C_2\xra{d_2^{C}} C_1\xra{d_1^{C}}
C_0\xra{d_0^{C}} C_{-1}\xra{d_{-1}^{C}} C_{-2}\to\cdots
\]
with $C_i$ finitely generated and free for each $i$ and H$(C)=0$.
\item If $M$ and $N$ are $R$-modules, then $\Hom_R(M,N)$ is the abelian group of all homomorphisms from $M$ to $N.$ Since $R$ is commutative, it is itself an $R$-module by the property $(rf)(m)=rf(m)=f(rm)$ for $r\in R$ and $f \in \Hom_R(M,N).$
\item If $$f:M \to N$$ is a homomorphism of $R$-modules, then we have the mapping $$f^*: \Hom (N,R) \to \Hom (M,R),$$ where
$$f^*(\theta)=\theta \circ f.$$ We set $M^*=\Hom(M,R)$ and call $M^*$ the {\it dual} of $M$ and $f^*$ the {\it dual} of $f.$ We have the dual $C^*=\Hom_R(C,R)$:
$$C^*= \cdots C_{n-1}^*\xra{d_n^*}C^*_n\xra{d_{n+1}^*} C^*_{n+1}\xra{}\cdots.$$
\item If $R^n \xra{f}R^m$ is represented by $A$ with respect to the dual bases of $R^n$ and $R^m,$ then $\Hom (R^m,R) \xra{f^*} \Hom(R^n,R)$ is represented with respect to the standard bases of $\Hom(R^m,R)$ and $\Hom(R^n,R)$ by $A^T.$
%\item If $\alpha:F'\xra{}F$ and $\beta:F \xra{}F''$ are maps of complexes, with $\beta\alpha=0,$ then we say that $$0 \xra{}F'\xra{\alpha}F\xra{\beta}F''\xra{}0$$ is a {\it short exact sequence of complexes} if for each $i$ the sequence $$0 \xra{}F_{i}'\xra{\alpha}F_{i}\xra{\beta}F_{i}''\xra{}0$$ is exact.
\item An acyclic complex of free $R$-modules $C$ satisfying $H(C^*)=0$, where $C^*=\Hom_R(C,R)$ is called {\it totally acyclic}, or a {\it complete resolution.}
\item For an acyclic complex $C,$ if we have $H_i(C^*)=0$ for $i \gg 0$ then $C$ is called a {\it sesqui-acyclic} complex.
\item Assume $(R,\m)$ is local with maximal ideal $\m,$ or assume $(R,\m)$ is graded with homogeneous maximal ideal $\m.$ An acyclic complex is {\it minimal} if $\im d_i\subseteq \m R^{d_{i-1}}$ for all $i$.
\item The {\it shift functor}, notated $\Sigma^r$, takes complexes over $R$ to complexes over $R$, acting on both the modules and the morphisms, i.e. $C \mapsto \Sigma^rC$, and ${f:C \to D} \mapsto \{\Sigma^r f:\Sigma^r C\to \Sigma^r D\}$, where
$(\Sigma^rC)_i=C_{i-r}$, and $d_i^{\Sigma^rC}=(-1)^rd_{i-r}^C$.
\item For a complex $C$ we define the {\it truncated complex} $C_{\geq r}$ to be the complex with $(C_{\geq r})_i=C_i$ if $i \geq r,$ and $0$ if $i < r.$
\end{itemize}
\subsection{Syzygies}
\begin{itemize}
\item The element $z \in M$ is called a {\it syzygy} of an $R$-module homomorphism $\phi:M\xra{}N$ if $z \in \Ker(\phi).$ If $\phi:R^n \xra{} R^m$ is represented by $[s_1\cdots s_n]$ then a syzygy takes the form $(c_1,\cdots,c_n) \in R^n$ such that $c_1f_1+\cdots +c_nf_n=0$ in $R^m.$
\item Let $M$ and $N$ be finitely generated $R$-modules. Then $M$ is called an
{\em $n$th syzygy module} (of $N$) if there is an exact sequence
\[
\cdots \to C_n \xra{d^C_n}C_{n-1}
\to \cdots \to C_1\xra{d^C_1} C_0 \to N \to 0
\]
with the $C_i$ finitely generated and free, and $M\cong \im d^C_n$.
\item We say that $M$ is an {\it infinite syzygy module}
if there exists a minimal acyclic complex of projective $R$-modules
\[
\cdots \to C_1 \xra{d^C_1} C_0 \xra{d^C_0} C_{-1} \to \cdots
\]
such that $M\cong\im d^C_i$ for some $i\in\mathbb Z$
.
\end{itemize}
%Given a module $N$, define $\Omega_n(N)$ as the {\it minimal $n$th syzygy of $N$}. It is the $\im d_i$ in a minimal free resolution of $N$, unique up to isomorphism.
\subsection{Koszul Homology}
\begin{itemize}
\item The {\it tensor product of two complexes} $$C: \cdots \xra{}C_i\xra{\alpha_i}C_{i+1}\xra{} \cdots$$ and $$D: \cdots \xra{} D_i\xra{\beta_i}D_{i+1}\xra{}\cdots$$ is defined to be the complex $$C \otimes D: \cdots \xra{} \bigoplus_{i+j=k} C_i \otimes D_j \xra{d_k} \bigoplus_{i+j=k-1}C_i \otimes D_j \xra{} \cdots$$ where the map $d_k$ on $C_i \otimes D_j$ (with $i+j=k$) is given by $$d_k: \bigoplus_{i+j=k} F_i \otimes D_j \to \bigoplus_{i+j=k-1} C_i \otimes D_j,$$ where for $a \otimes b \in C_i \otimes D_j,$ the differentials are given by $$d_k(a \otimes b)=d_i^C(a)\otimes b+(-1)^i a \otimes d_j^D(b).$$
\item Let $x$ be an element in $R.$ The {\it Koszul complex} $K(x;R)$ on $x$ is the complex $$K(x): 0\xra{}R\xra{x}R\xra{}0,$$ with $R$ situated in homological degrees $0$ and $1.$
\item Suppose we are given a sequence $\mathbf{x}=x_1,\ldots, x_n$ of elements in $R.$ The {\it Koszul complex on $\mathbf{x}$} is the complex $$K(\mathbf{ x};R)=K(x_1;R)\otimes_R\cdots \otimes_RK(x_n;R).$$ The nonzero modules in this complex are situated in degrees $0$ to $n.$
\item The {\it Koszul homology} of a local ring $R,$ is the homology of the Koszul complex on a minimal set of generators $\mathbf{x} = x_1,\dots,x_n$ of the maximal ideal: $$H(\mathbf{x};R)=H(K(\mathbf{x};R)).$$
\end{itemize}
\subsection{Linkage}
\begin{itemize}
\item Let $I$ and $J$ be ideals in a ring $R.$ Then $I$ and $J$ are said to be {\it linked,} written $I\sim J,$ if there exists a regular sequence $g_1,\ldots,g_d$ in $I \cap J$ such that $(g_1,\ldots,g_d):I=J$ and $(g_1,\ldots,g_d):J=I.$
%\begin{example} $I=(x,y)$ and $J=(y,z)$ are linked.
%\end{example}
%Let $(y^2,xz)$ be the regular sequence belonging to $(x,y)\int(y,z).$ Then $(y^2,xz):(x,y)=(y,z)$ and $(y^2,xz):(y,z)=(x,y)$ so $I\sim J.$
\item We also say $I$ is one link from a complete intersection if $I \sim J$ and $J$ is generated by a regular sequence.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Defining the Question}
We now pose the issue of branching in the form of a question, and attempt to answer it for specific types of rings and complexes throughout this paper.
\begin{question}
\label{question1}
Given a nonzero minimal acyclic complex of free modules \[A:\quad
\cdots \to A_1 \xra{d_1} A_0 \xra{d_0} A_{-1} \to \cdots,
\] does there exist a minimal acyclic complex of free modules\[
B:\quad \cdots \to B_1 \xra{f_1} B_0 \xra{f_0} B_{-1} \to \cdots
\]such that $A_{\ge s} \cong B_{\ge s}$ some $s\in \mathbb Z$, but $A \ncong B$?
\end{question}
\[
\xymatrixrowsep{2pc}
\xymatrixcolsep{1pc}
\xymatrix {\cdots \ar@{->}[r]&C_{2}\ar@{->}[rr]\ar@{->}[dd]^{\cong}&&C_{1}\ar@{->}[rr]\ar@{->}[dd]^{\cong}&&C_0\ar@{->}[rr]\ar@{->}[dd]^{\cong}\ar@{->>}[dr]&&C_{-1}\ar@{->}[rr]\ar@{->}[dd]^{\not{\cong}}&&C_{-2}\ar@{->}[r]\ar@{->}[dd]^{\text{{\normalsize or}}\,\,\,\,\,\not{\cong}}&\cdots \\&&\circlearrowleft&&\circlearrowleft&&M\ar@{^{(}->}[ur]\ar@{^{(}->}[dr]&&\text{or}\,\,\,\not{\circlearrowleft}&\\
\cdots \ar@{->}[r]&D_{2}\ar@{->}[rr]&&D_{1} \ar@{->}[rr]&&D_0\ar@{->}[rr]\ar@{->>}[ur]&&D_{-1}\ar@{->}[rr]&&D_{-2}\ar@{->}[r]&\cdots .}
\]
Before exploring this question, we give conditions for $R$ for the remainder of this paper.
