text:
Wrexham landlord lifts lid on It's Always Sunny spoof filmed in his pub
01 Nov 2021 5 minute read
Wayne (bottom left) and the lookalike cast of It's Always Sunny In Philadelphia in The Turf (Credit: Wayne Jones)
The landlord of The Turf in Wrexham, Wayne Jones, has revealed what it was like when two Hollywood stars and a Danny DeVito lookalike descended on his pub.
When Rob McElhenney and Ryan Reynolds hosted a press conference in the stands at Wrexham last week, Wayne and his pub got a mention.
Hours later the two stars, more than 60 production staff and a cast of lookalikes crammed into his hostelry to film a spoof It's Always Sunny In Philadelphia trailer to promote the forthcoming docuseries Welcome To Wrexham, to be screened on Disney+ in the UK.
The evening had seen social media alight when a video of what many believed was the actual Danny DeVito appeared showing him walking into the pub.
Danny DeVito's just turned up in Wrexham now! What an amazing story this week is turning into 😍 pic.twitter.com/xmFuGYDal4
— Ben Sheppard (@BenSheppard) October 28, 2021
It turned out to be an Irish lookalike called Mark, who had been filming with the It's Always Sunny In Philadelphia cast in Ireland.
However, it didn't stop every news outlet in the UK and many internationally reporting the story.
Unsurprisingly, Wayne says it's been a hugely positive and enjoyable, but very tiring few days.
Giving the inside track on his crazy week to Wrexham FC podcast, Fearless In Devotion, he said: "I'm absolutely shattered if I'm honest, it's been a bit of a whirlwind week. A very, very nice week but also a very tiring and exhausting week. But the pros certainly outweigh the cons.
"The publicity that the football club, first and foremost, has received is obviously very important and on the back of that, you know, a local little businessman and his pub has also received a lot of attention. So as a whole, it's been very positive."
When Wayne got the call that the production team behind the docuseries wanted to film a trailer in the pub advertising Welcome To Wrexham, he admitted he possibly wasn't ready for what came next.
"First and foremost, you know, I'm just a normal local lad who runs a pub," he said. "But on Thursday we had 60 crew turning up to film, which was insane.
"Then I'm told I have to go into hair and make up, which was hilarious, and then appear in a scene with Rob McElhenney.
"Then people see images of Danny DeVito coming into the pub, putting two and two together, thinking that he was here. It was a lot to take in, but I loved it. I'll be honest, I've absolutely loved it.
The real cast of It's Always Sunny In Philadelphia (Credit: Publicity pic)
"The film crew, every single one was absolutely lovely, and the actors and the extras. It was a sight to behold to see what goes into such a production, even just a small two, three minute, what is a commercial effectively.
"So you try to think what goes on in these multi billion pound movies but I've really enjoyed it and really tried to make everyone feel as welcome as they can."
As for meeting Rob McElhenney and Ryan Reynolds, Wayne said he was hugely impressed by the Hollywood stars, even if his wife almost fainted on hearing they were coming to the pub.
Rob McElhenney and Ryan Reynolds take questions from the press at The Racecourse
The pair visited The Turf on the Wednesday evening for a few drinks to meet Wayne, his family and a number of fans, before the production team moved in on the Thursday.
"We got the call a couple of days before they arrived and were told they wanted to come and say hello, introduce themselves, have two or three hours and have some drinks," he said. "Yeah, that was a bit of a surreal moment. I shouted up to my wife, 'you need to get your head on because you're meeting Ryan Reynolds in a few days' time'. She nearly fell down the stairs, fainting."
Wayne filming a scene with Rob McElhenney (Credit: Wayne Jones)
'When they walked in Rob came over and said 'nice to finally meet you Wayne, which blew me away that they knew my name. They spent 20 minutes talking to my son about university and it was just as natural as it could be meeting Ryan Reynolds and Rob McElhenney.
"I was so impressed by what they had to say and their commitment to the football club. They're everything you read about them. Just really nice down to earth guys."
Listen to the full story of Rob McElhenney and Ryan Reynolds' visit to Wrexham on the Fearless In Devotion podcast


OPINION: Will the Super League succeed where the County Championship has failed?
This year's County Championship and T20 Cup have recently concluded – in what looks set to be the last domestic season of its kind ever.
This time next year the inaugural T20 element of the Women's Cricket Super League will have taken place – assuming all goes according to plan over the next few months.
The suggestion is that the current domestic T20 competition has failed to secure the interest of the public in the same way that England Women have in recent years.
If that is the case, it is surely at least partly the fault of the ECB.
1. T20 Finals Day – one of the most exciting days in the domestic calendar, which in previous years has attracted a crowd of several hundred – was cancelled ahead of this season, the winner of the T20 Cup this season being calculated simply based on who had scored the most points in the initial four rounds. No explanation was offered as to why the change was made, and CRICKETher are still not entirely clear about the rationale behind this.
2. On the final day of T20 games, it came down to a three-way tie between Yorkshire, Sussex and Kent on points, after all three won their final games. The winner would be decided by Net Run Rate… but who was it? Frantic calculations by Martin Davies from Women's Cricket Blog initially suggested that Yorkshire had won… until the official scorecards were published on Play Cricket 24 hours later, and it was shown that Sussex were in fact the champions, by the tiniest of margins (0.02).
Was anyone from the ECB on hand to do the required calculations on Sunday evening? No – the whole situation was farcical.
3. Sussex were presented with the T20 Cup after their final 50-over game against Berkshire, by Tim Shutt of the Sussex Cricket Board. The ECB once again showed their commitment to the domestic competition, by… sending no representative to the presentation whatsoever.
This is all in addition to the ridiculous way in which the 50-over Cup was decided this season, whereby Yorkshire were able to win the entire Championship by refusing to play 3rd-placed Sussex, with nothing in the regulations to prevent this from happening.
Sadly all these things are indicative of the fact that, despite all the progress made at international level in recent years, very little attention (or finances) have been devoted to advancing the domestic game.
The Super League is an exciting prospect. If it succeeds where the County Championship has failed… if indeed it does bring in sponsorship revenue and grow crowds at domestic games… it will be at least partly because of the time, money and effort invested in it by the ECB.
It just seems a shame that it has taken this long for the domestic game to be given any kind of priority.
I think anyone interested in women's cricket would agree that the game deserves better on a number of levels considering the efforts of thousands of volunteers to bring the game to this point domestically.
The WSL is a chance to raise the profile of the game in a competitive market and bring a greater depth of professionalism that is supposed to provide a higher quality product at the top end. I have no doubt the will be sponsorship to bolster the ECB contribution to the franchise running costs. But I do wonder if this will be an increase in sponsorship or a reallocation of funds?
The Women's Big Bash has shown some of the challenges in a cricket mad country we in England have more as a cricketing community that the WSL should not cover-up or distract from.
iii) ECB's aims for WCSL are laudable but are they achievable? They are expecting an awful lot from the "hosts" with little or no guaranteed return.
iv) ECB cannot afford to forget County Champs – vital step in player development ladder.


text:
\section{Introduction}\label{sect:intro}

Let $S$ be a regular local ring of dimension $d+1$, and let $f$ be a non-zero
non-unit of $S$. Let $R = S/(f)$ be the $d$-dimensional hypersurface ring.  Fix an
integer $n \geq 2$ and set $R^\# = S[y]/(f+y^n)$. We refer to $R^\#$ as the
\emph{($n$-fold) branched cover} of $R$.

We consider maximal Cohen-Macaulay (MCM) modules over $R$ and $R^\#$.  Our starting
point is the following theorem of Kn\"orrer and its generalization by Herzog-Popescu.
Write $\Omega$ for the syzygy operator.

\begin{theorem}[{\cite{Knorrer}}]
  \label{thm:Knorrer}
  Suppose that $S = k[\![x_0, \dots, x_d]\!]$ and the characteristic of $k$ is not
  equal to $2$. Assume $n=2$, so that $R^\# = S[y]/(f+y^2)$.  Then for any MCM
  $R^\#$-module $N$, we have
  \[
  \Omega_{R^\#}(N/yN) \cong N \oplus \Omega_{R^\#}(N)\,.
  \]
\end{theorem}

\begin{theorem}[{\cite[Theorem 2.6]{Herzog-Popescu:1997}}]
  \label{thm:Herzog-Popescu}
  Suppose that $S = k[\![x_0, \dots, x_d]\!]$ and the characteristic of $k$ does not divide
   $n$. Let $n \geq 2$ be arbitrary.  Then for any MCM $R^\#$-module $N$,
  we have
  \[
  \Omega_{R^\#}(N/y^{n-1}N) \cong N \oplus \Omega_{R^\#}(N)\,.
  \]

\end{theorem}

We remark that each of these results is a special case of the more general results
in the respective papers, the proofs of which do require the additional assumption
that $S$ be a power series ring over a field.  As a byproduct of our results below,
we will see that both results continue to hold true when $S$ is an arbitrary regular
local ring.

In this note we consider the full subcategories of $R^\#$-modules of the form
$\Omega_{R^\#}(M)$, where $M$ is annihilated by $y^k$, for $k = 1, \dots,
n$. Specifically, set $A_k = S[y]/(f,y^k)$ for $1 \leq k \leq n$, and define
\[
\Sigma_k = \add\left\{\Omega_{R^\#}(M) \,\middle|\, M \text{ is a MCM
    $A_k$-module}\right\}\,.
\]

We are interested in the categories
$\Sigma_1 \subseteq \Sigma_2 \subseteq \cdots \subseteq \MCM(R^\#)$.  Observe that
the results of Kn\"orrer and Herzog-Popescu above imply $\Sigma_{1} = \MCM(R^\#)$
when $n=2$, and $\Sigma_{n-1}=\MCM(R^\#)$ for arbitrary $n$, respectively.

Our main result is the following.

{\defA{A}
\begin{theorem}\label{thm:A}
  The subcategories $\Sigma_1, \dots, \Sigma_n$ are functorially finite in
  $\MCM(R^\#)$, that is, every MCM $R^\#$-module $N$ admits both left and right
  approximations by modules in $\Sigma_k$ for each $k$.  Namely, there are short
  exact sequences
  \[
  0 \to \Omega_{R^\#}(N) \to \Omega_{R^\#}(N/y^kN) \to N \to 0
  \]
  and
  \[
  0 \to N \to \Omega_{R^\#}(\Omega_{R^\#}(N)/y^k\Omega_{R^\#}(N)) \to
  \Omega_{R^\#}(N) \to 0
  \]
  which are right and left $\Sigma_k$-approximations of $N$, respectively.
\end{theorem}
}

We give two applications of Theorem~\ref{thm:A}.

First we extend a result of Iyama, Leuschke, and Quarles to the effect that the
endomorphism ring of a representation generator for the MCM modules over a simple
(ADE) hypersurface singularity has finite global dimension.  When
$f \in k[\![x_0, \dots, x_d]\!]$ defines a simple singularity of dimension
$d \geq 1$, the polynomial $f+y^n$ does not in general define a simple singularity
for $n \geq 3$, so that in particular the branched cover does not have a
representation generator. However we show that the $n$-fold branched cover admits a
MCM module $M$ such that $\Lambda = \End_{R^\#}(M)$ behaves like a ``non-commutative
hypersurface ring,'' in the sense that $\Lambda$ is Iwanaga-Gorenstein and every
finitely generated $\Lambda$-module has a projective resolution which is eventually
periodic of period at most $2$.

Second we give upper bounds on the dimension, in the sense of Rouquier, of the stable
category of MCM modules over a complete hypersurface ring defined by
a polynomial of the form $x_0^{a_0} + \cdots + x_d^{a_d}$.  These bounds sharpen
the general bounds of Ballard-Favero-Katzarkov~\cite{Ballard-Favero-Katzarkov:2012}
for complete isolated hypersurface singularities.

\begin{notation*}
  Unless otherwise specified, all modules are left modules.

  For a ring $\Gamma$ and a left $\Gamma$-module $X$, we write $\Omega_\Gamma(X)$ for
  an arbitrary first syzygy of $X$. This is uniquely defined up to projective
  summands.  If $\Gamma$ is commutative local, $\Omega_\Gamma(X)$ is unique up to
  isomorphism.
\end{notation*}

\section{Approximating MCM modules over the branched cover}
\label{sect:approx}

We make essential use of the following general lemma.

\begin{lemma}
  \label{lem:ses}
  Let $\Gamma$ be a ring and $X$ a $\Gamma$-module. Let $z \in \Gamma$ be a
  non-zerodivisor on $X$. Then there is a short exact sequence
  \begin{equation}\label{eq:gen-ses}
    0 \to \Omega_\Gamma(X) \to \Omega_\Gamma(X/zX) \to X \to 0
  \end{equation}
  of $\Gamma$-modules.
\end{lemma}

\begin{proof}
  Let $0 \to \Omega_\Gamma(X) \to F \to X \to 0$ be a free presentation of
  $X$. Multiplication by $z$ on $X$ induces a pullback diagram
  \[
  \xymatrix{
            &    & 0 \ar[d]   & 0 \ar[d] \\
    0 \ar[r] & \Omega_\Gamma(X) \ar[r] \ar@{=}[d]  & P \ar[r] \ar[d] & X \ar[r]
    \ar[d]^{z} & 0 \\
    0 \ar[r] & \Omega_\Gamma(X) \ar[r]  & F \ar[r] \ar[d] & X \ar[r] \ar[d] &
    0\\
            &    & X/zX \ar@{=}[r] \ar[d]  & X/zX \ar[d] \\
            &    & 0          & 0
  }
  \]
  the middle column of which shows that $P \cong \Omega_\Gamma(X/zX)$.
\end{proof}

\begin{lemma}
  \label{lem:speed-kills}
  Let $Q$ be a commutative Noetherian ring, $X$ a finitely generated $Q$-module, and
  $z$ a non-zerodivisor on $X$. The following conditions are equivalent.
  \begin{enumerate}[\quad(i)]
  \item $\Omega_Q (X/zX) \cong X \oplus \Omega_Q(X)$;
  \item $z$ annihilates $\Ext_Q^1(X,\Omega_Q(X))$;
  \item $z$ annihilates $\Ext_Q^1(X,Y)$ for every finitely generated $Q$-module $Y$.
  \end{enumerate}
\end{lemma}

\begin{proof}
  Indeed, the short exact sequence~\eqref{eq:gen-ses} is the image in
  $\Ext_Q^1(X,\Omega_Q(X))$ of $z\cdot \id_X \in \End_Q(X)$. Therefore $z$ kills
  $\Ext_Q^1(X,\Omega_Q(X))$ if and only if~\eqref{eq:gen-ses} splits, which by
  Miyata's theorem~\cite{Miyata} happens if and only if
  $\Omega_Q(X/zX) \cong X \oplus \Omega_Q(X)$.  The equivalence of the second and
  third conditions is well known.
\end{proof}

