text:
\begin{document}
\begin{center}
{\Large{\bf A note on the injectivity of action by compact quantum groups on a class of  $C^{\ast}$-algebras}}\\
{\large {\bf Debashish Goswami\footnote{Partially supported by J C Bose Fellowship from D.S.T. (Govt. of India) and also acknowledges
 the Fields Institute, Toronto for providing hospitality for a brief stay when a small part of this work was done.} and \bf Soumalya Joardar \footnote{Acknowledges support from CSIR.}}}\\

Indian Statistical Institute\\
203, B. T. Road, Kolkata 700108\\
Email: goswamid@isical.ac.in,\\
Phone: 0091 33 25753420, Fax: 0091 33 25773071\vspace{2mm}\\
{\bf {\large Dedicated to Paul F. Baum on the occasion of his eightieth birthday.}}\\
\end{center}
\begin{abstract}
We give some sufficient conditions for the injectivity of actions of compact quantum groups on $C^{\ast}$-algebra. As an application,
 we prove that any faithful smooth  action by a compact quantum group on a compact smooth (not necessarily connected) manifold is
 injective. A similar result is proved for actions on $C^{\ast}$- algebras obtained by Rieffel-deformation of compact, smooth manifolds.
\end{abstract}
{\bf Subject classification :} 81R50, 81R60, 20G42, 58B34.\\
{\bf Keywords:} Compact quantum group, Riemannian manifold, smooth action.

 \section{Introduction}
 Quantum groups are natural generalization of groups and they are used as `generalized symmetry objects' in mathematics and physics.
  The pioneering work by  Drinfeld
and Jimbo (\cite{drinfeld}, \cite{drinfeldICM}, \cite{jimbo1}, \cite{jimbo}) and others  gave the formulation of  quantum groups
 in the algebraic setting as Hopf algebras typically obtained by
 deformations of the universal enveloping algebras of semisimple Lie algebras. This led  to a deep and successful theory having connections with
physics, knot theory, number theory, representation theory etc. On the other hand,   S.L. Woronowicz (see, e.g. \cite{Pseudogroup})  approached
 it from a point of view of harmonic analysis on locally compact groups and came up with
 a set of axioms for defining compact quantum groups (CQG for short) as
the generalization of compact topological groups. In this note, we will restrict ourselves to the framework of compact quantum groups only.
 It is natural to define quantum analogue of group action on spaces. This can be done in different ways: for the purely algebraic approach, this
  is defined as a co-action  of Hopf algebra. In the analytic theory, there are $C^{\ast}$ and von Neumann algebraic notions of action. We'll be concerned
   with Podles' formulation of $C^{\ast}$-action of compact quantum groups on $C^{\ast}$- algebras. A subtle point about this definition is that it does not assume
    the injectivity of the action. We mention here that some authors (e.g. \cite{free}) indeed prefer to include injectivity in the definition
     of a $C^{\ast}$-action, but this is not a universal practice. In fact, in \cite{Act_com}, Soltan discussed examples of non-injective $C^{\ast}$-actions. On the other hand,  group actions on
     spaces are always injective. Injectivity also follows in the algebraic setting for Hopf algebra co-actions as well as for von Neumann algebraic notion of
      actions of (von Neumann algebraic) quantum groups. Thus, it is an interesting and important problem to give sufficient conditions for injectivity of
       $C^{\ast}$-action of compact quantum groups in the sense of Podles. This is the aim of this short note.  We'll consider CQG actions on $C(M)$ and their Rieffel deformations where
        $M$ is a compact smooth manifold. Under a smoothness condition on the action on $C(M)$ ( in the sense of  \cite{gafa})  we can prove injectivity.
        For  the Rieffel-deformation of classical manifolds, we prove injectivity under a natural analogue of smoothness and compatibility of the action with the canonical toral action.

\section{ Preliminaries}
In this paper all the
Hilbert spaces are over $\mathbb{C}$ unless mentioned otherwise. For a vector space $V$, $V^{'}$ stands for its algebraic dual.
$\oplus$ and $\ot_{\rm alg}$ will denote the algebraic direct sum and
algebraic tensor
product respectively. On the other hand, the minimal $C^{\ast}$- algebra tensor product and tensor product of Hilbert spaces as well as Hilbert modules will be denoted by $\ot$. In particular, we consider Hilbert modules of the form $\clh \ot \clc$ where $\clc$ is a $C^*$ algebra.
We shall denote the $C^{\ast}$-
algebra of bounded operators on a Hilbert space $\clh$ by $\clb(\clh)$ and the
$C^{\ast}$- algebra
of compact operators on $\clh$ by $\clb_{0}(\clh)$. $Sp$, $\overline{Sp}$
stand for the
linear span and closed linear span of elements of a vector space respectively, whereas ${\rm Im}(A)$ denotes the image of a linear map $A$. Given a group action $\gamma$ on a locally convex space $Z$ we denote the fixed point subspace by $Z^\gamma$.