\section{Cohen-Macaulay Rings}
We have the classical chain of inclusion for classes of local rings: $${\text regular}\subset{\text complete\hspace {.1cm} intersections}\subset{\text Gorenstein}\subset{\text Cohen\-Macaulay}$$
In the case of a Gorenstein ring, any acyclic complex of free modules is totally acyclic. We point out in the final chapter that totally acyclic complexes do not have branching. Thus Question \ref{question1} is answered for Gorenstein rings. Gorenstein rings are part of the larger class of Cohen-Macaulay rings. Therefore, the next logical case to consider is that of Cohen-Macaulay rings.
\section{Dimension Zero}
A natural starting point is to investigate rings of dimension zero. One justification for this is the following fact.
Consider a minimal acyclic complex $C$ over a Cohen-Macaulay ring $R.$ If $\bold{x}=x_1,\cdots,x_d$ is a maximal $R$-sequence, then $R/(\bold{x})$ is a ring of dimension zero. If $C$ is a minimal acyclic complex over $R,$ then $C\otimes R/(\bold{x})$ is a minimal acyclic complex over $R/(\bold{x}).$
We note, however, that it is possible for two non-isomorphic acyclic complexes over $R$ to become isomorphic when modding out by an $R$-sequence. As an example, let $R=k[[x,y]]/{(x^2-y^2)}.$ Consider the minimal acyclic complexes over $R:$ $$C \hspace{.5 cm} \cdots\xra{}C_2\xra{x-y}C_1\xra{x+y} C_0 \xra{x-y}C_{-1}\xra{}\cdots,$$ and $$D \hspace{.5 cm} \cdots\xra{}D_2\xra{x+y}D_1\xra{x-y} D_0 \xra{x+y}D_{-1}\xra{}\cdots,$$ where $C_i \cong R$ and $D_i \cong R,$ all $i \in \mathbb Z.$
To see that $C$ and $D$ are not isomorphic, assume the following square commutes:
\[
\xymatrixrowsep{2pc}
\xymatrixcolsep{2pc}
\xymatrix{&
C_1\ar@{->}[r]^{x+y} \ar@{->}[d]^{v}&
C_{0} \ar@{->}[d]^{u}&\\
&
D_1 \ar@{->}[r]^{x-y}&
D_0 ,&}
\]
where $u$ and $v$ are units in $R.$ On the one hand, $$1\mapsto {x+y} \mapsto {u(x+y)},$$ on the other hand $$1 \mapsto v \mapsto {v(x-y)}.$$ However, for no units $u$ and $v$ do we have $${u(x+y)} = {v(x-y)},$$ which contradicts commutativity. Thus $C$ and $D$ are not isomorphic complexes.
Note that $y$ is a non-zero divisor on $R.$ Quotienting by $(y),$
\[
\xymatrixrowsep{2pc}
\xymatrixcolsep{2pc}
\xymatrix{C/(y):&\cdots\ar@{->}[r]&C_2/(y)\ar@{->}[rr]^{x}\ar@{->}[dd]^{1}&&C_1/(y)\ar@{->}[rr]^{x}\ar@{->}[dd]^{1}&&C_0/(y)\ar@{->}[r]&\cdots\\
&&&\circlearrowleft&&\circlearrowleft&
\\
D/(y):&\cdots\ar@{->}[r]&D_2/(y)\ar@{->}[rr]^{x}&&D_1/(y)\ar@{->}[rr]^{x}&&D_0/(y)\ar@{->}[r]&\cdots,}
\]
we obtain
$$C/{(y)} \cong D/{(y)},$$ giving the minimal acyclic complex $$\cdots \xra{}E_2\xra{x} E_1 \xra {x} E_0 \xra {x} E_{-1} \xra{} \cdots,$$ over $R/{(y)} \cong k[x]/{(x^2)}.$
Consider a minimal acyclic complex $C$ over a Cohen-Macaulay ring $R.$ If $\bold{x}=x_1,\cdots,x_d$ is a maximal $R$-sequence, then $R/(\bold{x})$ is a ring of dimension zero. If $C$ is a minimal acyclic complex over $R,$ then $C\otimes R/(\bold{x})$ is a minimal acyclic complex over $R/(\bold{x}).$ Many properties of $C$ over $R$ are transferred to those of $C\otimes R/(\bold{x})$ over $R/(\bold{x}).$ In particular, non-uniqueness for minimal acyclic complexes over $R/{(x)},$ implies the same for $R.$ If we can answer question \ref{question1} positively for $R/(x), x$ a non-zero divisor, then we can also answer it positively for $R.$
\section{$\m^3=0$}
Assuming $R$ has dimension zero, then $\m^n=0$ for some $n.$ It turns out, the first interesting case to investigate the uniqueness of minimal acyclic complexes is $\m^3=0.$ For iff $n=1,$ then $R$ is a field, and the only minimal acyclic complexes is the zero complex. For the $n=2$ case, nonzero minimal acyclic complexes also do not exist. To see this, suppose $(R,\m)$ has $\m^2=0.$ Let $\Omega$ be a finitely generated $R$-module with $\m\Omega=0,$ that is, $\Omega$ is a finite dimensional vector space over $k=R/\m,$ and $$0\to\Omega'\to F \to \Omega \to 0,$$ an exact sequence where $F$ is free, $\mu(F)=\mu(\Omega),$ where $\mu(X)$ denotes the minimal number of generators of the module $X.$ By minimality, $\Omega' \subseteq \m F.$ So $\m\Omega' \subseteq \m^2F=0,$ which gives $\m\Omega'=0.$
We know $\dim_k\Omega'=\length F - \dim_ k\Omega,$ and $\length F = \rank F(\length R),$ giving $$\dim_k\Omega'=\rank F(\length R)- \dim _k\Omega.$$ By exactness, $\rank F = \dim_k\Omega,$ so $$\dim_k \Omega' = (\dim_k\Omega)(\length R)-\dim_k\Omega$$ $$=\dim_k\Omega (\length R-1).$$ Now take a minimal acyclic complex $$C\hspace{.5 cm} \cdots\to C_{i+1} \to C_i \to C_{i-1} \to \cdots,$$ and apply inductively to $C,$ we have the short exact sequence $$0\xra{}\Omega_{i+1}\xra{}C_i\xra{}\Omega_i\xra{}0,$$ $$\rank C_i=\dim\Omega_i=\dim\Omega_{i-j}(\length R-1)^j,$$ for all $j\leq 0.$ This is absurd unless $\length R \leq 2.$ If the length of $R$ is $1,$ then $R$ is a field. If the length of $R$ is $2,$ then $R$ is isomorphic to the ring $k[x]/{(x^2)},$ which is Gorenstein. Any acyclic complex over a Gorenstein ring is totally acyclic, and therefore, unique.
For the $\m^3=0$ case, it is possible to construct minimal acyclic complexes over non-Gorenstein rings, therefore this is the first case where \ref{question1} is open. For example, over the ring $R=k[x,y,z]/(x^2,y^2,z^2,xy+yz), $ we have minimal acyclic complexes $$\cdots\xra{}R\xra{z}R\xra{z}R\xra{}\cdots,$$ and $$\cdots\xra{}R^2\xra{\left(\begin{matrix}x&y\\0&z\end{matrix}\right)}R^2\xra{\left(\begin{matrix}x&y\\0&z\end{matrix}\right)} R^2\xra{}\cdots.$$
\section{Required Properties}
The following theorem from [11] maintains that these complexes can exist only if $R$ has the following properties:
\begin{theorem}
Let $(R,\m,k)$ be a local ring that is not Gorenstein and has $\m^3=0\neq \m^2.$ If there exists a non-zero minimal acyclic complex $A$ of finitely generated free $R$-modules, then the ring has the following properties:
(a) $(0:\m)=\m^2.$
(b) $e=r+1$ with $\length R = 2e.$
(c) Poincare series $P_k^R(t)=1/{(1-t)(1-rt)}.$
\end{theorem}It is assumed that the rings we study in the following sections will be $\m^3=0$ and possess these properties.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Periodicity of Minimal Acyclic Complexes}
In the next chapter, we will prove, among other results, that periodic complexes have no branching. Once periodicity is established, it of course remains periodic to the left. It is not known, however, that a minimal complex with periodicity to the left must be periodic everywhere.
This first lemma builds a new complex from a family of isomorphic complexes. We will use the representative complex to establish isomorphisms among all complexes containing the given period.
\begin{lemma}
\label{lemma0}
Let $\{ A^i\}_{i \in \mathbb Z}$ be a family of complexes, and $\{f^i: A^i \to A^{i+1} \}_{i \in \mathbb Z}$ a family of chain isomorphisms.
For any sequence $\{n_j\}_{j \in \mathbb Z}$, define a new complex
$$ A\{n_j\}: \hspace {.25 in} \cdots \to A_2 \xra {d_2} A_1 \xra {d_1} A_0 \xra {d_0} \cdots$$
where $A\{n_j\}_i=A_i^{n_i}$ and
\begin{equation}
d_i=
\begin{cases}
(f_{i-1}^{n_{i-1}})^{-1}
(f_{i-1}^{n_{i-1}+1})^{-1}\cdots (f_{i-1}^{n_i-1})^{-1}d_i^{n_i} &\text{for } n_{i-1}< n_i\\
f_{i-1}^{n_{i-1}-1}f_{i-1}^{n_{i-1}-2}\cdots f_{i-1}^{n_i}d_i^{n_i}&\text{for } n_{i-1}\geq n_i
\end{cases}
\end{equation}
for $i \in \mathbb Z$. Then $A\{n_j\} \cong A^i$ for all $i\in \mathbb Z$.