Now we consider the $n$-fold branched cover $R^\#$ of a hypersurface ring $R$.  As in
the Introduction, let $S$ be a regular local ring of dimension $d+1$, let $f$ be a
non-zero non-unit of $S$, and let $R = S/(f)$ be the $d$-dimensional hypersurface
ring.  Fix $n \geq 2$ and set $R^\# = S[y]/(f+y^n)$. Then Lemma~\ref{lem:ses} yields
the following.

\begin{proposition}
  \label{prop:ses-mf}
  Let $N$ be a MCM $R^\#$-module. For each $k \geq 1$, we have:
  \begin{enumerate}[\quad(i)]
  \item\label{item:ses} There are short exact sequences
    \begin{equation}\label{eq:approx}
      0 \to \Omega_{R^\#}(N) \to \Omega_{R^\#}(N/y^kN) \to N \to 0
    \end{equation}
    and
    \begin{equation}
      \label{eq:otherapprox}
      0 \to N \to \Omega_{R^\#}(\Omega_{R^\#}(N)/y^k\Omega_{R^\#}(N)) \to
      \Omega_{R^\#}(N) \to 0\,.
    \end{equation}

  \item \label{item:speed-kills} We have
    $\Omega_{R^\#}(N/y^k N) \cong N \oplus \Omega_{R^\#}(N)$ if and only if $y^k$
    annihilates $\Ext_{R^\#}^1(N, \Omega_{R^\#}(N))$.
  \end{enumerate}
\end{proposition}

\begin{proof}
  Existence of \eqref{eq:approx} and the second statement both follow immediately
  from Lemma~\ref{lem:ses}.  For \eqref{eq:otherapprox} it suffices to observe that
  since $R^\#$ is a hypersurface, if $N$ has no free direct summands, $N$ is
  isomorphic to its own second syzygy. If on the other hand $N$ is free, then both
  exact sequences degenerate to the isomorphism $R^\# \cong \Omega_{R^\#}(R^\#/(y^k))$.
\end{proof}

We emphasize that $f \neq 0$ is required for this result, since we need $y$ to be a
non-zerodivisor on $R^\#$.

\begin{remark}
  \label{rmk:Ragnar} We observe that the results of Kn\"orrer and Herzog-Popescu from
  the Introduction hold true in our level of generality. The proof of~\cite[Theorem
  2.6]{Herzog-Popescu:1997} uses the \emph{Noether different} $\mathcal {N}_A^B$ of a
  ring homomorphism $A \to B$.  Let $\mu \colon B \otimes_A B \to B$ denote the
  multiplication map, and define
  \[
  \mathcal {N}_A^B = \mu\left(\Ann_{B\otimes_A B}( \ker \mu)\right)\,.
  \]
  The two results we need about the Noether different are contained
  in~\cite{Buchweitz:1987} in the generality required.

  First,~\cite[Theorem 7.8.3]{Buchweitz:1987} implies that if $A \to B$ is a
  homomorphism of commutative Noetherian rings of finite Krull dimension, with $A$
  regular and $B$ Gorenstein, and such that $B$ is a finitely generated free
  $A$-module, then $\mathcal {N}_A^B$ annihilates all $\Ext$-modules $\Ext_B^1(X,Y)$
  with $X$, $Y$ finitely generated $B$-modules.

  The second result is contained in~\cite[Example 7.8.5]{Buchweitz:1987}. If $A$ is
  an arbitrary commutative ring and $B = A[y]/(g(y))$ for some polynomial $g(y) \in
  A[y]$, then the (formal) derivative $g'(y)$ is contained in $\mathcal {N}_A^B$.
\end{remark}

\begin{proposition}
  \label{prop:newHP}
  Let $S$ be an arbitrary regular local ring, $f \in S$ a non-zero non-unit,
  $R = S/(f)$, and $R^\# = S[y]/(f+y^n)$ for some $n \geq 1$. Assume that $n$ is a
  unit in $S$. Then for any MCM $R^\#$-module $N$, we have
  \[
  \Omega_{R^\#}(N/y^{n-1} N) \cong
  N \oplus \Omega_{R^\#}(N)\,.
  \]
\end{proposition}

\begin{proof}
  By Proposition~\ref{prop:ses-mf}(\ref{item:speed-kills}), it is enough to show
  that $y^{n-1}$ annihilates $\Ext_{R^\#}^1(N, \Omega_{R^\#}(N))$.  Since
  $\frac{\partial} {\partial y}(f+y^n) = n y^{n-1}$ and $n$ is invertible, this
  follows from Remark~\ref{rmk:Ragnar}.
\end{proof}

Next we want to consider the relationship between the short exact
sequence~\eqref{eq:approx} and the module $N$.  We recall here some definitions from
``relative homological algebra''.

\begin{definition}
  \label{def:approx}
  Let ${\mathcal A}$ be an abelian category, ${\mathcal X} \subseteq {\mathcal A}$ a full subcategory, and
  $M$ an object of ${\mathcal A}$.
  \begin{enumerate}[\quad(i)]
  \item A \emph{right ${\mathcal X}$-approximation} (or \emph{${\mathcal X}$-precover}) of $M$ is a
    morphism $q \colon X \to M$, with $X \in {\mathcal X}$, such that any $f \colon X' \to
    M$ with $X' \in {\mathcal X}$ factors through $q$. Equivalently, the induced
    homomorphism $\Hom_{\mathcal A} (X',X) \to \Hom_{\mathcal A}(X',M)$ is surjective.
  \item A \emph{left ${\mathcal X}$-approximation} (or \emph{${\mathcal X}$-preenvelope}) is a
    morphism $j \colon M \to X$ with $X \in {\mathcal X}$ such that any $g \colon M \to X'$
    with $X' \in {\mathcal X}$ factors through $j$.  Equivalently the induced homomorphism
    $\Hom_{\mathcal A}(X,X') \to \Hom_{\mathcal A}(M,X')$ is surjective.
  \item A right ${\mathcal X}$-approximation $q \colon X \to M$ is \emph{minimal} (or a
    \emph{${\mathcal X}$-cover}) if every endomorphism $\phi \in \End_{\mathcal A}(X)$ with $\phi q = q$ is an
    isomorphism.  Dually, a left ${\mathcal X}$-approximation $j \colon M \to X$ is
    \emph{minimal} (a \emph{${\mathcal X}$-envelope}) if every endomorphism $\phi
    \in \End_{\mathcal A}(X)$ with $j \phi = j$ is an isomorphism.
  \item We say that ${\mathcal X}$ is \emph{contravariantly finite} (resp.\
    \emph{covariantly finite}) in ${\mathcal A}$ if every object $M$ of ${\mathcal A}$ has a right
    (resp.\ left) ${\mathcal X}$-approximation.
  \item A sequence $A \stackrel{f}{\to} B \stackrel{g}{\to} C$ is
    \emph{${\mathcal X}$-exact} (resp. \emph{${\mathcal X}$-coexact}) if the induced sequence
    $\Hom_{\mathcal A}(X,A) \stackrel{f_*}{\to} \Hom_{\mathcal A}(X,B) \stackrel{g_*}{\to}
    \Hom_{\mathcal A}(X,C)$ (resp.
    $\Hom_{\mathcal A}(C,X) \stackrel{g^*}{\to} \Hom_{\mathcal A}(B,X) \stackrel{f^*}{\to}
    \Hom_{\mathcal A}(A,X)$) is exact for all $X \in {\mathcal X}$.
  \end{enumerate}
\end{definition}

Approximations need not exist, and even when they do, minimal approximations need not
exist.

We routinely abuse language by referring to the object $X$ as an approximation of $M$
(on the appropriate side). Furthermore, when ${\mathcal A}$ is a module category and a right
(resp.\ left) ${\mathcal X}$-approximation happens to be surjective (resp.\ injective) we
will refer to the whole short exact sequence $0 \to \ker q \to X \xto{q} M \to 0$
(resp.\ $0 \to M \xto{j}  X \to \cok j \to 0$) as the approximation.

The next result contains Theorem~\ref{thm:A} from the Introduction.  Recall the
definition of $\Sigma_k$:
\[
\Sigma_k = \add\left\{\Omega_{R^\#}(M) \,\middle|\, M \text{ is a MCM
    $A_k$-module}\right\}\,.
\]

\begin{theorem}\label{thm:approx}
  Let $N$ be a MCM $R^\#$-module.  The short exact sequence \eqref{eq:approx} is a
  right $\Sigma_k$-approximation of $N$, and \eqref{eq:otherapprox} is a left
  $\Sigma_k$-approximation of $N$.
\end{theorem}

\begin{proof}
  To show that \eqref{eq:approx} is a $\Sigma_k$-approximation, it suffices to show
  that an arbitrary $R^\#$-homomorphism $\sigma \colon M \to N$, for
  $M \in \Sigma_k$, factors through $\Omega_{R^\#}(N/y^k N)$.  Equivalently, $\sigma$
  maps to zero in $\Ext_{R^\#}^1(M, \Omega_{R^\#}(N))$ in the long exact sequence
  \[
  \Hom_{R^\#}(M,\Omega_{R^\#}(N/y^k N)) \to \Hom_{R^\#}(M,N) \to
  \Ext_{R^\#}^1(M,\Omega_{R^\#}(N))
  \]
  obtained by applying $\Hom_{R^\#}(M,-)$ to \eqref{eq:approx}.

  Recall that \eqref{eq:approx} is the image of the short exact sequence
  $\chi = (0 \to \Omega_{R^\#}(N) \to F \to N \to 0) \in
  \Ext_{R^\#}^1(N,\Omega_{R^\#}(N))$ under multiplication by $y^k$.  It therefore
  suffices to show that $y^k\chi$ maps to zero in
  $\Ext_{R^\#}^1(M, \Omega_{R^\#}(N))$.  Since $M \in \Sigma_k$, we can write
  $M\oplus M' \cong \Omega_{R^\#}(X)$ for an $R^\#$-module $X$ annihilated by $y^k$
  and some $R^\#$-module $M'$. Then
  \[
  \begin{split}
    \Ext_{R^\#}^1(M\oplus M', \Omega_{R^\#}(N))
    &= \Ext_{R^\#}^1(\Omega_{R^\#}(X), \Omega_{R^\#}(N))\\
    &\cong \Ext_{R^\#}^2(X, \Omega_{R^\#}(N))
  \end{split}
  \]
  is annihilated by $y^k$ since $X$ is, which proves the claim.

  The statement about left approximation follows similarly, using the fact that
  \[
  \begin{split}
    \Ext_{R^\#}^1(N,M\oplus M')
    &= \Ext_{R^\#}^1(N, \Omega_{R^\#}(X))\\
    &\cong \Ext_{R^\#}^2(\Omega_{R^\#}(N), \Omega_{R^\#}(X))\\
    &\cong \Ext_{R^\#}^1(\Omega_{R^\#}(N), X)\
  \end{split}
  \]
  is also killed by $y^k$.
\end{proof}

Standard arguments about approximations now yield the following.

\begin{corollary}\label{cor:splitApprox} For a MCM $R^\#$-module $N$ and $k \geq 1$, the following are equivalent:
\begin{enumerate}
\item $N \in \Sigma_k$;
\item The sequence~\eqref{eq:approx} splits;
\item The sequence~\eqref{eq:otherapprox} splits;
\item $\Omega_{R^\#}(N) \in \Sigma_k$;
\item $y^k$ annihilates $\Ext^1_{R^\#}(N,\Omega_{R^\#}(N))$.
\end{enumerate}
\qed
\end{corollary}

\begin{remark}\label{rmk:relativeHomology} In \cite{AusSol:1993} Auslander and
  Solberg develop relative homological algebra from the perspective of sub-functors
  of $\Ext^1(-,-)$.  For any $k \geq 1$, $F_k(-,-) := y^k \Ext^1_{R^\#}(-,-)$ is such
  a sub-functor on $\MCM(R^\#)$.  The corresponding category of relatively
  $F_k$-projectives consists of all $N$ such that
  $F_k(N,-) = y^k \Ext^1_{R^\#}(N,-) = 0$, which coincides with $\Sigma_k$.
  Likewise, the category of relatively $F_k$-injectives, consisting of all $N$ such
  that $F_k(-,N) = y^k \Ext^1_{R^\#}(-,N) = 0$, also coincides with $\Sigma_k$, as
  $\Sigma_k$ is closed under syzygies and $\Omega^2_{R^\#}$ is isomorphic to the
  identity.  The fact that $\Sigma_k$ is functorially finite translates to the
  existence of enough relative $F_k$-projectives (and injectives).  Moreover, as the
  relative projectives and injectives coincide, we see that for each $k\geq 1$
  $\MCM(R^\#)$ has a (relative) Frobenius category structure with $\Sigma_k$ as the
  (relative) projectives.
\end{remark}

\begin{prop}
  Assume in addition that $S$ is complete and that $N$ is indecomposable.  Then
  \eqref{eq:approx} is a minimal right approximation of $N$ as long as it is not
  split.
\end{prop}

\begin{proof}
  Since MCM $R^\#$-modules satisfy the Krull-Remak-Schmidt uniqueness property for
  direct-sum decompositions, a right approximation $q \colon X \to M$ is non-minimal
  if and only if it vanishes on a direct summand of $X$.  When $N$ is indecomposable,
  the kernel of the approximation $\Omega_{R^{\#}}(N)$ is indecomposable as well, and
  it cannot contain a direct summand of $X$ other than itself, in which case the
  sequence splits.
\end{proof}

It does not seem possible in general to identify the first $k$ so that
\eqref{eq:approx} splits.  By Proposition~\ref{prop:ses-mf} and the long exact
sequence in $\Ext$, it coincides with the least $k$ so that multiplication $y^k$ on
$N$ factors through a free $R^\#$-module. Equivalently, multiplication by $y^k$ is in
the image of the natural map $N^* \otimes_{R^\#} N \to \End_{R^\#}(N)$.  In the
special case $n=2$, it is known (see for example~\cite[Proposition
8.30]{Leuschke-Wiegand:BOOK}) that for an indecomposable MCM $R$-module $M$,
$\Omega_{R^\#}(M)$ is indecomposable if and only if $M \cong \Omega_R(M)$.  As far as
we are aware, no criterion of this form is known for $n \geq 3$.