\end{lemma}
\begin{proof}Consider the diagram:
\[
\xymatrixrowsep{2pc}
\xymatrixcolsep{3pc}
\xymatrix{\vdots \ar@{->}[d] & & \vdots \ar@{->}[d] & \vdots \ar@{->}[d] & \vdots \ar@{->}[d] & \\ A^{-1}:\ar@{->}[d]^{f^{-1}} &
\cdots\ar@{->}[r] &
A_{1}^{-1}\ar@{->}[r]^{d_{1}^{-1}}\ar@{->}[d]^{f_{1}^{-1}}&
A_0^{-1}\ar@{->}[r]^{d_0^{-1}}\ar@{->}[d]^{f_0^{-1}}&
A_{-1}^{-1}\ar@{->}[r]^{d_{-1}^{-1}}\ar@{->}[d]^{f_{-1}^{-1}}&\cdots\\
A^0: \ar@{->}[d]^{f^0}&\cdots\ar@{->}[r] &
A_{1}^0\ar@{->}[r]^{d_{1}^0}\ar@{->}[d]^{f_{1}^0}&
A_0^0\ar@{->}[r]^{d_0^0}\ar@{->}[d]^{f_0^0}&
A_{-1}^0\ar@{->}[r]^{d_{-1}^0}\ar@{->}[d]^{f_{-1}^0}&\cdots \\
A^1: \ar@{->}[d]^{f^1}&\cdots\ar@{->}[r] &
A_{1}^1\ar@{->}[r]^{d_{1}^1}\ar@{->}[d]^{f_{1}^1}&
A_0^1\ar@{->}[r]^{d_0^1}\ar@{->}[d]^{f_0^1}&
A_{-1}^1\ar@{->}[r]^{d_{-1}^1}\ar@{->}[d]^{f_{-1} ^1}&\cdots\\
\vdots & & \vdots & \vdots & \vdots &.
}
\]
Since each of the $ A^i$ are isomorphic to each other, it suffices to show the new complex $ A$ is isomorphic to the complex $ A^0$. To simplify notation, let $A_i=( A\{n_j\})_i=A_i^{n_i}$. We need to define maps $f_i:A_i^0 \to A_i$ and show that the squares in the following diagram commute:
\[
\xymatrixrowsep{2pc}
\xymatrixcolsep{2pc}
\xymatrix{\cdots \ar@{->}[r] &
A_{i}^0 \ar@{->}[r]^{d_{i}^0} \ar@{->}[d]^{f_{i}}&
A_{i-1}^0 \ar@{->}[r] \ar@{->}[d]^{f_{i-1}}&\cdots\\
\cdots \ar@{->}[r]&
A_{i} \ar@{->}[r]^{d_{i}}&
A_{i-1} \ar@{->}[r]&\cdots.}\]
In other words, show $f_{i-1}d_i^0=d_i^{n_i}f_i$. For fixed $i$ we have six cases to consider:
$$(1)\hspace{.2cm} n_i\geq n_{i-1}\geq 0\hspace{1in}(4)\hspace{.2cm}n_{i-1}\geq n_i\geq 0$$
$$(2)\hspace{.2cm}n_i\geq 0 \geq n_{i-1}\hspace{1in}(5)\hspace{.2cm}n_{i-1}\geq 0 \geq n_i$$
$$(3)\hspace{.2cm}0 \geq n_i \geq n_{i-1}\hspace{1in}(6)\hspace{.2cm}0 \geq n_{i-1} \geq n_i.
$$
We examine the first three cases, and recognize the remaining three are symmetrically similar.
\noindent\textit {Case 1:}\hspace{.5 in} $n_i\geq n_{i-1}\geq0.$
From commutativity of
\[
\xymatrixrowsep{1pc}
\xymatrixcolsep{1pc}
\xymatrix{\cdots\ar@{->}[r]&A_i^{n_{i-1}}\ar@{->}[r]^{d_i^{n_i-1}}\ar@{->}[d]&A_{i-1}^{n_{i-1}}\ar@{->}[r]\ar@{->}[d]&\cdots\\&\vdots\ar@{->}[d]&\vdots\ar@{->}[d]\\
\cdots\ar@{->}[r]&A_i^{n_i}\ar@{->}[r]^{d_i^{n_i}}&A_{i-1}^{n_i}\ar@{->}[r]&\cdots.}\] we rewrite $d_i: A_i^{n_i} \to A_{i-1}^{n_{i-1}}$ as $$d_i=(f_{i-1}^{n_{i-1}})^{-1}
(f_{i-1}^{n_{i-1}+1})^{-1}\cdots (f_{i-1}^{n_i-1})^{-1}d_i^{n_i}=d_i^{n_{i-1}}(f_{i}^{n_{i-1}})^{-1}
(f_{i}^{n_{i-1}+1})^{-1}\cdots (f_{i}^{n_{i}-1})^{-1}.$$
The chain maps $f_i: A_i^0 \to A_i^{n_i}$ are given by $$f_i=f_i^{n_i-1}f_i^{n_i-2}\cdots f_i^0.$$ We need to show that $f_{i-1}d_i^0=d_if_i$:
$$f_{i-1}d_i^0=f_{i-1}^{n_{i-1}-1}f_{i-1}^{n_{i-1}-2}\cdots f_{i-1}^0d_i^0$$
$$=f_{i-1}^{n_{i-1}-1}f_{i-1}^{n_{i-1}-2}\cdots f_{i-1}^1d_i^1f_i^0$$
$$=f_{i-1}^{n_{i-1}-1}f_{i-1}^{n_{i-1}-2}\cdots d_i^2f_i^1f_i^0$$
$$\vdots$$
$$=f_{i-1}^{n_{i-1}-1}d_i^{n_{i-1}-1}f_i^{n_{i-1}-2}\cdots f_i^1f_i^0$$
$$=d_i^{n_{i-1}}f_i^{n_{i-1}-1}f_i^{n_{i-1}-2}\cdots f_i^1f_i^0.$$
Since $d_i^{n_{i-1}}=d_i^{n_i}f_i^{n_i-1}\cdots f_i^{n_{i-1}+1}f_i^{n_{i-1}}$, we then have
$$f_{i-1}d_i^0=d_i^{n_i}f_i^{n_i-1}\cdots f_i^{n_{i-1}+1}f_i^{n_{i-1}}f_i^{n_{i-1}-1}f_i^{n_{i-1}-2}\cdots f_i^1f_i^0=d_if_i.$$
\textit {Case 2:}\hspace{.5 in}$n_i \geq 0 \geq n_{i-1}.$
Define $f_i:A_i^0 \to A_i^{n_i}$ as $f_i=f_i^{n_i-1}f_i^{n_i-2}\cdots f_i^0.$ Since $n_{i-1}\leq 0$, define $$f_{i-1}=(f_{i-1}^{n_{i-1}})^{-1}
(f_{i-1}^{n_{i-1}+1})^{-1}\cdots (f_{i-1}^{-1})^{-1}.$$
We again need $f_{i-1}d_i^0=d_if_i$:
$$d_if_i=(f_{i-1}^{n_{i-1}})^{-1}
(f_{i-1}^{n_{i-1}+1})^{-1}\cdots (f_{i-1}^{-1})^{-1}(f_{i-1}^0)^{-1}\cdots(f_{i-1}^{n_i-1})^{-1}d_i^{n_i}f_i^{n_i-1}f_i^{n_i-2}\cdots f_i^0$$
$$=(f_{i-1}^{n_{i-1}})^{-1}
(f_{i-1}^{n_{i-1}+1})^{-1}\cdots (f_{i-1}^{-1})^{-1}(f_{i-1}^0)^{-1}\cdots(f_{i-1}^{n_i-1})^{-1}f_{i-1}^{n_i-1}d_i^{n_i-1}f_i^{n_i-2}\cdots f_i^0$$
$$=(f_{i-1}^{n_{i-1}})^{-1}
(f_{i-1}^{n_{i-1}+1})^{-1}\cdots (f_{i-1}^{-1})^{-1}(f_{i-1}^0)^{-1}\cdots (f_{i-1}^{n_i-1})^{-1}f_{i-1}^{n_i-1}f_{i-1}^{n_i-2}d_i^{n_i-2}\cdots f_i^0$$
$$\vdots$$
$$=(f_{i-1}^{n_{i-1}})^{-1}
(f_{i-1}^{n_{i-1}+1})^{-1}\cdots (f_{i-1}^{-1})^{-1}(f_{i-1}^0)^{-1}\cdots(f_{i-1}^0)d_i^0$$
$$=(f_{i-1}^{n_{i-1}}\cdots(f_{i-1}^{-1})^{-1}d_i^0=f_{i-1}d_i^0.$$
\textit {Case 3:} \hspace{.5 in}$0 \geq n_i \geq n_{i-1}.$
Define $f_i=(f_i^{n_i})^{-1}\cdots(f_i^{-2})^{-1}(f_i^{-1})^{-1}$ and
$$f_{i-1}=(f_{i-1}^{n_{i-1}})^{-1}
(f_{i-1}^{n_{i-1}+1})^{-1}\cdots (f_{i-1}^{n_i})^{-1}(f_{i-1}^{n_i+1})^{-1}\cdots (f_{i-1}^{-2})^{-1}(f_{i-1}^{-1})^{-1}.$$
So we have,
$$d_if_i=(f_{i-1}^{n_{i-1}})^{-1}
(f_{i-1}^{n_{i-1}+1})^{-1}\cdots (f_{i-1}^{n_i-1})^{-1}d_i^{n_i}(f_i^{n_i})^{-1}\cdots(f_i^{-2})^{-1}(f_i^{-1})^{-1}$$
$$=(f_{i-1}^{n_{i-1}})^{-1}
(f_{i-1}^{n_{i-1}+1})^{-1}\cdots (f_{i-1}^{n_i-1})^{-1}(f_{i-1}^{n_i})^{-1}d_i^{n_i+1}\cdots(f_i^{-2})^{-1}(f_i^{-1})^{-1}$$
$$\vdots$$
$$=(f_{i-1}^{n_{i-1}})^{-1}
(f_{i-1}^{n_{i-1}+1})^{-1}\cdots (f_{i-1}^{n_i-1})^{-1}(f_{i-1}^{n_i})^{-1}(f_{i-1}^{n_i+1})^{-1}\cdots d_i^{-1}(f_i^{-1})^{-1}$$
$$=(f_{i-1}^{n_{i-1}})^{-1}
(f_{i-1}^{n_{i-1}+1})^{-1}\cdots (f_{i-1}^{n_i-1})^{-1}(f_{i-1}^{n_i})^{-1}(f_{i-1}^{n_i+1})^{-1}\cdots (f_{i-1}^{-1})^{-1}d_i^0=f_{i-1}d_i^0.$$
\end{proof}
\noindent \textbf {Definition.} $(\Sigma^pC)_n=C_{n-p}$. A complex $C$ is {\it periodic} of period $p$ if there exists an isomorphism $f:C \to \Sigma^p C$ and $C \ncong \Sigma^s C$ for $0}[r]&(PC)_p\ar@{->}[r]^{d_p^{PC}}\ar@{->}[d]&
(PC)_{p-1}\ar@{->}[r]^{d_{p-1}^{PC}}\ar@{->}[d]&(PC)_{p-2}\ar@{->}[r]\ar@{->}[d]
&\cdots\ar@{->}[r]&(PC)_1\ar@{->}[r]^{d_1^{PC}}\ar@{->}[d]&(PC)_0\ar@{->}[r]^{d_0^{PC}}\ar@{->}[d]
&\cdots\\\cdots\ar@{->}[r]&C_0\ar@{->}[r]^{d^C_0f^{-1}}
&C_{p-1}\ar@{->}[r]^{d^C_{p-1}}&C_{p-2}\ar@{->}[r]&\cdots\ar@{->}[r] &C_1\ar@{->}[r]^{d^C_1}&C_{0}\ar@{->}[r]^{d^C_0f_p^{-1}}&\cdots.}
\]
\end{lemma}
\begin{proof} By definition, $C \xra{f} \Sigma^p C$ is an isomorphism, and $(\Sigma^pC)_n=C_{n-p}$.