\section{Branched covers of simple singularities} \label{sect:simple}

In this section we consider the case where $R = S/(f)$ is a hypersurface of
\emph{finite CM representation type} (of dimension $d\geq 1$). If $S$ is a power
series ring over an algebraically closed field of characteristic zero, this is
equivalent to $R$ being a \emph{simple singularity,} that is,
$R \cong k[\![x,y,z_2, \dots, z_d]\!]/(f)$, where $f$ is one of the following
polynomials:
\begin{enumerate}
  \setlength\itemindent{.2\linewidth}
\item[$(A_n)$:]  \qquad$x^2 + y^{n+1}+z_2^2+\cdots+z_d^2$ , \qquad $n \geq 1$
\item[$(D_n)$:]  \qquad$x^2y + y^{n-1}+z_2^2+\cdots+z_d^2$ , \qquad $n \geq 4$
\item[$(E_6)$:]  \qquad$x^3 + y^4+z_2^2+\cdots+z_d^2$
\item[$(E_7)$:]  \qquad$x^3 + xy^3+z_2^2+\cdots+z_d^2$
\item[$(E_8)$:]  \qquad$x^3 + y^5+z_2^2+\cdots+z_d^2$
\end{enumerate}
We point out that each of these is an iterated double ($2$-fold) branched cover of
the one-dimensional simple singularity; Kn\"orrer's Theorem~\ref{thm:Knorrer} allows
one to show that these hypersurface rings have finite CM representation type by
reducing to the $1$-dimensional case.  For $n \geq 3$, adding a summand of the form
$z^n$ to a simple singularity gives a non-simple singularity (except in type $A_1$
and some isolated examples for small $n$).

Since $R$ has finite CM representation type, the category
$\Sigma_1 \subset \MCM (R^\#)$ has only finitely many indecomposable objects up to
isomorphism.  Let $M$ be a representation generator for $\MCM R$, that is, $M$
satisfies $\add M = \MCM R$, so that in particular $\tilde{M} := \Omega_{R^\#}(M)$ is
a representation generator for $\Sigma_1$.  We note that $\tilde{M}$ has a direct
summand isomorphic to $R^\#$.

We set $\Lambda_0 = \End_R(M)$ and $\Lambda = \End_{R^\#}(\tilde{M})$.  Recall the
following result due independently to Iyama~\cite{Iyama:Aus-corr},
Leuschke~\cite{Leuschke:finrepdim}, and Quarles~\cite{Quarles:thesis}.

\begin{prop}\label{prop:gldimfinite}
  The endomorphism ring $\Lambda_0$ has global dimension at most $\max\{2, d\}$,
  with equality holding if $d \geq 2$. \qed
\end{prop}

The main result of this section is that $\Lambda$ can be thought of as a
``non-commutative hypersurface'', in that every finitely generated $\Lambda$-module
has a projective resolution which is eventually periodic of period at most $2$.
Furthermore $\Lambda$ is Iwanaga-Gorenstein, and we identify the relative
Gorenstein-projectives.

\begin{definition}\label{def:completeres}
  Let $A$ be a ring and $M$ a left $A$-module.  A \emph{complete resolution} of $M$
  is an exact sequence of projective $A$-modules
  \[
  {\mathbb P}_{\bullet} = \cdots \to P_{1} \stackrel{d_1}{\to} P_0 \stackrel{d_0}{\to} P_{-1}
  \to \cdots
  \]
  such that $\Hom_A({\mathbb P}_{\bullet},A)$ is exact and $M \cong \im (d_0)$.  A module $M$
  admitting a complete resolution is said to be \emph{Gorenstein projective} or
  \emph{totally reflexive}.  We will write $\GP(A)$ for the full subcategory of
  $\lmod{A}$ consisting of Gorenstein projective modules.
\end{definition}

We regard $\tilde{M}$ as a left $R^\#$-module and a right $\Lambda$-module, so that
we have functors $\Hom_{R^\#}(\tilde{M}, - ) \colon \lMod{R^\#} \to \lMod{\Lambda}$
and $\Hom_{R^\#}(-,\tilde{M}) \colon \lMod{R^\#} \to \rMod{\Lambda}$ which induce
equivalences from $\add{\tilde{M}}$ to the subcategories of finitely generated
projective left (respectively, right) $\Lambda$-modules.

\begin{lemma}\label{lemma:homfunctors}
  Restricted to $\MCM(R^\#)$, we have a commutative diagram of functors up to natural
  isomorphism.
  \[
    \xymatrixcolsep{5.0pc} \xymatrix{\MCM(R^\#) \ar[r]^{\Hom_{R^\#}(\tilde{M},-)}
    \ar[dr]_{\Hom_{R^\#}(-,\tilde{M})\ \ } & \Lambda\mbox{-}\mod
    \ar[d]^{\Hom_{\Lambda}(-,\Lambda)} \\ & \mod\mbox{-}\Lambda}
  \]
\end{lemma}

\begin{proof}
  By Theorem~\ref{thm:approx}, any MCM $R^\#$-module $N$ has a right
  $\Sigma_1$-approximation
  \begin{equation}\label{eq:rtApprox}
    0 \to \Omega_{R^\#} (N) \to T \to N \to 0\,,
  \end{equation}
  where in fact $T = \Omega_{R^\#}(N/yN)$. Furthermore $\Omega_{R^\#}(N)$ has a right
  $\Sigma_1$-approximation
  \begin{equation}\label{eq:ltApprox}
    0 \to N \to \Omega_{R^\#} (T) \to \Omega_{R^\#} (N)  \to 0\,.
  \end{equation}
  Since these sequences are $\Sigma_1$-exact and $\Sigma_1$-coexact, we obtain exact
  sequences of $\Lambda$-modules
  \[
    0 \to \Hom_{R^\#}(\tilde{M},N) \to \Hom_{R^\#}(\tilde{M},\Omega_{R^\#}(T)) \to
    \Hom_{R^\#}(\tilde{M}, T) \to \Hom_{R^\#}(\tilde{M},N) \to 0
  \]
  and
  \[
    0 \to \Hom_{R^\#}(N,\tilde{M}) \to \Hom_{R^\#}(T,\tilde{M}) \to
    \Hom_{R^\#}(\Omega_{R^\#}(T),\tilde{M}) \to \Hom_{R^\#}(N,\tilde{M}) \to 0\,,
  \]
  which yield projective presentations for the $\Lambda$-modules
  $\Hom_{R^\#}(\tilde{M},N)$ and $\Hom_{R^\#}(N,\tilde{M})$ respectively.  Thus these
  modules are finitely presented over $\Lambda$.  We now apply
  $\Hom_{\Lambda}(-,\Lambda)$ to the first of these and compare with the second to
  obtain an exact commutative diagram
  \[\small
    \xymatrix{ 0 \ar[r] & \Hom_{\Lambda}(\Hom_{R^\#}(\tilde{M},N),\Lambda) \ar[r] & \Hom_{\Lambda}(\Hom_{R^\#}(\tilde{M},T),\Lambda) \ar[r] & \Hom_{\Lambda}(\Hom_{R^\#}(\tilde{M},\Omega_{R^\#}(T)),\Lambda) \\
      0 \ar[r] & \Hom_{R^\#}(N,\tilde{M}) \ar@{-->}[u] \ar[r] &
      \Hom_{R^\#}(T,\tilde{M}) \ar[u]^{\cong} \ar[r] &
      \Hom_{R^\#}(\Omega_{R^\#}(T),\tilde{M}) \ar[u]^{\cong}}
  \]
  Here the vertical maps are induced by the functor $\Hom_{R^\#}(\tilde{M},-)$, and
  the two on the right are isomorphisms since
  $T, \Omega_{R^\#}(T) \in \add{\tilde{M}}$.  It follows that the induced map on the
  left is also an isomorphism.
\end{proof}

\begin{prop}\label{prop:completeres}
  Let $Z$ be a left (resp.\ right) $\Lambda$-module of the form
  $\Hom_{R^\#}( \tilde{M}, N)$ (resp. $\Hom_{R^\#}(N,\tilde{M})$), where $N$ is a MCM
  $R^\#$-module. Then $Z$ has a complete $\Lambda$-resolution which is periodic of
  period at most $2$.
\end{prop}

\begin{proof}
  As in the preceding proof we have exact sequences \eqref{eq:rtApprox} and
  \eqref{eq:ltApprox}, which we can splice together to form a doubly-infinite
  $2$-periodic exact sequence of $R^\#$-modules
 \begin{equation}\label{eq:completeSigmaRes}
  {\mathbb T}\colon \cdots \to \Omega_{R^\#} (T) \to T \to \Omega_{R^\#} (T) \to T \to \cdots\,.
  \end{equation}
  Since $\tilde{M}$ is in $\Sigma_1$, the induced sequences
  $\Hom_{R^\#}(\tilde{M},{\mathbb T})$ and $\Hom_{R^\#}({\mathbb T},\tilde{M})$ are still exact, and
  since both $T$ and $\Omega_{R^\#} (T)$ are in $\Sigma_1 = \add{\tilde{M}}$, each
  module in the sequence is a projective $\Lambda$-module. Thus
  $\Hom_{R^\#}(\tilde{M},{\mathbb T})$ (respectively, $\Hom_{R^{\#}}({\mathbb T},\tilde{M})$) is a
  projective resolution (and co-resolution) of $Z = \Hom_{R^\#}(\tilde{M}, N)$
  (respectively, of $Z' = \Hom_{R^\#}(N,\tilde{M})$).

  It remains to show that $\Hom_{R^\#}(\tilde{M},{\mathbb T})$ and
  $\Hom_{R^\#}({\mathbb T},\tilde{M})$ are complete resolutions, that is, remain exact upon
  dualizing into $\Lambda$.  However, by Lemma~\ref{lemma:homfunctors} we know that
  these resolutions are dual to each other under $\Hom_{\Lambda}(-,\Lambda)$, and the
  result follows since they are both exact.

\end{proof}

\begin{remark}
  In fact the resolution we build is something stronger than periodic of period $2$
  -- it is built out of a single map $f \colon \Omega (T) \to T$ and its syzygy
  $\Omega (f) \colon T \to \Omega(T)$.
\end{remark}

\begin{lemma}
  \label{lem:MCMsyz}
  Set $m = \max\{2,d+1\}$.  Let $X$ be an arbitrary finitely generated left (resp.\
  right) $\Lambda$-module.  Then there is a MCM $R^\#$-module $N$ such that
  $\Omega_\Lambda^m(X) \cong \Hom_{R^\#}(\tilde{M},N)$ (resp.
  $\Omega_{\Lambda}^m(X) \cong \Hom_{R^\#}(N,\tilde{M})$).
\end{lemma}

\begin{proof}
  First assume that $X$ is a left $\Lambda$-module.  Set $Z = \Omega_\Lambda^m(X)$, and let
  \[
  0 \to Z \to P_{m-1} \xto{\partial_{m-1}} P_{m-2} \to \cdots \to P_1
  \xto{\partial_1} P_0 \to X \to 0
  \]
  be the first $m-1$ steps of a resolution of $X$ by finitely generated projective
  $\Lambda$-modules. By the Yoneda Lemma, each $\partial_{j} \colon P_j \to P_{j-1}$
  is of the form $\Hom_{R^\#}( \tilde{M}, f_j)$ for some $R^\#$-linear
  $f_j\colon M_j \to M_{j-1}$, where each $M_j \in \add \tilde{M}$.  (Notice
  that since $m \geq 2$, there is indeed at least one $\partial_j$.)  We thus obtain
  a sequence of $R^\#$-modules and homomorphisms
  \begin{equation}\label{eq:Yoneda}
    0 \to \ker f_{m-1} \to M_{m-1} \xto{f_{m-1}} M_{m-2} \to \cdots \to M_1 \xto{f_1}
    M_0 \,.
  \end{equation}
  Since $\tilde{M}$ has an $R^\#$-free summand and the result of applying
  $\Hom_{R^\#}( \tilde{M},-)$ to \eqref{eq:Yoneda} is exact, in fact
  \eqref{eq:Yoneda} must be exact.  In particular $N := \ker f_{m-1}$  has depth at
  least $m \geq d+1$, and is hence a MCM $R^\#$-module. Furthermore, we have
  $\Hom_{R^\#}(\tilde{M},N) \cong Z$ by left-exactness of $\Hom$.