Consider the diagram:
\[
\xymatrixrowsep{2pc}
\xymatrixcolsep{1pc}
\xymatrix{
\vdots \ar@{->}[d]&&&&&&&&&&&\vdots\ar@{->}[d]^{(\Sigma^{-3p}f)_{-2p}}&&\\
\Sigma^{-2p}C:\ar@{->}[d]^{\Sigma^{-2p}f} && & && & && \cdots \ar@{->}[r] &C_p\ar@{->}[r]\ar@{->}[d]^{(\Sigma^{-2p}f)_{-p}} & \cdots\ar@{->}[r]&C_0\ar@{->}[r] &\cdots &
\\\Sigma^{-p}C:\ar@{->}[d]^{\Sigma^{-p}f} && & &&& \cdots \ar@{->}[r]& C_p\ar@{->}[r]\ar@{->}[d]^{(\Sigma^{-p}f)_{0}} &\cdots\ar@{->}[r]&C_0\ar@{->}[r] &\cdots&&&
\\C: \ar@{->}[d]^{f} & && & \cdots \ar@{->}[r]& C_p\ar@{->}[r]\ar@{->}[d]^{f_p} & \cdots\ar@{->}[r]&C_0\ar@{->}[r] &\cdots&&&&&
\\\Sigma^pC:\ar@{->}[d]^{\Sigma^{p}f} && \cdots \ar@{->}[r]&C_p\ar@{->}[r]\ar@{->}[d]^{(\Sigma^{p}f)_{2p}} & \cdots\ar@{->}[r]&C_0\ar@{->}[r] &\cdots&&&&&&&
\\\Sigma^{2p}C: \ar@{->}[d]^{\Sigma^{2p}f}&C_p\ar@{->}[r]\ar@{->}[d]^{(\Sigma^{2p}f)_{3p}}f & \cdots\ar@{->}[r]&C_0\ar@{->}[r] &\cdots&&&&&&&&&
\\\vdots&\vdots&&&&&&&&&&&&.
}
\]
By Lemma \ref{lemma0}, $C$ is isomorphic to the complex
$$\cdots \to C_0 \xra {d_0^Cf_p^{-1}} C_{p-1} \xra {d_{p-1}^C} C_{p-2} \xra {d_{p-2}^C} \cdots \to C_1 \xra {d_1^C} C_0 \xra {d_0^Cf_p^{-1}} C_{p-1} \xra {d_{p-1}^C} C_{p-2} \xra {d_{p-2}^C} \cdots $$ which is what we wanted to show.
\end{proof}
Finally, we show that once periodicity is established, the length of the period does not change.
\begin{lemma}\label{lemma1}
If $D$ is a periodic complex of period $p$, then for all $r \in \mathbb Z$, there is an isomorphism $D_{\geq r} \to (\Sigma^p D)_{\geq r}$ and $D_{\geq r} \ncong (\Sigma^q D)_{\geq r}$ for $q
}[r]&D_{r+i}\ar@{->}[r]\ar@{->}[d]^{f_{r+i}}&\cdots\ar@{->}[r]&D_{r+1}\ar@{->}[r]\ar@{->}[d]^{f_{r+1}}
&D_{r}\ar@{->}[d]^{f_{r}}\ar@{->}[r]&0\ar@{->}[r]&\cdots\\\cdots\ar@{->}[r]&D_{r-p+i}\ar@{->}[r]&\cdots\ar@{->}[r]&D_{r-p+1}\ar@{->}[r]&D_{r-p}\ar@{->}[r]&0\ar@{->}[r]&\cdots.}
\]
There exists a chain isomorphism $g:D_{\geq {r}} \to (\Sigma^qD)_{\geq r},$ and we have the following diagram:
\[
\xymatrixrowsep{3pc}
\xymatrixcolsep{2pc}
\xymatrix{\cdots \ar@{->}[r]&D_{r-p+1} \ar@{->}[r]\ar@{->}[d]^{f^{-1}_{r+1}}&D_{r-p} \ar@{->}[d]^{f^{-1}_{r}}\ar@{->}[r]&0\ar@{->}[r]&\cdots\\
\cdots \ar@{->}[r]&D_{r+1} \ar@{->}[r]\ar@{->}[d]^{g_{r+1}}&D_{r} \ar@{->}[d]^{g_{r}}\ar@{->}[r]&0\ar@{->}[r]&\cdots\\
\cdots \ar@{->}[r]&D_{r-q+1} \ar@{->}[r]\ar@{->}[d]^{f_{r-q+1}}&D_{r-q} \ar@{->}[d]^{f_{r-q}}\ar@{->}[r]&0\ar@{->}[r]&\cdots\\
\cdots \ar@{->}[r]&D_{r-q-p+1}\ar@{->}[r]&D_{r-q-p}\ar@{->}[r]&0\ar@{->}[r]&\cdots.}
\]
Since each of the squares commute, $D_{\geq r-p} \cong (\Sigma^qD)_{r-p},$ which contradicts the choice of $r$.
\end{proof}
%**********************************************************************************************************
\chapter{Uniqueness Results that Follow from Periodicity}
%\begin{question}
%\label{question1}
%Given a nonzero minimal acyclic complex of free modules \[A:\quad
%\cdots \to A_1 \xra{d_1} A_0 \xra{d_0} A_{-1} \to \cdots,
%\] does there exist an acyclic complex \[
%B:\quad \cdots \to B_1 \xra{f_1} B_0 \xra{f_0} B_{-1} \to \cdots
%\]such that $A_{\ge s} \cong B_{\ge s}$ some $s\in \mathbb Z$, but $A \ncong B$?
%\end{question}
In this chapter, we first answer \ref{question1} in the negative for periodic complexes. From this result, we additionally prove that complexes over $k$-algebras with $\m^3=0$ and Hilbert series $H_R(t)=1+et+(e-1)t^2$ such that $k$ is a finite field, and complexes over rings of codimension $\leq 3$ are also unique to the right.
\section{Periodic Complexes}
\begin{theorem}
\label{pertheorem} Let $C$ be a periodic complex of periodicity $p$ and $D$ be a periodic complex of periodicity $q$ such that $C_{\geq r} \cong D_{\geq r}$. Then $C \cong D$.
\end{theorem}
\begin{proof}
Since $C$ is periodic, $C \cong \Sigma^p C$. Likewise, $D \cong \Sigma^q D$. We have
$$D_{\geq {r+p}} \cong C_{\geq {r+p}} \cong (\Sigma^pC)_{\geq {r+p}} \cong (\Sigma^pD)_{\geq {r+p}}.$$
By \ref{lemma1}, this implies $q \leq p$ since $q$ is the smallest $q$ such that $D_{\geq r+p} \cong \Sigma^q D_{\geq r+p}$ for all $r.$ Symmetrically, $$C_{r+p} \cong D_{r+p} \cong (\Sigma^q D)_{\geq {r+p}} \cong (\Sigma^q C)_{\geq {r+p}}.$$ Again, by \ref{lemma1}, $p\leq q$. Thus, $p=q$ and the complexes have the same period. This gives by Lemma $4.0.3$ $C$ isomorphic to the complex $$\cdots \to C_r \to C_{r+p} \xra{d^C_{r+2}} C_{r+1} \xra{d^C_{r+1}} C_r \xra{d_r} C_{r+p} \to \cdots,$$ with $D$ isomorphic to $$\cdots \to D_r \to D_{r+p} \to \cdots \xra{d^D_{r+2}} D_{r+1} \xra{d^D_{r+1}} D_r \to D_{r+p} \to \cdots.$$ Since $D_r \cong C_r$, and the complexes have the maps and modules repeated:
\[
\xymatrixrowsep{2pc}
\xymatrixcolsep{1pc}
\xymatrix {\cdots \ar@{->}[r]&C_{r}\ar@{->}[r]\ar@{->}[d]^{\cong}&C_{r+1}\ar@{->}[r]\ar@{->}[d]^{\cong}&\cdots
\ar@{->}[r]&C_{r+1}\ar@{->}[r]^{d^C_{r+1}}\ar@{->}[d]^{\cong}&C_{r}\ar@{->}[r]\ar@{->}[d]^{\cong}&C_{r+p}\ar@{->}[r]\ar@{->}[d]^{\cong}& \cdots \\
\cdots \ar@{->}[r]&D_{r}\ar@{->}[r]&D_{r+1}\ar@{->}[r]&\cdots\ar@{->}[r]&D_{r+1} \ar@{->}[r]^{d_{r+1}^D}&D_{r}\ar@{->}[r]&D_{r+p}\ar@{->}[r]& \cdots, }
\]
they are isomorphic everywhere.
\end{proof}
\section{$\mathbf{k}$-algebras with $\mathbf{\m^3=0}$ and $\mathbf{k}$ finite}
\begin{lemma}
\label{lemmaff}
If $\m^3=0$ and $|k|<\infty,$ then any minimal acyclic complex $C$ is periodic.
\end{lemma}
\begin{proof}For a minimal acyclic complex with $\m^3=0$, the negative Betti numbers are constant by [11], say equal to $n$. The ``negative differentials" are thus represented by $n \times n$ matrices with linear entries. For the codimension of $R$ being $e$, there are $|k|^e$ possible linear forms, allowing $(k^e)^{n^2}$ possible matrices representing the $d^i$. Since there are infinitely many $d_i$ for $i\ll 0,$we must have $d_i^C = d_j^C$ for some $i\neq j$. Assume $i }[r]&A_{r-2}^*\ar@{->}[r]\ar@{->}[d]^{\cong}&A_{r-1}^*\ar@{->}[r]\ar@{->}[d]^{\cong}&A_r^*\ar@{->}[r]\ar@{->}[d]^{\cong}&A_{r+1}^*\ar@{->}[r]\ar@{->}[d]^{\cong}&\cdots \ar@{->}[r]&A_{p-1}^*\ar@{->}[r]\ar@{->}[d]^{\cong}&A_p^*\ar@{->}[r]\ar@{->}[d]^{\cong}&\cdots\\B^*:\hspace{.3cm}\cdots \ar@{->}[r] &B_{r-2}^*\ar@{->}[r]&B_{r-1}^*\ar@{->}[r]&B_r^*\ar@{->}[r]&B_{r+1}^*\ar@{->}[r]&\cdots\ar@{->}[r]&B_{p-1}^*\ar@{->}[r]&B_p^*\ar@{->}[r]&\cdots.}$$
Dualize back to get $A \cong B.$
\end{proof}
\begin{corollary}
Let $A$ and $B$ be totally acyclic complexes such that $A_{\geq r} \cong B_{\geq r}.$ Then $A \cong B.$
\end{corollary}
\begin{proof}
Since totally acyclic complexes are a special case of sesqui-acyclic complexes, by Theorem \ref{sesqui}, $A \cong B.$
\end{proof}
\chapter{Conclusion}
Although we have not determined that branching of a minimal acyclic complex is possible, we have found certain scenarios where these complexes are decidedly unique. Periodic minimal acyclic complexes do not branch. From this result, we additionally conclude that rings over finite residue fields with $m^3=0,$ rings of codepth $\leq 3,$ and rings that are one link from a complete intersection are unique to the right as well. We have learned conclusively that conca rings have no branching, and that branching cannot occur over sesqui-acyclic complexes in general.
In addition to determining circumstances where branching cannot happen, we have found possible affirmative scenarios via the push-forward method. Specifically, this occurs with the semi-conca $e=4$ monomial algebra case as well as with conca-generated non-monomial algebras.
Possible future results include:
\begin{itemize}
\item Pushing a syzygy module forward an infinite number of times. (Achieving branching of a minimal acyclic complex.)
\item Examining uniqueness of minimal acyclic complexes for monomial algebras for $e\geq 5.$
\item Answering the question: If $M$ is an $n$th syzygy module for all $n$, is $M$ an infinite syzygy?
\end{itemize}
%\chapter{$N$th Syzygies}
%\begin{question}
%\label{question3}
%If $M$ is an $n$th syzygy module for all $n$, is $M$ an infinite syzygy?
%\end{question}
%\begin{corollary}
%For any module $N$ over $(R,m,k)$ with $m^2=0$, $b_1^R(N)=e^{i-1}b_1^R(N)$ for all $i>0$.
%\end{corollary}
%\begin{corollary}
%If $M$ is an $n$th syzygy for all $n$, over $(R,n,k)$ with $m^2=0$, then $M$ is free.
%\end{corollary}
%\begin {proof} For all $n$, we have $M \cong \Omega_n(N(n))\oplus F_n$
%\end {proof}
%First, consider the case for $m^2=0$. Let $e=emb \dim R$. Let $M$ be the $i$th syzygy module for $N$ and the $j$th syzygy module for $N^{'}$ with $j>i$, giving the presentations
%\[
%\cdots \to 0 \to M\to R^{b_i(N)}
%\to \cdots \to R^{b_1(N)} \to N \to 0
%\]
%and
%\[
%\cdots \to 0 \to M \to R^{b_j(N')}
%\to \cdots \to R^{b_1(N')} \to N' \to 0
%\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\pagebreak
%\appendix
%\addcontentsline{toc}{section}{Polynomials of $T_A$ }
%\begin{center}
%{\bf APPENDIX}
%\end{center}
%\section{Polynomials of $T_A$ from Macaulay2}
%\footnotesize{
%\begin{verbatim}
%\end{verbatim}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\chapter{Lax-Pair}
%Let $L$ be the linear differential operator and $M$ be a time evolution operator.
%\begin{equation} \label{1}
%Lv=\lambda v,
%\end{equation}
%\begin{equation}
%v_t=Mv.
%\label{2}
%\end{equation}
%
%Consider the time derivative of (\ref{1}):
%\begin{equation}
%L_tv+Lv_t=\lambda_t v+\lambda v_t,
%\label{3}
%\end{equation}
%
%the spectral parameter $\lambda$ is assumed not to change in time; i.e. $\lambda _t=0$. Therefore,
%\begin{equation}
%L_tv+Lv_t=\lambda v_t.
%\label{4}
%\end{equation}
%
%Then since $v_t=Mv$ we can simplify (\ref{4}),
%\begin{equation}\nonumber
%L_tv+LMv =\lambda Mv,
%\end{equation}
%\begin{equation}\nonumber
%L_t+LM-\lambda MV=0,
%\end{equation}
%\begin{equation}\nonumber
%L_t+LMv-MLv=0,
%\end{equation}
%\begin{equation}
%L_t+[L,M]v=0,
%\label{5}
%\end{equation}
%
%where $[L,M]:=LM-ML$.
%
%Now consider the $n\times n$ matrix-valued operators $T$ and $X$, where $v$ is an $n$-dimensional vector,
%\begin{equation}
%v_x=Xv,
%\label{6}
%\end{equation}
%\begin{equation}
%v_t=Tv.
%\label{7}
%\end{equation}
%
%We can then take the $t$ derivative of (\ref{6}) and the $x$ derivative of (\ref{7}) and we then have,
%\begin{equation}
%v_{xt}=X_tv+Xv_t,
%\label{8}
%\end{equation}
%\begin{equation}
%v_{tx}=T_xv+Tv_x.
%\label{9}
%\end{equation}
%
%If $v$ is in $C^2$ in its argument $(x,t)$, then $v_{xt}=v_{tx}$. Thus,
%\begin{equation} \nonumber
%X_tv+Xv_t=T_xv+Tv_x,
%\end{equation}
%\begin{equation}\nonumber
%X_tv-T_xv+Xv_t-Tv_x=0,
%\end{equation}
%\begin{equation}\nonumber
%X_tv-T_xv+XTv-TXv=0,
%\end{equation}
%\begin{equation}
%X_tv-T_xv+[X,T]v=0.
%\label{10}
%\end{equation}
%
%Since $v$ is arbitrary, (\ref{10}) implies that,
%\begin{equation}
%X_t-T_x+XT-TX=0,
%\label{10a}
%\end{equation}
%
%which can be interpreted as nonlinear evolution equation.
%
%Now given $X$ you can find $T$ that satisfies the nonlinear evolution equation given in (\ref{10a}). The operators $X$ and $T$ are called the Lax pair associated to the system of differential equations given in (\ref{1}) and (\ref{2}). The appropriate choice or $X$ and $T$ will then determine which partial differential equation that the system of ordinary differential equations that defines $L$ will solve. But then $X$ should have a value, $\lambda$, that plays the role of and eigenvalue such that $\lambda _t=0$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Definition of $a, \overline{a}, b, \overline{b}$ and Properties}
%
%Since the potentials $q(x)$, $r(x)\rightarrow 0$ as $|x|\rightarrow \infty$, there are basic solutions,
%
%as $|x|\rightarrow-\infty$:
%\begin{equation}
%\varphi =
%\left[\begin{array}{clrr}
% 1 \\ 0
%\end{array}
%\right] e^{-i\lambda x} +o(1) \hspace{.5in} \overline{\varphi} =
%\left[\begin{array}{clrr}
% 0 \\ -1
%\end{array}
%\right] e^{i\lambda x}+o(1),
%\label{36}
%\end{equation}
%
%as $|x|\rightarrow\infty$:
%\begin{equation}
%\psi =
%\left[\begin{array}{clrr}
% 0 \\ 1
%\end{array}
%\right] e^{i\lambda x}+o(1) \hspace{.5in} \overline{\psi} =
%\left [ \begin{array}{clrr}
% 1 \\ 0
%\end{array}
%\right] e^{-i\lambda x}+o(1).