  Similarly, if $X$ is a right $\Lambda$-module, we can take a projective resolution
  ${\mathbb P} \cong \Hom_{R^\#}({\mathbb M},\tilde{M})$ of $X$, where ${\mathbb M}$ denotes a sequence
  $M_0 \stackrel{f_1}{\to} M_1 \to \cdots \to M_{m-2} \stackrel{f_{m-1}}{\to} M_{m-1}
  \to \cdots$ in $\add{\tilde{M}}$. Since $R^\#$ is a summand of $\tilde{M}$ and
  ${\mathbb P}$ is exact, we have an exact sequence
  \[
    0 \to K \to M_{m-1}^* \stackrel{f_{m-1}^*}{\to} M_{m-2}^* \to \cdots \to M_1^*
    \stackrel{f_1^*}{\to} M_0^*\,,
  \]
  where we set $K := \ker(f_{m-1}^*)$, writing $(-)^*$ for the exact duality
  $\Hom_{R^\#}(-,R^\#)$. Since each $M_i^*$ is a MCM $R^{\#}$-module, $K$ has depth
  at least $m \geq d+1$, and is hence a MCM $R^\#$-module.  Taking the $R^\#$-dual
  now shows that $K^* \cong \cok(f_{m-1})$, and the left-exactness of
  $\Hom_{R^\#}(-,\tilde{M})$ yields that
  $\Hom_{R^\#}(K^*,\tilde{M}) \cong \ker \Hom_{R^\#}(f_{m-1},\tilde{M}) \cong
  \Omega_{\Lambda}^{m}(X)$.
\end{proof}

\begin{theorem}\label{thm:hypersurface}
  Let $R = S/(f)$ be a hypersurface of finite CM representation type, with
  representation generator $M$. Let $R^\#  = S[y]/(f+y^n)$ for some $n \geq 1$, and
  set $\Lambda = \End_{R^\#}(\Omega_{R^\#}(M))$. Then every finitely
generated left (resp.\ right) $\Lambda$-module has a projective resolution which is
  eventually periodic of period at most $2$.
\end{theorem}

\begin{proof}
  Let $X$ be a finitely generated left (resp.\ right) $\Lambda$-module. By
  Lemma~\ref{lem:MCMsyz}, $\Omega_\Lambda^m(X)$ is of the form
  $\Hom_{R^\#}(\tilde{M},N)$ (resp. $\Hom_{R^\#}(N,\tilde{M})$) for some MCM
  $R^\#$-module $N$, where $m = \max\{2, \dim R + 1\}$. It then follows from
  Proposition~\ref{prop:completeres} that $\Omega_\Lambda^m(X)$ has a $2$-periodic
  complete resolution.  Splicing the left-hand half of this complete resolution
  together with the first $m-1$ steps of the projective resolution of $X$, we obtain
  an eventually periodic resolution.
\end{proof}

\begin{remark}\label{rem:notreally}
  While we think of the ring $\Lambda^\#$ as a ``non-commutative hypersurface'' in a
  homological sense, on the basis of Theorem~\ref{thm:hypersurface}, we should
  emphasize that as far as we know there does not exist a ring $\Gamma$ of finite
  global dimension such that $\Lambda^\# \cong \Gamma/(g)$ for some element $g$.
\end{remark}

\begin{remark}\label{rem:manyobjects}
  It seems likely that some version of many of the results in this section hold more
  generally (that is, without the assumption that $R$ is a simple singularity) if one
  uses the formalism of ``rings with several objects'' as in, for
  example,~\cite{Holm:2015}.  We don't pursue this direction.
\end{remark}

Recall that a noetherian ring $A$ is said to be Iwanaga-Gorenstein of dimension at
most $d$ if $\id_A A \leq d$ and $\id A_A \leq d$.

\begin{proposition}\label{prop:injectivedimension}
  The ring $\Lambda = \End_{R^\#}(\Omega_{R^\#}(M))$ constructed above is
  Iwanaga-Gorenstein of dimension at most $m = \max\{2,d+1\}$.
\end{proposition}

\begin{proof}
  Set $m = \max\{2,d+1\}$ and observe that for any finitely generated left (resp.\ right)
  $\Lambda$-module $X$, we have
  \[
  \Ext^{m+1}_{\Lambda}(X,\Lambda) \cong \Ext^1_{\Lambda}(\Omega^m_{\Lambda}(X),\Lambda) \cong
  0\,,
  \]
  since $\Omega_\Lambda^m(X)$ is Gorenstein projective by Lemma~\ref{lem:MCMsyz}
  and Proposition~\ref{prop:completeres}, and
  Gorenstein projectives are Ext-orthogonal to $\Lambda$ by definition.  It follows
  that $\id {}_{\Lambda} \Lambda \leq m$ and $\id \Lambda_{\Lambda} \leq m$.
\end{proof}

\begin{theorem}\label{thm:Gprojectives} The functor $F := \Hom_{R^\#}(\tilde{M},-)$
  induces an equivalence of categories from $\MCM(R^\#)$ to $\GP(\Lambda)$.  In
  particular, a left $\Lambda$-module $X$ is Gorenstein projective if and only if
  $X \cong \Hom_{R^\#}(\tilde{M},N)$ for a MCM $R^\#$-module $N$.
\end{theorem}

\begin{proof} We begin with the second claim, which establishes denseness of $F$.  By
  Proposition~\ref{prop:completeres} we know that every $\Lambda$-module of the form
  $\Hom_{R^\#}(\tilde{M},N)$ with $N$ in $\MCM(R^\#)$ is Gorenstein projective.
  Conversely, suppose that $X$ belongs to $\GP(\Lambda)$.  By Lemma~\ref{lem:MCMsyz}
  $\Omega_{\Lambda}^m(X) \cong \Hom_{R^\#}(\tilde{M},N)$ for some $N$ in
  $\MCM(R^\#)$, and thus by Proposition~\ref{prop:completeres},
  $\Omega_{\Lambda}^m(X)$ has a complete resolution of the form
  $\Hom_{R^\#}(\tilde{M},{\mathbb T})$ for an exact sequence ${\mathbb T}$ as in
  \eqref{eq:completeSigmaRes}. Here, as there, $m = \max\{2, d+1\}$.  Comparing this
  complete resolution to that of $X$ we see that, up to projective direct summands,
  $X \cong \Hom_{R^\#}(\tilde{M},N)$ or
  $X \cong \Hom_{R^\#}(\tilde{M},\Omega_{R^\#}N)$ depending on whether $m$ is even or
  odd, respectively.

  The functor $F$ is easily seen to be faithful since $R^\#$ is a direct summand of
  $\tilde{M}$.  To see that it is full, consider a map $g \colon F(N) \to F(N')$ for
  $N, N'$ in $\MCM(R^\#)$.  Then $g$ can be lifted to a map of projective
  presentations as in the diagram,
  \[
    \xymatrix{F(M_1) \ar[r]^{F(f_1)} \ar[d]^{g_1} & F(M_0) \ar[r]^{F(f_0)} \ar[d]^{g_0}
      & F(N) \ar[d]^g \ar[r] & 0 \\ F(M'_1) \ar[r]^{F(f'_1)} & F(M'_0) \ar[r]^{F(f'_0)}
      & F(N') \ar[r] & 0}
  \]
  with $M_i, M'_i \in \add{\tilde{M}}$.  Since $F$ induces an equivalence from
  $\add{\tilde{M}}$ to $\add(\Lambda)$, we can write $g_i = F(h_i)$ for suitable maps
  $h_i$ (for $i=1,2$), and we obtain a commutative exact diagram in $\MCM(R^\#)$
  \[
    \xymatrix{M_1 \ar[r]^{f_1} \ar[d]^{h_1} & M_0 \ar[r]^{f_0} \ar[d]^{h_0} & N
      \ar@{-->}[d]^h \ar[r] & 0 \\ M'_1 \ar[r]^{f'_1} & M'_0 \ar[r]^{f'_0} & N'
      \ar[r] & 0}
  \]
  yielding a map $h \colon N \to N'$.  It then follows that $g = F(h)$.
\end{proof}

\begin{remark}\label{rmk:FrobeniusCategories}
  The preceding Proposition and Theorem can also be deduced from recent work of
  Iyama, Kalck, Wemyss and Yang \cite{IKWY:2015}.  To see this, recall from Remark~\ref{rmk:relativeHomology}
  that the category $\MCM(R^\#)$ has a
  (relative) Frobenius category structure obtained by taking
  $\Sigma_1 =\add{\tilde{M}}$ as the subcategory of (relative) projective objects,
  and $\Sigma_1$-exact sequences as the exact sequences.

   Since the functor categories $\rmod{\MCM(R^\#)}$ and
  $\rmod{\MCM(R^\#)^{\op}}$ have global dimension at most $m = \max\{2,d+1\}$
  by~\cite{Holm:2015}, Theorem 2.8 of \cite{IKWY:2015} applies.  Part (1) of that
  theorem corresponds to the Iwanaga-Gorenstein property of our
  $\Lambda = \End_{R^\#}(\tilde{M})$, while part (2) gives the equivalence between
  $\MCM(R^\#)$ and $\GP(\Lambda)$.  Of course, it also follows from this realization
  of a Frobenius structure, that we have an equivalence of triangulated categories
  induced by
  \[
    \Hom_{R^\#}(\tilde{M},-) \colon  \MCM(R^\#)/[\Sigma_1] \approx
  \underline{\GP}(\Lambda)\,,
  \]
  where $[\Sigma_1]$ denotes the ideal of morphisms that
  factor through an object in the subcategory $\Sigma_1$ and we write
  $\underline{\GP}(\Lambda)$ for the stable category of Gorenstein projective
  $\Lambda$-modules.
\end{remark}

\section{Generation of MCM module categories}
\label{sect:generate}

When $R$ is a Gorenstein local ring, $\MCM(R)$ is a Frobenius category, and thus the
stable category $\uMCM(R)$, whose objects are the same as $\MCM(R)$ and whose
$\Hom$-sets are obtained by killing those morphisms factoring through a free
$R$-module, is triangulated with suspension given by the co-syzygy functor
$\Omega^{-1}$~\cite{Buchweitz:1987}.  This stable category is also
equivalent~\cite{Orlov:DerCatsCohTriang} to the \emph{singularity category}
$D_{\text{sg}}(R)$, that is, the Verdier quotient of the bounded derived category
$D^b(\mod R)$ by the subcategory of perfect complexes. In this section we investigate
the dimension of this triangulated category, in the sense of Rouquier, when $R$ is an
isolated hypersurface singularity.  In this setting, Ballard, Favero and Katzarkov
have found a general upper bound for the dimension of $\uMCM(R)$, showing in
particular that it is always finite \cite{Ballard-Favero-Katzarkov:2012}.  We focus
on a special class of hypersurfaces, where we improve upon this upper bound.

We begin by reviewing the definition of the dimension of a triangulated category
introduced by Rouquier~{\cite{Rouquier:2008}}.  Let ${\mathcal T}$ be a triangulated
category and ${\mathcal I}$ a full subcategory of ${\mathcal T}$.  (For ease of exposition,
we will assume all full subcategories are closed under isomorphisms.)
We let $\gen{{\mathcal I}}$ denote the
smallest full subcategory of ${\mathcal T}$ that contains ${\mathcal I}$ and is closed under
isomorphisms, direct summands, finite direct sums and shifts.  For two full
subcategories ${\mathcal I}_1$ and ${\mathcal I}_2$ we also write ${\mathcal I}_1 * {\mathcal I}_2$ for the full
subcategory of all $Y$ for which ${\mathcal T}$ contains a distinguished triangle
$X_1 \to Y \to X_2 \to X_1[1]$ with $X_i \in {\mathcal I}_i$, and we further set
${\mathcal I}_1 \diamond {\mathcal I}_2 = \gen{{\mathcal I}_1 * {\mathcal I}_2}$.  Observe that if ${\mathcal T} = \uMCM(R)$, then $Y \in {\mathcal I}_1 * {\mathcal I}_2$
if and only if there is a short exact sequence $0 \to X_1 \to Y \oplus F \to X_2 \to 0$ in $\MCM(R)$ with $X_i \in {\mathcal I}_i$ for $i=1,2$ and $F$ free.  Following the conventions of
\cite{Ballard-Favero-Katzarkov:2012}, we inductively define
$\gen{{\mathcal I}}_0 = \gen{{\mathcal I}}$ and
$\gen{{\mathcal I}}_{n} = \gen{{\mathcal I}}_{n-1} \diamond \gen{{\mathcal I}}$ for $n \geq 1$.

\begin{definition}[Rouquier]
  \label{def:DimTriCat} The \emph{dimension} of ${\mathcal T}$ is
  the smallest integer $n$ such that ${\mathcal T} = \gen{X}_n$ for some object $X$ in
  ${\mathcal T}$, or else $\infty$ if no such $n$ exists.
\end{definition}

If $R = k[\![x_0,\ldots,x_d]\!]/(f)$ is a complete isolated hypersurface singularity
over an algebraically closed field $k$ of characteristic zero, Ballard, Favero and
Katzarkov bound the dimension of $\uMCM(R)$ in terms of the Tjurina algebra
$k[\![x_0,\ldots,x_d]\!]/\Delta_f$, where $\Delta_f$ is the ideal in
$k[\![x_0,\ldots,x_d]\!]$ generated by the partial derivatives of $f$ with respect to
the $x_i$.  Recall that this algebra is Artinian and local.

\begin{theorem}[{\cite{Ballard-Favero-Katzarkov:2012}}]
  \label{thm:HypersurfaceCatDim}
  Let $R$ be a $d$-dimensional isolated hypersurface singularity as above, with $\ell$
  denoting the Loewy length of the Tjurina algebra of $R$.  Then
  $\uMCM(R) = \gen{\Omega^d(k)}_{2\ell-1}$, and in particular
  $\dim(\uMCM(R)) \leq 2 \ell - 1$.
\end{theorem}

In particular, if $R$ is defined by the polynomial
$f = x_0^{a_0} + \cdots + x_d^{a_d}$, where each $a_i$ is at least $2$ and not
divisible by $\charac(k)$, then the Tjurina algebra of $R$ is isomorphic to
$k[x_0,\ldots,x_d]/(x_0^{a_0-1}, \ldots, x_d^{a_d-1})$, and it is easy to see that
its Loewy length is
\[
\ell = \sum_{i=0}^d (a_i - 2) + 1 = \sum_{i=0}^d a_i - 2d - 1\,.
\]
In this case, we have $\dim(\underline{\MCM}(R)) \leq 2\sum_{i=0}^d a_i - 4d - 3$.

We now return to the context explored in the previous section.  Thus assume that $R$
is an isolated hypersurface singularity given by a non-zero power series $f$ in
$x_0, \ldots, x_d$ and set $R^\# = k[\![x_0,\ldots,x_d,y]\!]/(f+y^n)$.  For
$1 \leq i \leq n$, we set
\[
  \Sigma_i = \add \left\{\Omega_{R^\#}(M) \,\middle|\, M \in
    \MCM(R^\#/(y^i))\right\}\,,
\]
and continue to write $\Sigma_i$ for its image in the stable category.  Note that $\Sigma_i$ is closed under isomorphisms, finite direct sums and direct summands by definition, and is closed under shifts by Corollary~\ref{cor:splitApprox}.

The following lemma, which is a slight generalization of \cite[Lemma
2.4]{Herzog-Popescu:1997}, will be used repeatedly in what follows.  As the lemma can
be stated over any ring $\Gamma$, we recall that syzygies are only defined up to
projective summands.  Thus for each left $\Gamma$-module $M$ we fix a projective resolution $(F_{i}^M,\partial_{i}^M)_{i\geq 0}$ of $M$ and define
$\Omega^i_\Gamma(M) = \ker \partial_{i-1}^M$.  We say that two $\Gamma$-modules $X$ and $Y$ are stably isomorphic if $X \oplus P \cong Y \oplus P'$ for some projective $\Gamma$-modules $P$ and $P'$.

\begin{lemma}\label{lemma:doubleSyzygy}
  Let $\Gamma$ be any ring, let $I$ be a proper two-sided ideal of $\Gamma$ such that
  $\bar{\Gamma} = \Gamma/I$ has finite projective dimension $r \geq 1$ as a left
  $\Gamma$-module.  For any left $\bar{\Gamma}$-module $X$ we have
\[
\Omega^{r+1}_\Gamma(X) \oplus P \cong \Omega^{r}_\Gamma(\Omega_{\bar{\Gamma}}(X)) \oplus P'
\]
for projectives $P$ and $P'$.  More generally, for any $m \geq 1$ there exist
projectives $P, P'$ such that
\[
\Omega^{r+m}_\Gamma(X) \oplus P \cong \Omega^{r}_\Gamma(\Omega^{m}_{\bar{\Gamma}}(X)) \oplus P'\,.
\]
\end{lemma}

\begin{proof} Fix a projective resolution
  $\cdots \to F_1 \stackrel{\partial_1}{\to} F_0 \stackrel{\partial_0}{\to} X \to 0$ so that
  $\Omega^i_\Gamma(X) = \ker \partial_{i-1}$ for each $i \geq 1$.
  Tensoring down to $\bar{\Gamma}$, we obtain an exact commutative diagram of
  $\Gamma$-modules
  \[
    \xymatrix{
      & 0 \ar[d] & 0 \ar[d] \\
      & IF_0 \ar[d] \ar@{=}[r] & IF_0 \ar[d] \\
      0 \ar[r] & \Omega_\Gamma X \ar[d] \ar[r] & F_0 \ar[d] \ar[r] & X \ar@{=}[d] \ar[r] & 0 \\
      0 \ar[r] & K \ar[r] \ar[d] & F_0/IF_0 \ar[r] \ar[d] & X \ar[r] & 0 \\
      & 0  & 0
    }
  \]
  where $K \oplus Q_1 \cong \Omega_{\bar{\Gamma}}X \oplus Q_2$ for projective
  $\bar{\Gamma}$-modules $Q_1, Q_2$ by Schanuel's Lemma.  Since $\pd ({}_{\Gamma}\bar{\Gamma}) = r$, $\Omega^{r}_{\Gamma}(Q_i)$ is a projective $\Gamma$-module for each $i$.  Thus, $\Omega^{r}_{\Gamma}K$ and $\Omega^{r}_\Gamma(\Omega_{\bar{\Gamma}} X)$ will be stably isomorphic over $\Gamma$.

  We then have a second diagram
  as below.
  \[
    \xymatrix{
      & & & 0 \ar[d] \\
      & 0 \ar[d] & 0 \ar[d] & IF_0 \ar[d] \\
      0 \ar[r] & \Omega^2_\Gamma X \ar[d] \ar[r] & F_1 \ar@{=}[d] \ar[r] & \Omega_\Gamma X \ar[d] \ar[r] & 0 \\
      0 \ar[r] & L \ar[d] \ar[r] & F_1 \ar[d] \ar[r] & K \ar[r] \ar[d] & 0 \\
      & IF_0 \ar[d] & 0  & 0 \\
      & 0
    }
  \]
  Considering the vertical sequence on the left, we can use the horseshoe lemma to obtain a projective resolution of $L$ and a short exact sequence of syzygies
 $$0 \to \Omega^{r+1}_\Gamma X \to \Omega^{r-1}_\Gamma L \oplus P \to \Omega^{r-1}_\Gamma (IF_0) \oplus P' \to 0$$
for projective $\Gamma$-modules $P$ and $P'$.  Since $IF_0 \in \add(I)$ and ${}_\Gamma I$ has projective dimension $r-1$, $\Omega^{r-1}_\Gamma(IF_0)$ is projective and this sequence splits.  It follows that $\Omega^{r+1}_\Gamma X$ is stably isomorphic to $\Omega^{r-1}_{\Gamma} L$.   Since $L$ is stably isomorphic to $\Omega_\Gamma (K)$, we see that $\Omega^{r+1}_\Gamma X$ is stably isomorphic to $\Omega^{r}_\Gamma K$ and thus to $\Omega^{r}_\Gamma(\Omega_{\bar{\Gamma}}X)$.

  The second claim now follows via induction on $m\geq 1$.
\end{proof}

In particular, when the ideal $I$ is generated by a central regular sequence of length $r$, we have $\pd ({}_\Gamma \Gamma/I) = r$ and thus the preceding lemma applies in this situation.   Furthermore, for such ideals $I$ in a commutative ring $\Gamma$, the Eagon-Northcott resolution shows that $\pd ({}_\Gamma \Gamma/I^j) = r$ for each $j \geq 1$ \cite{Eagon-Northcott}, and thus the lemma also applies for the ideal $I^j$, as will be needed a bit later on.

\begin{prop}\label{prop:SigmastarSigma}
  We have $\Sigma_k = \Sigma_{k-j} \diamond \Sigma_{j}$ for every
  $1 \leq j < k \leq n$.  In particular,
  $\MCM(R^\#) = \Sigma_{n-1} = \gen{\Sigma_1}_{n-2}$.
\end{prop}

\begin{proof}
  First let $M \in \Sigma_k$, say $M \oplus M' = \Omega_{R^\#}(X)$ with $X$ an
  $R^\#$-module annihilated by $y^k$. Then we have a short exact sequence
  \begin{equation}\label{eq:nonMCM}
    0 \to y^j X \to X \to X/y^j X \to 0\,,
  \end{equation}
  in which the leftmost, resp.\ rightmost, term is annihilated by $y^{k-j}$, resp.\
  $y^j$. Observe, however, that neither of these need be MCM over $R^\#/(y^{k-j})$,
  resp.\ $R^\#/(y^{j})$.  Fix an odd integer $m > d$, and apply $\Omega_{R^\#}^m$ to
  \eqref{eq:nonMCM}, obtaining
  \begin{equation}\label{eq:nowMCM}
    0 \to \Omega_{R^\#}^m(y^j X) \to \Omega_{R^\#}^m(X) \oplus F \to
    \Omega_{R^\#}^m(X/y^j X) \to 0
  \end{equation}
  for some free $R^\#$-module $F$.  Since $m$ is odd, the middle term is isomorphic
  to $\Omega_{R^\#}^1(X) \oplus F \cong M \oplus M' \oplus F$. By
  Lemma~\ref{lemma:doubleSyzygy}, since powers of $y$ are regular in $R^\#$, we may rewrite the outer terms as the first
  syzygies, over $R^\#$, of $\Omega_{R^\#/(y^{k-j})}^{m-1}(y^j X)$ and
  $\Omega_{R^\#/(y^{j})}^{m-1}(X/y^j X)$, respectively. For $m$ large enough, these
  are both MCM over the appropriate quotient of $R^\#$, and it follows that
  $\Omega^m_{R^\#}(y^j X) \in \Sigma_{k-j}$ and $\Omega^m_{R^\#}(X/y^j X) \in
  \Sigma_{j}$. Hence $M \in \Sigma_{k-j} \diamond \Sigma_{j}$.

  For the other containment, it suffices to complete a diagram of the form
  \[
    \xymatrix{
      & 0  \ar[d]    & & 0  \ar[d]\\
      0 \ar[r] & \Omega_{R^\#}(X_1) \ar[r] \ar[d] & M\oplus M' \ar[r] &
      \Omega_{R^\#}(X_2) \ar[r]
      \ar[d] & 0 \\
      & F_1 \ar[d] & & F_2 \ar[d] \\
      & X_1 \ar[d] & & X_2 \ar[d] \\
      & 0 & & 0 }
  \]
  where $y^{k-j}X_1 = y^j X_2 = 0$ and $F_1$, $F_2$ are free $R^{\#}$-modules.  Apply
  $\Hom_{R^\#}(-,F_1)$ to the top row, obtaining
  \[
  \cdots \to \Hom_{R^\#}(M\oplus M',F_1) \to \Hom_{R^\#}(\Omega_{R^\#}(X_1),F_1) \to
  \Ext_{R^\#}^1(\Omega_{R^\#}(X_2),F_1) \,.
  \]
  Since $\Omega_{R^\#}(X_2)$ is MCM and $F_1$ is free,
  $\Ext_{R^\#}^1(\Omega_{R^\#}(X_2),F_1)=0$. This implies that the map
  $\Omega_{R^\#}(X_1) \to F_1$ extends to a map $M \oplus M' \to F_1$, and we therefore obtain
  a commutative diagram of exact sequences
  \[
    \xymatrix{ 0 \ar[r] & \Omega_{R^\#}(X_1) \ar[r] \ar[d] & M \oplus M'\ar[r] \ar[d]
      & \Omega_{R^\#}(X_2) \ar[r]
      \ar[d] & 0 \\
      0 \ar[r] & F_1 \ar[r] & F_1 \oplus F_2 \ar[r] & F_2 \ar[r] & 0 \,.  }
  \]
  Set $X = \cok (M\oplus M' \to F_1\oplus F_2)$. Then of course
  $M \oplus M' = \Omega_{R^\#}(X)$ and we have a short exact sequence
  $0 \to X_1 \to X \to X_2 \to 0$. Since $y^{k-j}X_1=0$ and $y^jX_2 =0$, we have
  $y^k X=0$, as required.
\end{proof}

We now extend this result to iterated branched covers, as considered by O'Carroll and
Popescu in \cite{OCarroll-Popescu:2000JPAA}.
 Let $S$ be a $d+1$-dimensional regular local
ring, and set $R = S/(f)$ and
\[
R_r = S[\![y_1,\ldots,y_r]\!]/(f+y_1^{a_1} + \cdots y_r^{a_r})
\]
for positive integers $a_i \geq 2$.  Let $I_r$ be the ideal
$(y_1, \ldots, y_r)R_r$, and define
\[
  \Sigma^r_j := \add \{\Omega^r_{R_r}(X) \ |\ X \in \MCM({R_r}/I_r^j)\} \,,.
\]
Note that $y_1, \dots, y_r$ is a regular sequence on $R_r$, so that $R_r/I_r^j$ is a
CM ring of dimension $d$ for any $j$ (by, for example, Eagon-Northcott~\cite[Theorem
2]{Eagon-Northcott}). It follows that any $X \in \MCM({R_r}/I_r^j)$ has depth $d$ as
an $R_r$-module, and thus that $\Sigma_j^r \subseteq \MCM(R_r)$.

The argument outlined in Remark~\ref{rmk:Ragnar} has been extended by
Takahashi~\cite{Takahashi:2010} to recover the following result of O'Carroll and
Popescu~\cite{OCarroll-Popescu:2000JPAA}:
\begin{prop}\label{prop:takahashi}
  Assume that each $a_i$ is invertible in $S$.  Then for any MCM $R_r$-module
  $N$, we have
  \begin{equation}
    \Omega_{R_r}^r (N/(y_1^{a_1-1}, \dots, y_r^{a_r-1})N)
    \cong \bigoplus_{j=0}^r \Omega_{R_r}^j (N)^{r \choose j}\,.\tag*{\qed}
  \end{equation}
\end{prop}
The necessary ingredients in proving Proposition~\ref{prop:takahashi} are $(i)$ to
observe that the Jacobian ideal $(y_1^{a_1-1}, \dots, y_r^{a_r-1})$ annihilates
$\Ext_{R_r}^1(N, \Omega_{R_r}(N))$, and $(ii)$ to induct over the number
$r$ of new variables by applying Lemma~\ref{lemma:doubleSyzygy}. We omit the details.

We set $m := \sum_{i=1}^r(a_i-2) + 1$, so that
$I_r^m \subseteq (y_1^{a_1-1}, \ldots, y_r^{a_r-1})$. By
Proposition~\ref{prop:takahashi}, every MCM $R_r$-module $N$ is a direct
summand of $\Omega_{R_r}^r(N/(y_1^{a_1-1}, \ldots, y_r^{a_r-1})N)$.  Since
$N/(y_1^{a_1-1}, \ldots, y_r^{a_r-1})N$ is a MCM $R_r/I_r^m$-module, this yields
$\Sigma^r_m = \MCM(R_r)$.

\begin{theorem}\label{thm:newSigmaStarSigma} For each $1 \leq j < k$ we have
  $\Sigma^r_k = \Sigma^r_{k-j} \diamond \Sigma^r_j = \gen{\Sigma^r_1}_{k-1}$.  In
  particular, $\MCM(R_r) = \Sigma^r_{m} = \gen{\Sigma^r_1}_{m-1}$.
\end{theorem}

\begin{proof}
  Step 1.  We first show the inclusions
  $\Sigma^r_k \subseteq \Sigma^r_{k-j} \diamond \Sigma^r_j \subseteq
  \gen{\Sigma^r_1}_{k-1}$.  The argument is similar to the first part of the proof of Proposition~\ref{prop:SigmastarSigma}.  Let $M \in \Sigma^r_k$, say
  $M\oplus M' = \Omega^r_{R_r}(X)$ with $X$ an $R_r$-module annihilated by
  $I_r^k$. Then, for any even integer $n>d$, the short exact sequence $0 \to I_r^j X \to X \to X/I_r^j X \to 0$
  induces a short exact sequence
  $0 \to \Omega^{r+n}_{R_r}(I_r^j X) \to M\oplus M' \oplus F \to \Omega^{r+n}_{R_r}(X/I_r^j
  X) \to 0$ for some free module $F$. By Lemma~\ref{lemma:doubleSyzygy}, $\Omega^{r+n}_{R_r}(X/I_r^j X)$ is stably isomorphic to $\Omega^r_{R_r}(\Omega^n_{R_r/I_r^j} (X/I_r^j X))$ with $\Omega^n_{R_r/I_r^j} (X/I_r^j X)$ MCM over $R_r/I_r^j$.  Similarly,  $\Omega^{r+n}_{R_r}(I_r^j X)$ is stably isomorphic to $\Omega^r_{R_r}(\Omega^n_{R_r/I_r^{k-j}} (I_r^j X))$ with $\Omega^n_{R_r/I_r^{k-j}} (I_r^j X)$ MCM over $R_r/I_r^{k-j}.$  In particular,
  $M \in \Sigma^r_{k-j} \diamond \Sigma^r_{j}$.  The second inclusion now follows by
  definition of $\gen{\Sigma^r_1}_{k-1}$ and induction on $k$.