%\label{37}
%\end{equation}
%
%We know that if there are two functions, $u(x,\lambda)$ and $v(x,\lambda)$, that are solutions to (\ref{35}) then the $x$ derivative of the Wronskian of $u$ and $v$ is zero. Where the Wronskian is defined as $W(u,v)=u_1v_2-u_2v_1$. We then want to consider the Wronskian of our eigenfunctions,
%\begin{equation}
%W(\varphi, \overline{\varphi})=
%\left|\begin{array}{clrr}
% e^{-i\lambda x} & 0 \\
% 0 & e^{i\lambda x}
%\end{array}
%\right|=-e^{-i\lambda x}e^{i\lambda x}=-1,
%\label{29}
%\end{equation}
%\begin{equation}
%W(\psi, \overline{\psi})=
%\left|\begin{array}{clrr}
% 0 & e^{-i\lambda x} \\
% e^{i\lambda x} & 0
%\end{array}
%\right|=-e^{-i\lambda x}e^{i\lambda x}=-1.
%\label{30}
%\end{equation}
%
%Therefore the derivatives of these are in fact zero and then $\psi$ and $\overline{\psi}$ are linearly independent and $\vpbar$ and $\pbar$ are linearly independent, and also independent of $x$. We can then express $\varphi$ and $\overline{\varphi}$ in terms of $\psi$ and $\overline{\psi}$.
%\begin{equation}\nonumber
%\varphi=a(\lambda)\overline{\psi}+b(\lambda)\psi,
%\end{equation}
%\begin{equation}
%\overline{\varphi}=\overline{a}(\lambda)\psi+\overline{b}(\lambda)\overline{\psi}.
%\label{31}
%\end{equation}
%
%From this we can form the scattering matrix,
%\begin{equation}
%S=\left[\begin{array}{clrr}
% a(\lambda) & b(\lambda) \\
% \overline{b}(\lambda) & -\overline{a}(\lambda)
%\end{array}
%\right].
%\label{32}
%\end{equation}
%
%From (\ref{31}) since $W(\varphi, \overline{\varphi})=1$ we can say,
%\begin{equation}
%W(\varphi, \overline{\varphi})=
%\left|
%\begin{array}{clrr}
% a(\lambda) & b(\lambda) \\
% \overline{b}(\lambda) & -\overline{a} (\lambda)
%\end{array} \right| =-a(\lambda)\overline{a}(\lambda)-b(\lambda)\overline{b}(\lambda)=-1.
%\label{33}
%\end{equation}
%
%We then get a very useful identity,
%\begin{equation}
%a(\lambda)\overline{a}(\lambda)+b(\lambda)\overline{b}(\lambda)=1.
%\label{34}
%\end{equation}
%
%Recall that the Wronskian of any two solutions to (\ref{35}) are independent of $x$ and for our basic solutions is defined as,
%\begin{equation}
%W(\varphi, \psi)=
%\left|\begin{array}{clrr}
% \varphi _1 & \psi_1 \\
% \varphi_2 & \psi_2
%\end{array}\right|
%=\varphi_1\psi_2-\varphi_2\psi_1.
%\label{38}
%\end{equation}
%
%Now consider the $x$ derivative of (\ref{38}),
%\begin{equation}
%\displaystyle\frac{d}{dx}W(\varphi, \psi)=
%\left|\begin{array}{clrr}
% \varphi_{1}^{'} & \psi_{1} \\
% \varphi_{2}^{'} & \psi_{2}
%\end{array}
%\right|
%+
%\left|\begin{array}{clrr}
% \varphi_{1} & \psi_{1}^{'} \\
% \varphi_{2} & \psi_{2}^{'}
%\end{array}
%\right|,
%\label{39}
%\end{equation}
%
%where $'$ is the $x$ derivative.
%
%But we know from (\ref{36}) that we have expressions for $\varphi^{'}$, $\psi^{'}$ in terms of $\varphi$, $\psi$,
%\begin{equation}\nonumber
%\varphi_{1}^{'}=-i\lambda\varphi_{1}+q(x)\varphi_{2},
%\end{equation}
%\begin{equation}
%\varphi_{2}^{'}=i\lambda\varphi_{2}+r(x)\varphi_{1},
%\label{40}
%\end{equation}
%\begin{equation}\nonumber
%\psi_{1}^{'}=-i\lambda\psi_{1}+q(x)\psi_{2},
%\end{equation}
%\begin{equation}
%\psi_{2}^{'}=i\lambda\psi_{2}+r(x)\psi_{1}.
%\label{41}
%\end{equation}
%
%By Substituting (\ref{40}) and (\ref{41}) into (\ref{39}) we find,
%\begin{equation}
%\begin{array}{l}
%\displaystyle\frac{d}{dx}W(\varphi, \psi) =
%\left|\begin{array}{cc}
% -i\lambda\varphi_{1}+q(x)\varphi_{2} & \psi_{1} \\
% i\lambda\varphi_{2}+r(x)\varphi_{1}& \psi_{2}
%\end{array}
%\right|
%+
%\left|\begin{array}{cc}
% \varphi_{1} & -i\lambda\psi_{1}+q(x)\psi_{2} \\
% \varphi_{2} & i\lambda\psi_{2}+r(x)\psi_{1}
%\end{array}
%\right|\\
% =
%-i\lambda \left|\begin{array}{cc}
% \varphi_{1} & \psi_{1} \\
% -\varphi_{2} & \psi_{2}
%\end{array}
%\right| +
%\left|\begin{array}{cc}
% q(x)\varphi_{2} & \psi_{1} \\
% r(x)\varphi_{1} & \psi_{2}
%\end{array}
%\right| -i\lambda
%\left|\begin{array}{cc}
% \varphi_{1} & \psi_{1} \\
% \varphi_{2} & -\psi_{2}
%\end{array}
%\right| +
%\left|\begin{array}{cc}
% \varphi_{1} & q(x)\psi_{2} \\
% -\varphi_{2} & r(x)\psi_{1}
%\end{array}
%\right| \\
% =
%-i\lambda(\varphi_{1}\psi_{2}+\varphi_{2}\psi_{1})+q\varphi_{2}\psi_{2}-r\varphi_{1}\psi_{1}-i\lambda(-\varphi_{1}\psi_{2} -\varphi_{2}\psi_{1})+r\varphi_{1}\psi_{1}-q\varphi_{2}\phi_{2} \\
% = 0.
%\end{array}
%\end{equation}
%
%Then from the (\ref{31}) and (\ref{32}) we know,
%\begin{equation}
%\left[\begin{array}{cc}
% \varphi \\
% \overline{\varphi}
%\end{array}
%\right]=
%\left[\begin{array}{clrr}
% a(\lambda) & b(\lambda) \\
% \overline{b}(\lambda) & -\overline{a}(\lambda)
%\end{array}
%\right]
%\left[\begin{array}{clrr}
% \overline{\psi} \\
% \psi
%\end{array}
%\right].
%\label{42}
%\end{equation}
%
%Form this we can begin to discuss the properties of the spectral coefficients with respect to the eigenfunctions. Let $W(f,g)=[f,g]$, then consider,
%\begin{eqnarray}
%[\psi, \varphi] & = & \nonumber
%[\psi, a\overline{\psi} +b\psi]
%\\
%& = & \nonumber
%a[\psi, \overline{\psi}]+b[\psi,\psi]
%\\
%& = & \nonumber
%-a.
%\end{eqnarray}
%
%Therefore $[\varphi, \psi]=a$. Similarly consider,
%\begin{eqnarray}
%[\varphi, \overline{\psi}]] & = & \nonumber
%[a\overline{\psi}+b\psi, \overline{\psi}]
%\\
%& = & \nonumber
%a[\overline{\psi}, \overline{\psi}]+b[\psi,\overline{\psi}]
%\\
%& = & \nonumber
%-b.
%\end{eqnarray}
%
%Therefore $[\overline{\psi}, \varphi]=b$. Similarly consider,
%\begin{eqnarray}
%[\psi, \overline{\varphi}]] & = & \nonumber
%[\psi, \overline{b}\overline{\psi}\overline{a}\psi]
%\\
%& = & \nonumber
%\overline{b}[\psi, \overline{\psi}]-\overline{a}[\psi,\psi]
%\\
%& = & \nonumber
%-\overline{b}.
%\end{eqnarray}
%
%Therefore $[\overline{\varphi}, \psi]=\overline{b}$. Similarly consider,
%\begin{eqnarray}
%[\overline{\psi}, \overline{\varphi}]] & = & \nonumber
%[\overline{\psi},\overline{b}\overline{\psi}-\overline{a}\psi]
%\\
%& = & \nonumber
%\overline{b}[\overline{\psi}, \overline{\psi}]-\overline{a}[\overline{\psi},\psi]
%\\
%& = & \nonumber
%-\overline{a}.
%\end{eqnarray}
%
%Therefore $[\overline{\varphi},\overline{\psi}]=\overline{a}$.
%
%Notice that since $a(\lambda)$ is defined by $\varphi (\lambda, x)$ and $\psi (\lambda, x)$ which are analytic in the upper half complex plane, it to is analytic in the upper half complex plane. Similarly, $\overline{a}(\lambda)$ is defined by $\overline{\varphi} (\lambda, x)$ and $\overline{\psi} (\lambda, x)$ which are analytic in the lower half complex plane, so it too is analytic in the lower half complex plane. Now consider $b(\lambda)$ which is defined by $\varphi (\lambda, x)$ and $\overline{\psi} (\lambda, x)$ and $\overline{b}(\lambda)$ which is defined by $\overline{\varphi} (\lambda, x)$ and $\psi (\lambda, x)$. Therefore each will be analytic on a band about the real axis, the size of which depends on the functions $r(x)$ and $q(x)$. It is important to note that if the functions $r(x)$ and $q(x)$ vanish faster than an exponential function as $|x| \rightarrow \infty$, then $a(\lambda)$, $\overline{a}(\lambda)$, $b(\lambda)$, $\overline{b}(\lambda)$, $\varphi (\lambda, x)$, $\overline{\varphi} (\lambda, x)$, $\psi (\lambda, x)$ and $\overline{\psi} (\lambda, x)$ are entire functions in $\lambda$.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\section{Asymptotic Behavior of Solutions}
%
%To discuss the asymptotic behavior of the solutions $\psi$, $\overline{\psi}$, $\varphi$, and $\overline{\varphi}$ we have to consider the cases of each solution at both infinities. Recall,
%
%as $|x|\rightarrow-\infty$:
%\begin{equation}
%\varphi =
%\left[\begin{array}{clrr}
% 1 \\ 0
%\end{array}
%\right] e^{-i\lambda x} +o(1) \hspace{.5in} \overline{\varphi} =
%\left[\begin{array}{clrr}
% 0 \\ -1
%\end{array}
%\right] e^{i\lambda x}+o(1),
%\label{36a}
%\end{equation}
%
%as $|x|\rightarrow\infty$:
%\begin{equation}
%\psi =
%\left[\begin{array}{clrr}
% 0 \\ 1
%\end{array}
%\right] e^{i\lambda x} +o(1)\hspace{.5in} \overline{\psi} =
%\left[\begin{array}{clrr}
% 1 \\ 0
%\end{array}
%\right] e^{-i\lambda x}+o(1).
%\label{37a}
%\end{equation}
%
%We can also define these solutions at the opposite infinities in general. Since the potentials $q(x)$, $r(x)\rightarrow 0$ as $|x|\rightarrow \infty$, there are basic solutions,
%
%as $|x|\rightarrow-\infty$:
%\begin{equation}
%\psi =
%\left[\begin{array}{clrr}
% Ae^{-i\lambda x} \\ Be^{i\lambda x}
%\end{array}
%\right] +o(1) \hspace{.5in} \overline{\psi} =
%\left[\begin{array}{clrr}
% \overline{A}e^{-i\lambda x} \\ \overline{B}e^{i\lambda x}
%\end{array}
%\right] +o(1),
%\label{36b}
%\end{equation}
%
%as $|x|\rightarrow\infty$:
%\begin{equation}
%\varphi =
%\left[\begin{array}{clr}
% Ce^{-i\lambda x} \\ De^{i\lambda x}
%\end{array}
%\right] +o(1) \hspace{.5in} \overline{\varphi} =
%\left[\begin{array}{clrr}
% \overline{C}e^{-i\lambda x} \\ \overline{D}e^{i\lambda x}
%\end{array}
%\right] +o(1).
%\label{37b}
%\end{equation}
%
%Where $A$, $\overline{A}$, $B$, $\overline{B}$, $C$, $\overline{C}$, $D$, $\overline{D}$ are scattering coefficients. We then wish to establish a relation with these values and $a$, $\overline{a}$, $b$, $\overline{b}$. Consider,
%\begin{equation}
%-a=[\psi, \varphi]_{x=-\infty}=
%\left|\begin{array}{cc}
% Ae^{-i\lambda x} & e^{-i\lambda x} \\
% Be^{i\lambda x} & 0
%\end{array}
%\right|=-B.
%\label{37c}
%\end{equation}
%
%Similarly,
%
%\begin{equation}
%-a=[\psi, \varphi]_{x=\infty}=
%\left|\begin{array}{cc}
% 0 & Ce^{-i\lambda x} \\
% e^{i\lambda x} & De^{i\lambda x}
%\end{array}
%\right|=-C,
%\label{37d}
%\end{equation}
%
%therefore we find that $a=B=C$. Now consider,
%\begin{equation}
%-b=[\psi, \overline{\varphi}]_{x=-\infty}=
%\left|\begin{array}{cc}
% Ae^{-i\lambda x} & 0 \\
% Be^{i\lambda x} & -e^{i\lambda x}
%\end{array}
%\right|=-A.
%\label{37e}
%\end{equation}
%
%Similarly,
%\begin{equation}
%-b=[\psi, \overline{\varphi}]_{x=\infty}=
%\left|\begin{array}{cc}
% 0 & \overline{C}e^{-i\lambda x} \\
% e^{i\lambda x} & \overline{D}e^{i\lambda x}
%\end{array}
%\right|=-\overline{C},
%\label{37f}
%\end{equation}
%
%therefore we find $\overline{b}=A=\overline{C}$. Now consider,
%\begin{equation}
%-b=[\varphi, \overline{\psi}]_{x=-\infty}=
%\left|\begin{array}{cc}
% e^{-i\lambda x} & \overline{A}e^{-i\lambda x} \\
% 0 & \overline{B}e^{i\lambda x}
%\end{array}
%\right|=\overline{B}.
%\label{37g}
%\end{equation}
%
%Similarly,
%\begin{equation}
%-b=[\psi, \overline{\varphi}]_{x=\infty}=
%\left|\begin{array}{cc}
% Ce^{-i\lambda x} & e^{-i\lambda x} \\
% De^{i\lambda x} & 0
%\end{array}
%\right|=-D,
%\label{37h}
%\end{equation}
%
%therefore we find $b=-\overline{B}=D$. Again consider,
%\begin{equation}
%\overline{a}=[\overline{\varphi}, \overline{\psi}]_{x=-\infty}=
%\left|\begin{array}{cc}
% 0 & \overline{A}e^{-i\lambda x} \\
% -e^{i\lambda x} & \overline{B}e^{i\lambda x}
%\end{array}
%\right|=\overline{A}.
%\label{37i}
%\end{equation}
%
%Similarly,
%\begin{equation}
%\overline{a}=[\overline{\varphi}, \overline{\psi}]_{x=\infty}=
%\left|\begin{array}{cc}
% \overline{C}e^{-i\lambda x} & e^{-i\lambda x} \\
% \overline{D}e^{i\lambda x} & 0
%\end{array}
%\right|=-\overline{D},
%\label{37j}
%\end{equation}
%
%therefore we find $\overline{a}=\overline{A}=\overline{D}$. We can now represent $\psi$, $\overline{\psi}$, $\varphi$, and $\overline{\varphi}$ in terms of $a$, $\overline{a}$, $b$, $\overline{b}$ at both infinities.
%\begin{equation}
%\psi =
%\left[\begin{array}{clrr}
% 0 \\ e^{i\lambda x}
%\end{array}
%\right] +o(1)\,, \quad x\rightarrow \infty \hspace{.5in} \psi =
%\left[\begin{array}{clrr}
% \overline{b}e^{-i\lambda x} \\ ae^{i\lambda x}
%\end{array}
%\right] +o(1)\,, \quad x\rightarrow -\infty,
%\label{36k}
%\end{equation}
%\begin{equation}
%\overline{\psi} =
%\left[\begin{array}{clrr}
% e^{-i\lambda a} \\ 0
%\end{array}
%\right] +o(1) \,,\quad x\rightarrow \infty \hspace{.5in} \overline{\psi} =
%\left[\begin{array}{clrr}
% \overline{a}e^{-i\lambda x} \\ -be^{i\lambda x}
%\end{array}
%\right] +o(1) \,,\quad x\rightarrow -\infty,
%\label{36l}
%\end{equation}
%\begin{equation}
%\varphi =
%\left[\begin{array}{clrr}
% ae^{-i\lambda x} \\ be^{i\lambda x}
%\end{array}
%\right] +o(1)\,, \quad x\rightarrow \infty \hspace{.5in} \varphi =
%\left[\begin{array}{clrr}
% e^{-i\lambda x} \\ 0
%\end{array}
%\right] +o(1)\,, \quad x\rightarrow -\infty,
%\label{36m}
%\end{equation}
%\begin{equation}
%\overline{\psi} =
%\left[\begin{array}{clrr}
% \overline{b}e^{-i\lambda x} \\ -\overline{a}e^{i\lambda x}
%\end{array}
%\right] +o(1)\,, \quad x\rightarrow \infty \hspace{.5in} \psi =
%\left[\begin{array}{clrr}
% 0 \\ -e^{i\lambda x}
%\end{array}
%\right] +o(1) \,,\quad x\rightarrow -\infty.
%\label{36n}
%\end{equation}
%
%Now if we define the transmission coefficients as $T=\frac{1}{a}$ and $\overline{T}=\frac{1}{\overline{a}}$ and the reflection coefficients as $R=\frac{b}{a}$ and $\overline{R}=-\frac{\overline{b}}{\overline{a}}$. To see the physical reasoning for this consider $\frac{1}{a}\varphi$,
%\begin{equation}
%\frac{1}{a} \varphi=\varphi T=\left\{\begin{array}{l}\left[\begin{array}{c}e^{-i\lambda x} \\ \frac{b}{a}e^{i\lambda x}
% \end{array} \right]+o(1)\,,\quad x\rightarrow \infty \vspace{.1in}
% \\
%\left[\begin{array}{c}\frac{1}{a}e^{-i\lambda x} \\ 0 \end{array} \right]+o(1)\,, \quad x\rightarrow -\infty. \end{array} \right .