  \vspace{3mm} Step 2.  We show that
  $\Omega_{R_r}(\Sigma^{r-1}_i) \subseteq \Sigma^r_i$ for each $r \geq 2$ and
  $i \geq 1$.  Let $M \in \Sigma^{r-1}_i$, so that
  $M \oplus M' \cong \Omega^{r-1}_{R_{r-1}}(X)$ for some $M' \in \MCM(R_{r-1})$ and
  $X \in \MCM(R_{r-1}/I_{r-1}^i)$.  Then
  \[
    \Omega_{R_r}M \oplus \Omega_{R_r}M' \cong \Omega_{R_r}(\Omega^{r-1}_{R_{r-1}}(X))
    \cong  \Omega^r_{R_r}(X)\,,
  \]
  where the last isomorphism holds up to free summands by
  Lemma~\ref{lemma:doubleSyzygy}.  Thus $\Omega_{R_r}M$ belongs to $\Sigma^r_i$ since
  $X$ may also be viewed as a MCM module over $R_{r}/I_r^i$.

  \vspace{3mm} Step 3.  We show that $\Sigma_1^r = \Omega_{R_r}(\Sigma_1^{r-1})$ for
  each $r \geq 2$.  Let $M \in \Sigma_1^r$.  Then $M \oplus M' \cong \Omega^r_{R_r}X$
  for a MCM $R_r/I_r$-module $X$.  But $R_r/I_r \cong R \cong R_{r-1}/I_{r-1}$, so
  $X$ is an $R_{r-1}$-module and $\Omega^{r-1}_{R_{r-1}}X \in \Sigma^{r-1}_1$.  Thus,
  up to free summands,
  $M \oplus M' \cong \Omega_{R_r}(\Omega^{r-1}_{R_{r-1}}X) \in
  \Omega_{R_r}(\Sigma^{r-1}_1)$ as required.

  \vspace{3mm} Step 4.  For each $j \geq 1$, consider the following two statements:
  \begin{equation}\label{eqn:S1}
    \forall\ r \geq 2 \ \ \Sigma^r_j = \Omega_{R_r}(\Sigma^{r-1}_j)\,,
  \end{equation}
  and
  \begin{equation}\label{eqn:S2}
    \forall\ r \geq 1 \ \ \add(\Sigma^r_1 * \Sigma^r_j) = \Sigma^r_{j+1}\,.
  \end{equation}
  For any fixed $j \geq 1$, we show that \eqref{eqn:S1} implies \eqref{eqn:S2}.  To
  see this, we apply induction on $r$, noting that the $r=1$ case of \eqref{eqn:S2}
  is handled by Proposition~\ref{prop:SigmastarSigma}.  By Step 1, it suffices to
  show $\add(\Sigma^r_1 * \Sigma^r_j) \subseteq \Sigma^r_{j+1}$.  Let $r \geq 2$.
  For any $M \in \add(\Sigma^r_1 * \Sigma^r_j)$, we have an extension
  $0 \to N \to M \oplus M' \to N' \to 0$ for
  $N \in \Sigma^r_1= \Omega_{R_r}(\Sigma^{r-1}_1)$ and
  $N' \in \Sigma^r_j = \Omega_{R_r}(\Sigma^{r-1}_j)$.  Now, as in the second half of
  the proof of Proposition~\ref{prop:SigmastarSigma}, we obtain
  $M \in \Omega_{R_r}(\Sigma^{r-1}_1 * \Sigma^{r-1}_j) =
  \Omega_{R_r}(\Sigma^{r-1}_{j+1}) \subseteq \Sigma^r_{j+1}$, where we have used the
  inductive hypothesis and the result of Step 2.

  \vspace{3mm} Step 5.  We now prove \eqref{eqn:S1}, and hence also \eqref{eqn:S2},
  for all $j \geq 1$ by induction on $j$.  For $j=1$, \eqref{eqn:S1} was established
  in Step 3.  Now, assume $\eqref{eqn:S1}$ holds for some $j \geq 1$, and let
  $M \in \Sigma^r_{j+1} = \add(\Sigma^r_1 * \Sigma^r_j)$.  The argument in the
  previous step shows that $M \in \Omega_{R_r}(\Sigma^{r-1}_{j+1})$, thus
  establishing (\ref{eqn:S1}) for $j+1$ with the help of Step 2.

  \vspace{3mm} Step 6.  We finally show that $\Sigma^r_j = \gen{\Sigma^r_1}_{j-1}$
  for all $r, j \geq 1$.  It then follows that
  $\Sigma^r_k = \Sigma^r_{k-j} \diamond \Sigma^r_j$ for all $1 \leq j < k$.  By Step
  1, it suffices to show that $\gen{\Sigma^r_1}_{j-1} \subseteq \Sigma^r_j$.  For
  $j=1$, this amounts to saying that $\Sigma^r_1$ is closed under syzygies.  For
  $r=1$, we know that $\Sigma^r_j$ is closed under syzygies from
  Corollary~\ref{cor:splitApprox}.  By induction on $r$, using (\ref{eqn:S1}) and
  Lemma~\ref{lemma:doubleSyzygy}, we see that all $\Sigma^r_j$ are in fact closed
  under syzygies.  Now using induction on $j$, we have
  \[
    \gen{\Sigma^r_1}_{j} = \gen{\Sigma^r_1 * \gen{\Sigma^r_1}_{j-1}} = \gen{\Sigma_1^r *
      \Sigma^r_j} = \gen{\Sigma^r_{j+1}} = \Sigma^r_{j+1}\,.
  \]
\end{proof}

\begin{corollary}\label{cor:findim}
  With notation as above, $\dim (\MCM(R_r)) \leq \sum_{i=1}^r(a_i-2)$ whenever $R$
  has finite CM-type.
\end{corollary}

\begin{remark}
  We emphasize again that the Corollary above requires $f \neq 0$. Indeed, it is
  known that the category of MCM modules over the $A_\infty$ singularity
  $k[\![x,y]\!]/(y^2)$ has dimension 1~\cite[Proposition
  3.7]{Dao-Takahashi:dimension}.
\end{remark}

\begin{question}  We have shown that $\MCM(R_r) = \Sigma^r_{m} = \gen{\Sigma^r_1}_{m-1}$ and $\gen{\Sigma^r_1}_{m-2} = \Sigma^r_{m-1}$.  But we do not know if  $m-1$ is the smallest number of steps in which $\Sigma^r_1$ generates $\MCM(R_r)$, or equivalently whether $\Sigma^r_{m-1}$ must be a proper subset of $\Sigma^r_m = \MCM(R_r)$.  Properness here would follow from the existence of an MCM $R_r$-module $M$ for which $\Ext^1_{R_r}(M,-)$ is not annihilated by $\displaystyle y := \prod_{i=1}^r y_i^{a_i-2}$.  To see this, note that if such an $M$ is a direct summand of some $\Omega_{R_r}^r(X)$, then $\Ext^1_{R_r}(M,-)$ is isomorphic to a direct summand of $\Ext^{r+1}_{R_r}(X,-)$, and the latter must not be killed by $y$.  Since $y \in I_r^{m-1}$, $X$ is not annihilated by $I_r^{m-1}$ and thus $M \notin \Sigma^r_{m-1}$.
\end{question}

\section{Examples}
\label{sect:egs}

We illustrate the constructions in this paper in a couple of examples.  For
computational purposes it is convenient to work with matrix factorizations, on which
the necessary background can be found in Yoshino's book \cite{Yoshino:book}.  For a
power series $f$ contained in the maximal ideal of $S = k[\![\underline{x}]\!]$, we write $\MF_S(f)$
for the category of reduced matrix factorizations of $f$ over $R$ and recall that the
functor $\cok \colon \MF_R(f) \to \MCM(R/(f))$, sending a matrix factorization
$(\phi, \psi)$ to $\cok(\phi)$, induces equivalences of categories
$\MF_R(f)/[(1,f)] \approx \MCM(R/(f))$ and
$\MF_R(f)/[(1,f),(f,1)] \approx \uMCM(R/(f))$.  We also write
$\Omega(\phi,\psi) = (\psi,\phi)$.

We begin with a description of the functor
$\Omega_{R^\#}\colon \MCM(R) \rightarrow \MCM(R^\#)$ in terms of matrix
factorizations.  On this level, it turns out that this functor is a special case of
Yoshino's tensor product of matrix factorizations \cite{Yoshino:1998}.  Let
$R = k[\![x_0,\ldots,x_d]\!]$ and $R' = k[\![y_0,\ldots,y_{d'}]\!]$, and set
$S= k[\![x_0,\ldots,x_d,y_0,\ldots,y_{d'}]\!]$.  If $X = (\phi, \psi)$ and
$X' = (\phi',\psi')$ are matrix factorizations of $f \in R$ and $g \in R'$
respectively, of sizes $n$ and $m$, then Yoshino defines the matrix factorization
\[
  X \hat{\otimes} X' = \left( \begin{bmatrix} \phi \otimes I_m & I_n \otimes \phi' \\
      -I_n \otimes \psi' & \psi \otimes I_m \end{bmatrix}, \begin{bmatrix} \psi
      \otimes I_m & -I_n \otimes \phi' \\ I_n \otimes \psi' & \phi \otimes
      I_m \end{bmatrix}\right) \in \MF_S(f+g)\,.
\]
For a fixed $X'$, Yoshino shows that $-\hat{\otimes} X'\colon \MF_R(f) \to \MF_S(f+g)$ is
an exact functor that preserves trivial matrix factorizations, and behaves well with
respect to syzygies.  In particular,
$X \hat{\otimes} \Omega X' \cong \Omega(X \hat{\otimes} X') \cong \Omega X
\hat{\otimes} X'$.  In addition, if $X'$ is reduced, $-\hat{\otimes} X'$ is a faithful
functor (although, it is typically very far from being full).  We also have a
reduction functor $-\otimes_S S/(\underline{y})\colon \MF_S(f+g) \to \MF_R(f)$, which sends
$(\Phi, \Psi)$ to $(\Phi \otimes_S S/(\underline{y}), \Psi \otimes_S S/(\underline{y}))$.

We now specialize to the setting considered in this paper: $R' = k[\![y]\!], g = y^n$
and $Y = (y,y^{n-1})$.  Then the functor $-\hat{\otimes} Y$ sends an $m \times m$
matrix factorization $(\phi, \psi)$ of $f$ over $R$ to the $2m \times 2m$ matrix
factorization
\[
(\phi, \psi) \hat{\otimes} (y,y^{n-1}) = \left( \begin{bmatrix} \phi & yI_m \\
    -y^{n-1}I_m & \psi \end{bmatrix}, \begin{bmatrix} \psi & -yI_m \\ y^{n-1}I_m &
    \phi \end{bmatrix} \right)\,.
\]

\begin{prop}\label{prop:syzygyMF} Let $R = k[\![x_0,\ldots,x_d]\!]/(f)$ be an
  isolated hypersurface singularity and set
  $R^\# = k[\![x_0,\ldots,x_d,y]\!]/(f + y^n)$.  If $(\phi, \psi)$ is a reduced matrix
  factorization of $f$ corresponding to the MCM $R$-module $M = \cok (\phi, \psi)$,
  then we have $\Omega_{R^\#}(M) \cong \cok \Omega( (\phi,\psi) \hat{\otimes} Y))$.
\end{prop}

\begin{proof}
The proof given in~\cite[Lemma 8.17]{Leuschke-Wiegand:BOOK} or \cite[Lemma
12.3]{Yoshino:book} in the case $n=2$ applies equally well for arbitrary $n$.
\end{proof}

\begin{example}
  Consider $R = k[\![x]\!]/(x^4)$ and $R^\# = k[\![x,y]\!]/(x^4+y^3)$, which is a
  simple curve singularity of type ${\mathbb E}_6$.  Of course $R$ is a finite-dimensional
  $k$-algebra of finite representation type, and $\lmod{R}$ is easily pictured via
  its Auslander-Reiten quiver
\[
  \xymatrix{R/(x) \ar[r]<0.5ex> & R/(x^2) \ar[r]<0.5ex> \ar[l]<0.5ex> & R/(x^3)
    \ar[r]<0.5ex> \ar[l]<0.5ex> & R \ar[l]<0.5ex>}\,.
\]

For reference we also provide the Auslander-Reiten quiver of $R^\#$, following the
notation of \cite{Yoshino:book}, Chapter 9.
\[
  \xymatrixrowsep{0.75pc} \xymatrix{ & & B \ar[dl]<0.5ex> \ar[r] \ar@{--}[dd] & M_1
    \ar[ddl] \ar@{--}[dd] \\ M_2 \ar[r]<0.5ex> \ar@{--}@(dl,dr) & X \ar@{--}@(dl,dr)
    \ar[ur]<0.5ex> \ar[dr]<0.5ex> \ar[l]<0.5ex> & & & R \ar[ul] \\ & & A
    \ar[ul]<0.5ex> \ar[r] & N_1 \ar[uul] \ar[ur]}
\]
We have
\[
  \Omega_{R^\#}(R/(x)) \cong \cok((x^3,x) \hat{\otimes} (y,y^2)) = \cok
  \left( \begin{bmatrix} x^3 & -y \\ y^2 & x \end{bmatrix}, \begin{bmatrix} x & y \\
      -y^2 & x^3 \end{bmatrix} \right) \cong \cok(\psi_1, \phi_1) \cong N_1\,;
\]
 and
\[
\Omega_{R^\#}(R/(x^3)) \cong \cok((x,x^3) \hat{\otimes} (y,y^2)) \cong \Omega(N_1)
\cong M_1\,.
\]
Furthermore

\[
  \Omega_{R^\#}(R/(x^2)) \cong \cok ((x^2,x^2) \hat{\otimes} (y,y^2)) = \cok
  \left( \begin{bmatrix} x^2 & -y \\ y^2 & x^2 \end{bmatrix}, \begin{bmatrix} x^2 & y
      \\ -y^2 & x^2 \end{bmatrix} \right) \cong \cok(\psi_2, \phi_2) \cong N_2 \cong
  M_2\,.
\]

Thus $\Sigma_1 = \add(M_1 \oplus N_1 \oplus M_2 \oplus R^\#)$.  We compute the
minimal $\Sigma_1$-approximations of the remaining indecomposables in $\MCM(R^\#)$.
From the matrix factorizations for $A$, $B$ and $X$ given in \cite{Yoshino:book} (with $x$ and $y$ swapped), it
is easy to see that $A/yA \cong (R/(x^3))^2 \oplus R/(x^2)$,
$B/yB \cong (R/(x))^2 \oplus R/(x^2)$ and
$X/yX \cong (R/(x^2))^2 \oplus R/(x) \oplus R/(x^3)$.  Applying $\Omega_{R^\#}$ now
yields the middle terms in the following short exact sequences, which are left and
right $\Sigma_1$-approximations of the modules on the ends:
\begin{align*}
& 0 \to B \to M_1^2 \oplus M_2 \to A \to 0 & \\
& 0 \to A \to N_1^2 \oplus M_2 \to B \to 0 &\\
& 0 \to X \to M_1 \oplus N_1 \oplus M_2^2 \to X \to 0&
\end{align*}

In light of Proposition~\ref{prop:SigmastarSigma}, there must also be short exact
sequences with end terms in $\Sigma_1$ and each of $B, A$ and $X$ as direct summands
of the middle term.  In fact these extensions are evidenced by the following almost
split sequences

\begin{align*}
& 0 \to M_1 \to A \to N_1 \to 0 & \\
& 0 \to N_1 \to B \oplus R^\# \to M_1 \to 0 & \\
& 0 \to M_2 \to X \to M_2 \to 0. &
\end{align*}

Set $\tilde M = M_1 \oplus N_1 \oplus M_2 \oplus R^\#$.  The ring
$\Lambda = \End_{R^\#}(\tilde M)$ is isomorphic to the corner ring of the Auslander
algebra of $\MCM(R^\#)$ corresponding to the 4 vertices $M_1, N_1, M_2$ and $R^\#$ in
the AR-quiver above.  To describe this algebra explicitly in terms of quivers and
relations, we identify $R^\#$ with $k[\![t^3,t^4]\!]$, and note that all the
indecomposables in $\Sigma_1$ can be represented as fractional ideals (for example,
see \cite{Yoshino:book}):
\[
M_1 \cong (t^3,t^8), \ \ N_1 \cong (t^3, t^4), \ \ M_2 \cong (t^6,t^8)\,.
\]
The irreducible morphisms between the indecomposables in $\Sigma_1$ are easily seen to correspond to the following paths in the
AR-quiver of $\MCM(R^\#)$.  Moreover each is realized by multiplication by a suitable power of $t$.
\begin{eqnarray*}
	t^5 & : & N_1 \to B \to X \to M_2, \\
 	t^4 & : & N_1 \to B \to X \to B \to M_1, \\
  	1 & : & N_1 \to R^\#, \\
   	t^3 & : & R^\# \to M_1, \\
   	t^3 & : &M_1 \to A \to X \to M_2,  \\
   	1 & : &M_1 \to A \to X \to A \to N_1, \\
    	t^{-2} & : & M_2 \to X \to A \to N_1, \\
    	1 & : &M_2 \to X \to B \to M_1, \\
     	t^2 & : & M_2 \to X \to A \to X \to M_2.
\end{eqnarray*}
It follows that $\Lambda$ can be described as a factor of the completed path algebra of the
following quiver\footnote{with the convention that arrows are composed left to right}

\[
  \xymatrix{& M_2 \ar@(ul,ur)^{t^2} \ar[dr]<0.5ex>^{t^{-2}} \ar[dl]<0.5ex>^1 \\ M_1
    \ar[ur]<0.5ex>^{t^3} \ar[rr]<0.5ex>^1 & & N_1 \ar[ul]<0.5ex>^{t^5}
    \ar[ll]<0.5ex>^{t^4} \ar[dl]^{1} \\ & R^\# \ar[ul]^{t^3}}
\]

While we don't list all the relations here, we note that they can be easily
identified from the above quiver, as they correspond to parallel paths that compose
to identical powers of $t$.  For instance, among the minimal relations we find the
difference between the paths $M_2 \stackrel{1}{\to} M_1 \stackrel{1}{\to} N_1$ and
$M_2 \stackrel{t^2}{\to} M_2 \stackrel{t^{-2}}{\to} N_1$ since both paths compose to
$1$, as well as the difference between $N_1 \to R^\# \to M_1 \to N_1$ and
$N_1 \to M_2 \to N_1$ since both compose to $t^3$.

Next, we apply results from Section 3 to describe the minimal projective resolutions
of the simple $\Lambda$-modules.  Recall that these will become periodic of period
$2$ after the first two terms (since $m = \max \{2,d+1\} = 2$ here).  If $S(i)$ is a
simple left $\Lambda$-module, corresponding to a vertex $i$ of the quiver of
$\Lambda$, its minimal projective presentation is given by
$\bigoplus_{\alpha : j \to i} P(j) \stackrel{\pi(i)}{\to} P(i) \to S(i) \to 0$, where
the sum ranges over all arrows $\alpha$ ending at $i$, and the $\alpha$ component of
$\pi(i)$ is just the map $\alpha \colon P(j) \to P(i)$.  In general, we can find a map
$d(i)$ between $X, Y \in \add(\tilde{M})$ so that $\pi(i)$ is realized as
$\Hom_{R^\#}(\tilde{M},d(i)) \colon \Hom_{R^\#}(\tilde{M},X) \to \Hom_{R^\#}(\tilde{M},Y)$
and $\ker d(i) \in \MCM(R^\#)$ with
$\Hom_{R^\#}(\tilde{M},\ker d(i)) \cong \Omega^2_{\Lambda}(S(i))$.  Furthermore the
$\add(\tilde{M})$-resolution of $\ker d(i)$ will induce the remaining terms of the
minimal projective resolution of $S(i)$ over $\Lambda$.  For example, for $S(N_1)$,
the minimal projective presentation has the form
$P(M_1) \oplus P(M_2) \stackrel{\pi(N_1)}{\to} P(N_1) \to S(N_1) \to 0$ where
$\pi(N_1)$ is the map induced by
$\begin{pmatrix} 1 \\ t^{-2} \end{pmatrix} : M_1 \oplus M_2 \to N_1$, whose kernel is
isomorphic to $B$.  Using the $\Sigma_1$-approximations of $B$ and $A$, we now obtain
the minimal projective resolution
\[
\cdots \to P(M_1)^2 \oplus P(M_2) \to P(N_1)^2 \oplus P(M_2) \to
P(M_1) \oplus P(M_2) \to P(N_1) \to S(N_1) \to 0\,.
\]
Similarly for $S(M_2)$, we compute $\Omega^2_{\Lambda}(S(M_2)) \cong \Hom_{R^\#}(\tilde{M},\ker d(M_2))$, where
\[
d(M_2) = \begin{pmatrix} t^3 \\ t^5 \\ t^2 \end{pmatrix} : M_1 \oplus N_1 \oplus M_2
\to M_2\,.
\]
One can compute $\ker d(M_2) \cong X$.  Then using the $\Sigma_1$-approximation sequence for $X$, we will get the minimal projective resolution
\[
\begin{split}
\cdots \to & P(M_1) \oplus P(N_1) \oplus P(M_2)^2 \to P(M_1) \oplus P(N_1) \oplus
P(M_2)^2 \to \\ & \hspace{2cm} P(M_1) \oplus P(N_1) \oplus P(M_2) \to P(M_2) \to S(M_2) \to 0.
\end{split}
\]
\end{example}

\begin{example}
Similar computations can be made for the ${\mathbb E}_8$ curve singularity
$R^{\#} = k[\![x,y]\!]/(x^5+y^3) \cong k[\![t^3, t^5]\!]$.  We now set
$R = k[x]/(x^5)$, and see that in the notation of \cite{Yoshino:book} (except with $x$ and $y$ swapped)
\[
  \Omega_{R^\#}(R/(x)) \cong \cok((x^4,x) \hat{\otimes} (y,y^2)) = \cok
  \left( \begin{bmatrix} x^4 & -y \\ y^2 & x \end{bmatrix}, \begin{bmatrix} x & y \\
      -y^2 & x^4 \end{bmatrix} \right) \cong \cok(\psi_1, \phi_1) \cong N_1\,;
\]
 and
\[
\Omega_{R^\#}(R/(x^2)) \cong \cok((x^3,x^2) \hat{\otimes} (y,y^2)) = \cok
  \left( \begin{bmatrix} x^3 & -y \\ y^2 & x^2 \end{bmatrix}, \begin{bmatrix} x^2 & y \\
      -y^2 & x^3 \end{bmatrix} \right) \cong \cok(\psi_2, \phi_2) \cong N_2 ;
 \]
 from which it follows that $\Omega_{R^\#} (R/(x^3)) \cong \Omega_{R^\#}(N_2) \cong M_2$ and $\Omega_{R^\#} (R/(x^4)) \cong \Omega_{R^\#}(N_1) \cong M_1$.  Thus $\Sigma_1$
contains $R^\#$ along with the indecomposables
$M_1 \cong (t^3, t^{10}), N_1 \cong (t^3, t^5), M_2 \cong (t^6,t^{10})$ and
$N_2 \cong (t^5, t^6)$.  As before, the irreducible morphisms between these
indecomposable $R^{\#}$-modules can all be realized as multiplication by powers of
$t$, and we obtain the following quiver for $\Lambda$.
\[
  \xymatrixrowsep{3.0pc} \xymatrix{ M_2 \ar[d]<0.5ex>^{1} \ar@{=>}[rr]<0.5ex>^{1,
      t^{-1}} & & N_2 \ar@{=>}[ll]<0.5ex>^{t^4, t^5} \ar[d]<0.5ex>^1 \\ M_1
    \ar[u]<0.5ex>^{t^3} \ar[rr]<0.5ex>^1 & & N_1 \ar[u]<0.5ex>^{t^3}
    \ar[ll]<0.5ex>^{t^5} \ar[dl]^{1} \\ & R^\# \ar[ul]^{t^3}}
\]
Thus, $\Lambda$ is isomorphic to a quotient of the completed path algebra of
the above quiver, with relations defined by the labels as in the previous example.

We also list the $\Sigma_1$-approximation sequences for the remaining indecomposable MCM $R^\#$-modules.  As in the previous example, for each $M \in \MCM(R^\#)$ the corresponding sequence is computed by first calculating $M/yM \in \MCM(R)$, which is easily done by looking at the matrix factorization associated to $M$.  We obtain the following sequences (and their syzygies):
\begin{align*}
& 0 \to B_1 \to M_1^2 \oplus N_2 \to A_1 \to 0 &\\
& 0 \to B_2 \to M_1 \oplus M_2^2 \to A_2 \to 0 & \\
& 0 \to D_i \to M_1 \oplus M_2 \oplus N_1 \oplus N_2 \to C_i \to 0 \ \ (i = 1,2) &\\
& 0 \to Y_1 \to M_1 \oplus M_2^2 \oplus N_1 \oplus N_2^2 \to X_1 \to 0 & \\
& 0 \to Y_2 \to M_1^2 \oplus M_2 \oplus N_2^2 \to X_2 \to 0 &
\end{align*}

Finally, we mention that the short exact sequences realizing each indecomposable
$M \in \MCM(R^\#)$ as a direct summand of an extension of modules in $\Sigma_1$ are
not as apparent here as they were in our previous example.  Here, only the modules
$A_1, B_1, C_2$ and $D_2$ arise in the middle terms of almost split sequences ending
with objects in $\Sigma_1$.  We sketch the construction of this sequence in one other
example.  Consider the module
$$A_2 \cong \cok \begin{bmatrix} x & -y & 0 \\ 0 & x^2 & -y \\ y & 0 &
  x^2 \end{bmatrix}.$$ As in the proof of Proposition~\ref{prop:SigmastarSigma}, we
obtain the desired short exact sequence by applying $\Omega_{R^\#}$ to the sequence
$$0 \to yA_2/y^2A_2 \to A_2/y^2A_2 \to A_2/yA_2 \to 0.$$
An easy computation shows that both $yA_2/y^2A_2$ and $A_2/yA_2$ are isomorphic to $R/(x) \oplus (R/(x^2))^2$ as $R$-modules.  Hence, using Proposition~\ref{prop:newHP} we obtain the short exact sequence
$$0 \to N_1 \oplus N_2^2 \to A_2 \oplus B_2 \oplus F \to N_1 \oplus N_2^2 \to 0$$
for some free module $F$.  Since $A_2, N_1$ and $N_2$ have rank $1$, while $B_2$ has rank $2$, we have $F \cong (R^\#)^3$.
\end{example}

\section*{Acknowledgments}
This work was begun during the authors' participation in the research program
``IRTATCA: Interactions between Representation Theory, Algebraic Topology and
Commutative Algebra'', at the Centre de Recerca Matem\`atica in Barcelona in
2015. The authors thank the CRM for a pleasant and productive experience.  GJL was
supported by NSF grant DMS-1502107, and ASD was supported by NSF conference grant DMS-58502086.

The authors are grateful to Tokuji Araya for pointing out a gap in an earlier version
of this paper.