%\label{36o}
%\end{equation}
%
%With the time dependence $e^{-i \lambda t}$, consider the wave $e^{-i\lambda(x+t)}$ sent from $x=\infty$. Then $\frac{b}{a}$ is the reflection coefficient from the right and $\frac{1}{a}$ is the transmission coefficient from the right. Similarly consider,
%\begin{equation}
%\frac{1}{\overline{a}} \overline{\varphi}=\overline{\varphi} \overline{T}=\left\{\begin{array}{l}\left[\begin{array}{c} -\frac{\overline{b}}{\overline{a}}e^{-i\lambda x} \\ e^{i\lambda x}
% \end{array} \right] +o(1)\,, \quad x\rightarrow \infty \vspace{.1in}
% \\
%\left[ \begin{array}{c} 0 \\ \frac{1}{\overline{a}}e^{i\lambda x} \end{array} \right] +o(1) \,,\quad x\rightarrow -\infty, \end{array} \right .
%\label{36p}
%\end{equation}
%\begin{equation}
%\frac{1}{a} \psi= \psi T=\left\{\begin{array}{l}\left[\begin{array}{c} \frac{\overline{b}}{a}e^{-i\lambda x} \\ e^{i\lambda x}
% \end{array} \right] +o(1)\,, \quad x\rightarrow -\infty \vspace{.1in}
% \\
%\left[\begin{array}{c} 0 \\ \frac{1}{a}e^{i\lambda x} \end{array} \right] +o(1)\,, \quad x\rightarrow \infty ,\end{array} \right .
%\label{36q}
%\end{equation}
%\begin{equation}
%\frac{1}{\overline{a}} \overline{\psi}= \overline{\psi} \overline{T}=\left\{\begin{array}{l}\left[\begin{array}{c} e^{-i\lambda x} \\ -\frac{b}{a}e^{i\lambda x}
% \end{array} \right] +o(1)\,, \quad x\rightarrow -\infty \vspace{.1in}
% \\
%\left[\begin{array}{c} \frac{1}{a}e^{-i\lambda x} \\ 0 \end{array} \right] +o(1)\,, \quad x\rightarrow \infty .\end{array} \right .
%\label{36r}
%\end{equation}
%
%Notice that the transmission coefficients from the right are the same as the transmission coefficients from the left. We can now define the left reflection coefficients, $L=\frac{\overline{b}}{a}$ and $\overline{l}=-\frac{b}{\overline{a}}$. Now using what we know about the transmission and reflection coefficients we can derive some useful identities. Recall,
%\begin{equation}
%S=\left [ \begin{array}{cc} a & b \\ \overline{b} & -\overline{a} \end{array} \right]
%= \left [ \begin{array}{cc} \frac{1}{T} & -\frac{\overline{L}}{\overline{T}} \\ -\frac{\overline{R}}{\overline{T}} & -\frac{1}{\overline{T}} \end{array} \right]
%=\left [ \begin{array}{cc} \frac{1}{T} & -\frac{R}{T} \\ -\frac{L}{T} & -\frac{1}{\overline{T}} \end{array} \right].
%\label{36s}
%\end{equation}
%
%But then we can see that $\frac{R}{T}=-\frac{\overline{L}}{\overline{T}}$ and $\frac{L}{T}=-\frac{\overline{R}}{\overline{T}}$. These equalities can be reduced to $L\overline{L}=R\overline{R}$. Then since we know that $det(S)=1$ it can be shown that $1=T\overline{T}+R\overline{R}$ and also $1=T\overline{T}+L\overline{L}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Now consider the most general linear time dependence, where $A,B,C,D$ are scalar functions,
%\begin{equation}\nonumber
%\xi _t=A \xi +B\eta,
%\end{equation}
%\begin{equation}
%\eta _t=C \eta +D\xi,
%\label{12}
%\end{equation}
%
%with the assumptions,
%\begin{equation}\nonumber
%(\xi_x)_t=(\xi_t)_x,
%\end{equation}
%\begin{equation}\nonumber
%(\eta_x)_t=(\eta_t)_x,
%\end{equation}
%\begin{equation}
%\lambda _t=0.
%\label{13}
%\end{equation}
%
%From the assumptions in (\ref{13}) we find the following restrictions on our scalar functions,
%\begin{equation} \nonumber
%A_x=qC-rB,
%\end{equation}
%\begin{equation} \nonumber
%B_x+2i\lambda B=r_t+(A-D)q,
%\end{equation}
%\begin{equation} \nonumber
%C_x-2i\lambda C=r_t+ (A-D)r,
%\end{equation}
%\begin{equation}
%(-D)_x=qC-rB.
%\label{14a}
%\end{equation}
%
%Then if we let $-A=D$, we then find the following restrictions on our scalar functions,
%\begin{equation}\nonumber
%A_x=qC-rB,
%\end{equation}
%\begin{equation}\nonumber
%B+x+2i\lambda B=q_t-2Aq,
%\end{equation}
%\begin{equation}
%C_x-2i\lambda C=r_t+2Ar.
%\label{15}
%\end{equation}
%
%We now want to solve for $A,B,C$. To do this we first consider a power series for each in terms of $\lambda$.
%\begin{equation}\nonumber
%A=A_2\lambda ^2+A_1\lambda +A_0,
%\end{equation}
%\begin{equation}\nonumber
%B=B_2\lambda ^2+B_1\lambda +B_0,
%\end{equation}
%\begin{equation}
%C=C_2\lambda ^2+C_1\lambda +C_0.
%\label{16}
%\end{equation}
%
%Using the conditions in (\ref{14a}) we find,
%\begin{equation} \nonumber
%A_2=a_2 \hspace{.3in} A_1=a_1 \hspace{.3in} A_0=a_2\displaystyle\frac{qr}{2}+a_0,
%\end{equation}
%\begin{equation} \nonumber
%B_2=0 \hspace{.3in} B_1=ia_2q \hspace{.3in} B_0=-a_2\displaystyle\frac{q_x}{2},
%\end{equation}
%\begin{equation}
%C_2=0 \hspace{.3in} C_1=ia_2r \hspace{.3in} C_0=a_2\displaystyle\frac{r_x}{2}.
%\label{17}
%\end{equation}
%
%From this we obtain,
%\begin{equation}\nonumber
%-\frac{1}{2}a_2q_{xx}=q_t-a_2q^2r,
%\end{equation}
%\begin{equation}
%\frac{1}{2}a_2r_{xx}=r_t+a_2qr^2.
%\label{18a}
%\end{equation}
%
%We can see that if $r= \mp q^*$ we then form the nonlinear Schr\"odinger equation. Then if the formula in (\ref{18a}) are compatible then $a_2=i\alpha$ for all real $\alpha$. We also notice that (\ref{16}) is the compatibility condition for the 2$\times$2 problem with the time dependence.
%
%If the nonlinear Schr\"odinger equation holds for $q(x,t)$ we find the following,
%\begin{equation}\nonumber
%A=2i\lambda ^2 \pm iqq^*,
%\end{equation}
%\begin{equation}\nonumber
%B=2q\lambda +iq_{xx},
%\end{equation}
%\begin{equation}
%C=\mp2q^*\lambda \pm i(q_x)^*.
%\label{19}
%\end{equation}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%\section{Special Cases}
%
%The procedure of representing $A,B,C$ as expansions of $\lambda$ can be done for larger powers of $\lambda$. If this is done up to powers of $\lambda ^3$, we get three special cases:
%
%Special Case 1: The KdV equation, $q_t+6qq_x+q_{xxx}=0$, is obtained if
%\begin{equation}
%a_0=a_1=a_2=0 \hspace{.3in} a_3=-4i \hspace{.3in} r=-1.
%\label{20}
%\end{equation}
%
%Special Case 2: The modified KdV equation, $q_t\pm6q^2q_x+q_{xxx}=0$, is obtained if
%\begin{equation}
%a_0=a_1=a_2=0 \hspace{.3in} a_3=-4i \hspace{.3in} r=\mp q.
%\label{21}
%\end{equation}
%
%Special Case 3: The differential equation, $iq_t=q_{xx}\pm2q^2q^*$, is obtained if
%\begin{equation}
%a_0=a_1=a_3=0 \hspace{.3in} a_2=-2i \hspace{.3in} r=\mp q^*.
%\label{22}
%\end{equation}
%
%Similarly, consider,
%\begin{equation}
%A=\displaystyle\frac{a(x,t)}{\lambda} \hspace{.3in} B=\displaystyle\frac{b(x.t)}{\lambda} \hspace{.3in} C=\displaystyle\frac{c(x.t)}{\lambda}.
%\label{23}
%\end{equation}
%
%We then find that,
%\begin{equation}
%a_x=\frac{1}{2}qr \hspace{.3in} q_{xt}=-4iaq \hspace{.3in} r_{xt}-4iar.
%\label{24}
%\end{equation}
%
%We again have two special cases when $a,b,c,q,r$ are defined,
%
%Special case 1: The Sine-Gordon equation, $u_{xt}=\sin u$, is obtained if
%\begin{equation}
%a=\displaystyle\frac{i\cos u}{4} \hspace{.3in} b=-c=\displaystyle\frac{i\sin u}{4} \hspace{.3in} q=-r=-\displaystyle\frac{u_x}{2}.
%\label{25}
%\end{equation}
%
%Special case 2: the sinh-Gordon equation, $u_{xt}=\sinh u$, is obtained if
%\begin{equation}
%a=\displaystyle\frac{i\cosh u}{4} \hspace{.3in} b=-c=\displaystyle\frac{i\sinh u}{4} \hspace{.3in} q=r=\displaystyle\frac{u_x}{2}.
%\label{26}
%\end{equation}
\end{document}