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general application in an operational setting as was done in Chapter 3. Here, they are considered within the major functional areas of communications, reconnaissance, intelligence, surveillance, and target acquisition, weather, terrain, and environmental monitoring, position and navigation, and missile warning. These capabilities reinforce the importance of space systems as force multipliers, increasing the effectiveness and efficiency of the force. The advantages gained through the use of space systems are vital and integral to the success of any Army operation; however, the limitations associated with the use of space systems must also be considered.
Accurately assess the current situation.
Adapt to the demands of the situation, that is, to modify plans.
Act and react faster than the enemy.
Exploit opportunities and enemy vulnerabilities.
Identify targets for fire support systems.
Command and control its forces.
However, access to and availability of space capabilities depend to a degree on the echelon of command. The staff's responsibility is to know what capabilities are available and to optimize them. As forces are tailored to satisfy mission requirements, space system user terminals may have to be redistributed and space support teams created. To ensure adequate space support for the deploying force, a USARSPACE ARSST can form the nucleus of a tailored space support team.
Each space system consists of three segments: a space segment--the satellites; a control segment--ground control stations and managers; and a user segment--the equipment necessary to receive the satellite signals (see Figure 4-1). Following is a general discussion of space systems and associated terminals available to support the Army.
High ground has always played an important role in effective military communications. Communicating with forces dispersed across the battlefield or deployed great distances from their home bases has always been a major challenge. The importance of communications increases as the Army's mission becomes more complex and force levels decrease. Combat net radios for ultrahigh frequency (UHF) and very high frequency (VHF) are limited by line of sight (LOS) and have to rely on radio relays located on high ground to overcome terrain restrictions. Satellite communications systems are less hampered by LOS restrictions and can significantly enhance Army communications capabilities by extending ranges and reliability. Satellites offer an effective means of overcoming the physical limitations of LOS radios, extending the range of terrestrial communications systems, such as MSE, throughout the area of operations.
Communications satellites are the cornerstone of the Army's battle command architecture. They operate in a wide band of radio frequencies and provide the link between theater and CONUS for split-based operations and range extension between subordinate commands in the theater (see Figure 4-2). UHF satellite communications systems, such as the FLTSAT and AFSATCOM systems, are used to support battle command requirements for high-priority users, to include EAM dissemination, force direction, and JCS/CINC netting. SHF communications systems, such as the DSCS, support worldwide, long-haul, secure-voice and high-data-rate communications for battle command, early warning, crisis management, and internetting between the NCA/JCS and the combatant commanders. Milstar, the next generation military satellite communications system, is an EHF system capable of jam-resistant secure worldwide communications during all levels of conflict.
Greater freedom from siting restrictions.
Extended range, capacity, and coverage.
Stand-alone capability and reduced logistical support on the battlefield.
Freedom from rigid network configurations.
Military and commercial satellite communications systems are invaluable assets that can be used at all echelons of command. Military battle command nets are usually routed through military satellites due to the low probability of intercept and the increased communications capability (for example, to pass higher data rates and imagery) that they afford. Administrative and logistics networks may use either military or commercial satellites, which routinely support force projection and split-based operations around the world. These same systems may also be available to support selected training exercises. Military satellite communications systems, however, are considered joint assets and are controlled by the JCS, who allocate the resources based on need. JCS Memorandum of Policy 37 specifies access priorities. More often than not, demand exceeds the capabilities of the systems; therefore, access to military satellite channels is at a premium and closely monitored. Consequently, Army forces must clearly define and articulate their requirements for critical battle command connectivity. The Army also must consider the use of commercial satellites, such as the INTELSAT and the INMARSAT terminals, as an alternative means of satisfying their communications requirements. Space is not the panacea for all communications shortfalls; however, it is an important element of the flexible, responsive. and integrated battle command system required for Army operations.
First, because GPS receivers do not transmit any signals, they are not electronically detectable.
Second, the number of simultaneous GPS users is unlimited.
Third, although the accuracy of GPS is subject to certain influences, it does not vary over time as does the accuracy of inertial navigation aids. Ongoing technical refinements promise to strengthen the robustness and reliability of the GPS signal. GPS can quickly regain precision accuracy after momentary disruptions without further operator actions.
GPS greatly improves the commander's ability to locate units and to control them on the battlefield, facilitating situational awareness and improving agility and the ability to synchronize forces. GPS receivers increase the ability to accurately locate assets and to move them long distances over difficult terrain, especially during poor weather or lighting conditions. The GPS, when used in conjunction with maps or image products, can assist in all-weather, day-or-night rendezvous at designated points and times under radio silence. Accurate positioning data on friendly troops and on the location of minefields emplaced by friendly units can simplify the passage of lines and reduce fratricide.
GPS output can be in any one of the three commonly used coordinate systems: geographic, universal transverse mercator, or military grid reference. Joint and combined operations benefit from a common user grid. This grid allows units to use their specific coordinate position system and then to convert it rapidly to common coordinates or into similar systems used by other services or nations. Common positioning also reduces the minimum separation distance requirements for artillery, naval fire support, or close air support. The GPS's inherent accuracy increases the accuracy of requested fires, reducing the expenditure of ammunition as well as the risk of casualties caused by friendly fire.
The commander can also use GPS to manage critical assets that support the battle, to include weapon convoys, fuel, and ammunition supply points. The true value of POS/NAV equipment is that, for the first time, ground forces have access to small, lightweight navigational tools--that is, the SLGR and its replacement, the PLGR, AN/PSN-11--that are capable of providing very accurate information. This capability can provide POS/NAV information under most terrain and weather conditions. It will also profoundly influence azimuth determination, air traffic control, munitions guidance, and gun-laying operations.
The Army also has a limited number of commercial off-the-shelf (COTS) geodetic-quality receivers that provide positioning accuracies within millimeters. While engineers primarily use these systems to support terrain analysis and mapping requirements, they can also be used to support battle command and targeting. The cost and the necessity for specially trained operators limit the use of these receivers.
Space-based sensors have the advantage of unrestricted access over battlefields and other areas that are difficult to observe due to political or military reasons. Space systems allow the commander to see his area of operations and battle space, often far better than terrestrial systems, and permit near-real-time exploitation of favorable situations. This capability improves the agility, flexibility, and synchronization of the force--important aspects of current Army doctrine. Commanders can receive deep operations information as quickly and as accurately as close operations information. When information derived from space-based RISTA sensors is merged with information from ground and airborne systems through the ASAS, IPB is enhanced, situational awareness improved, and uncertainty reduced.
The Army's TENCAP Program, as executed by the ASPO, focuses on tactical applications of national space systems. The program provides Army commands with equipment (see Figure 4-3) that can receive and process data from these systems. The result has been the development and fielding of limited production equipment as operational systems that provide a valuable adjunct to organic sensors.
The commander benefits from access to data on enemy troop movements, lines of communications, and terrain conditions. He can use such information to determine the enemy's intent through direction and mass of enemy movement. The favored avenues of approach and the direction of the main attack can point to the most effective time and location for targeting friendly fires. Such information also offers opportunities for target damage assessment after deep attack by fire. See The Joint Tactical Exploitation of National Systems (J-TENS) manual for further information on the TENCAP Program and system capabilities. The J-TENS manual contains the procedures tactical commanders follow to obtain TENCAP support.
The DSP offers an early warning missile surveillance capability during operations. The DSP satellite constellation recognizes the launch of strategic and certain tactical missiles using infrared sensors to detect heat from missile plumes. Warning information consists of an assessment of the time and place of launch, the type of missile launched, and the missile's estimated course/direction. This information is provided to supported CINCs via voice and data communications. SATCOM is used to disseminate voice warning, and the TERS is used for data. The TERS is a worldwide distribution system currently made up of the Tactical and Related Applications (TRAP), Tactical Information Broadcast Service (TIBS), and Joint Operation Tactical System Communications Network. This launch warning data is communicated to Army forces within a theater to support TMD operations. Today, warning information is both centrally processed in CONUS and transmitted to the user via JTAGS prototypes that are actively serving tactical users with direct in-theater downlinks today. They will be replaced by van-mounted objective systems in the near future.
Detailed analysis of the environment, that is, weather and terrain, is a critical step in the IPB process. Weather and terrain conditions impact on friendly and enemy capabilities to move, shoot, and communicate. To optimize the capabilities of modern weapons systems, the tactical commander requires real-time weather and terrain information about his battle space. Satellites with weather and terrain monitoring sensors are a vital component in the information collection system. Weather and terrain information must be collected and downlinked to a ground processing unit where it can be used to prepare tailored products to support decision making by tactical commanders.
Military and civil weather satellites provide worldwide, near-real-time weather information. DMSP satellites obtain comprehensive information on weather phenomena and atmospheric data. They can image weather phenomena in both visual and infrared spectral bands. Simultaneously, they record the temperature and moisture data throughout the swath width and at various altitudes. Large DMSP receiver systems, known as Air Force Mark IV vans, normally deploy to rear areas in mature theaters where they receive data and relay products to corps and division SWOs via facsimile. There it is analyzed and combined with other local weather information that may be received through the IMETS to forecast conditions throughout the battle space.
Civil geostationary satellites such as the GOES provide a hemispherical view of weather patterns, while polar-orbiting satellites, such as the television infrared observation satellite (TIROS), provide a low earth view of the weather as they pass overhead. Wraase weather receivers receive weather facsimile (WEFAX) and automatic picture transmission images from US, Russian, Japanese, and European civil weather satellites. They do not receive DMSP or other data, for example, atmospheric temperatures and moisture content, transmitted by civil satellites. Using the Wrasse weather receiver to exploit near-real-time satellite-gathered weather data and making it available down to division level has significantly improved forecasting at the tactical level of operations. The public and private sectors have jointly made great advances in exploiting space technologies, resulting in the rapid accumulation and dissemination of weather data in immediately usable form to Army forces.
Understanding the limitations and opportunities of terrain is a fundamental military skill. Terrain forms the natural structure of the battlefield. Commanders must recognize the its drawbacks and potential to protect friendly operations and to put the enemy at a disadvantage. Terrain analysis is critical to current and projected operational uses of specific terrain.
Much of the world is not adequately mapped to support Army operations. Satellites collecting MSI can provide reasonably accurate, medium-resolution data (see Figure 4-4) to aid in mapping and terrain analysis. The DMA and some engineer topographic (TOPO) units can create MSI maps primarily to be used as map substitutes in areas of the world that are not adequately mapped to large scale. Satellite image mapping capabilities can provide the most current data worldwide. Terrain-sensing satellites using MSI can provide accurate 5-to-80-meter resolution terrain data to support mapping and other analytical requirements. In the near future, satellites will be capable of providing HSI data that will further enhance map-producing capabilities.
Merging MSI data with digital TOPO data and digital terrain elevation data produced by the DMA provides three-dimensional perspective views. These views highlight observation and fields of fire from both the aerial and ground perspectives. A rough analysis of these images will show potential ingress and egress routes and aid in the development of trafficability assessments. This information could include soil trafficability, mobility corridors, and perspective views of denied areas, including enemy-controlled territory, contaminated areas, minefields, smoke, forest fires, and many other conditions that may arise in the operational environment that may critically impact battle space awareness. In the IPB process, MSI is useful in determining maneuverability and possible areas of enemy concealment and operations.
While satellites can provide the Army many valuable capabilities, planners and users must understand some of their general limitations. Though not all-inclusive, the following limitations represent areas that must be considered when planning and requesting space support.
Launch operations are complex, time-consuming, manpower-intensive, and costly. For these reasons, military satellites are national resources supporting the NCA, CINCs, other services, and tactical users. As a result, requirements may exceed system capacity and capabilities. A validation process to determine what requirements will be satisfied is based on priority and system availability. Potentially, this process limits the Army's accessibility to satellite capabilities. Not only are these systems limited, most are owned, controlled, or dedicated to exclusively supporting other missions and may not be available to support Army requirements. Furthermore, many satellites do not provide direct links to the ultimate user, often requiring significant processing time by a third party to convert data into usable media. This conversion further delays distribution to Army users. As a result of having limited access, Army planners should explore the use of commercial space systems when developing military operations plans.
Satellites are designed to survive the harsh space environment and have a degree of hardiness that many ground systems do not have. While in orbit, they may be affected by temperature extremes, radiation, solar flares, meteoroids, and space debris. Imaging systems used for reconnaissance can be affected by clouds, fog, and smoke.
METT-T affects employment of satellite ground terminals. They are also affected by line-of-sight disruptions such as high foliage areas, low take-off angles, placement in fringe areas of coverage, high usage in small and close areas, and susceptibility to whatever military capability--such as destruction, denial, disruption--an enemy force may have to use against any other ground system in the area of operation. The ground control and user segments represent the most likely targets for an adversary. When planning the use of satellite systems, the planner must consider alternatives in the event these systems are lost.
Satellites can be attacked, but they are not easy targets. Russia has demonstrated a limited capability to attack and destroy satellites in low earth orbit. Jamming satellite systems or the link between the satellite and the ground segment of the system is also a threat. Not lost on Operations Desert Shield/Storm observers was how much dependence was placed on the use of satellite systems. Therefore, jam-resistant satellite capabilities such as Milstar must become the backbone of our satellite systems. Also imperative is that the Army should place more emphasis on influencing their design and survivability. Military use of commercial satellites must be expanded; however, risk as well as benefits must be considered, given the vulnerability of these systems.
Operationally, the Army is dependent upon systems currently in orbit, although these systems may or may not be suited to a particular Army mission. Many satellites do not provide continuous coverage; for example, Landsat sensors revisit equatorial points on the earth approximately every 16-18 days. Moving a satellite to a more advantageous orbit or position takes time and is limited to the amount of fuel on board since satellites cannot be refueled. Satellites are normally built with a high degree of survivability and redundancy, but once damaged or having experienced component failure, their utility and reliability are degraded. Furthermore, they are difficult to repair.
Generally, the orbital characteristics of a space system are related to the function of the satellite (see Figure 4-5). Satellites may be at a relatively fixed altitude (circular orbits) or vary in altitude (elliptical orbits). Low orbits, being closer to the earth, best support sensing requirements. The disadvantage of a low orbit is a limited view of the earth and a relatively short time over any particular location. As altitude increases, the field of view increases, but the ability to resolve small objects decreases.
Another factor affecting the use of satellites is inclination--the angle the satellite's orbital plane makes with the earth's orbital plane. A higher inclination generally means that more of the earth is covered. For example, a satellite in polar orbit (90 degrees inclination) will observe the entire globe as the earth rotates through the orbital plane. Inclinations from 0 to 90 degrees cover increasingly higher latitudes for low-altitude satellites. The length of time between satellite coverage of a particular location (that is, revisit time) depends upon the number of satellites in a constellation and the capabilities of the payload, such as direct versus slant view, type of frequency, band width, data transmission rates, and sensor footprint size.


The Movistar Team achieved a great result today in the team time trial. The stage took place on 38 km between Vannes and Plumelec, where Alejandro won a stage of the French race in 2008.
Eusebio Unzué's squad took the third place four seconds only behind Team BMC and three behind Team Sky.
On the eve of the first rest day, Alejandro is now the seventh overall, 1' 50" behind the still leader Chris Froome.