  We call a
locally convex space Fr\'echet if the family of seminorms is countable
(hence the space
is metrizable) and the space is complete with respect to the metric given by the family
of seminorms. There are many ways to equip the algebraic
tensor product of two locally convex spaces with a locally convex topology. Let
$E_{1}$, $E_{2}$ be two
locally convex spaces with family of seminorms $\{||.||_{1,i}\}$ and
$\{||.||_{2,j}\}$ respectively. Then one wants a family $\{||.||_{i,j}\}$ of
seminorms for $E_{1}\ot_{\rm alg} E_{2}$ such that $||e_{1}\ot
e_{2}||_{i,j}=||e_{1}||_{1,i}||e_{2}||_{2,j}$. The problem is that such a choice
is far from unique and there is a maximal and a minimal choice giving the
projective and injective tensor product respectively. Let us denote the projective tensor product by $E_1 \hat{\ot} E_2$. A Fr\'echet locally convex space is called nuclear if its projective
and
injective tensor products with any other Fr\'echet space coincide as a
locally convex space. It is known that closed subspaces and quotients by closed subspaces of a nuclear Fr\'echet space are again nuclear.
 We do not go into further details
of this topic
here but refer the reader to \cite{TVS} for a comprehensive discussion.
Furthermore if the space is a $\ast$ algebra then we demand that its $\ast$
algebraic structure is compatible with its locally convex topology i.e.
the involution $\ast$ is continuous and multiplication is jointly continuous with
respect to the topology. Projective and
injective tensor product of two such topological $\ast$ algebras are again
topological $\ast$ algebra. We shall mostly use unital $\ast$
algebras. Henceforth all the topological $\ast$-algebras will be unital unless
otherwise mentioned. Consider a locally convex algebra $\cla$ for which each of  the defining  seminorms, say $\| \cdot \|_i$, satisfies
  \be \label{1} \| xy\|_i \leq C_i\| x \|_i \| y\|_i\ee for some constant $C_i$ and all $x,y \in \cla$. Then it is easy to see from the definition of
   the projective tensor product that the algebra multiplication map (say $m$) lifts to a continuous map from $\cla \hat{\ot} \cla$ to $\cla$.

We mainly need a particular class of nuclear locally convex $\ast$-algebra, which is $C^\infty(M)$, where $M$ is any compact smooth manifold.  The natural Fr\'echet topology on $C^{\infty}(M)$ is
given by the seminorms of the form $p^{U, K,\alpha}$,
\begin{displaymath}
p^{U,K,\alpha}(f)={\rm sup}_{x\in K}|\partial^{\alpha}(f)(x)|,
\end{displaymath}
where $K$ is a compact subset contained in the domain of some coordinate chart $(U,(x_{1},...,x_{n}))$, $\alpha=(i_{1},...,i_{k})$ a multi index and
$\partial^{\alpha}=\frac{\partial}{\partial x_{i_{1}}}...\frac{\partial}{\partial x_{i_{k}}}$, $i_j \in \{1, \ldots, n\}$.
We can similarly define a Fr\'echet topology on $C^{\infty}(M,E)$, the space of smooth $E$-valued functions on $M$ for any Fr\'echet space $E$. We refer the reader to \cite{gafa} for more details.
One can verify condition (\ref{1}) for the family of seminorms defining the Fr\'echet algebra  $C^\infty(M,\cla)$ where $\cla$ is a Banach algebra, by Leibniz rule.

\subsection{Compact quantum groups and $C^{\ast}$-actions}
\bdfn
A compact quantum group (CQG for short) is a  unital $C^{\ast}$-algebra $\clq$ with a
coassociative coproduct
(see \cite{VanDaele}, \cite{Pseudogroup}) $\Delta$ from $\clq$ to $\clq \ot \clq$
such that each of the linear spans of $\Delta(\clq)(\clq\ot 1)$ and that
of $\Delta(\clq)(1\ot \clq)$ is norm-dense in $\clq\ot \clq$.
\edfn

 A unitary representation of a CQG $(\clq,\Delta)$ on a Hilbert
space $\clh$ is a unitary $U \in \cll(\clh \ot \clq)$ such that the $\mathbb{C}$-linear map $V$ from $\clh$ to the Hilbert module
$\clh \ot
\clq$ given by $V(\xi)=U(\xi \ot 1)$ satisfies
 $(V \ot {\rm id})\circ V=({\rm id} \ot \Delta)\circ V.$
Here,
the map $(V \ot {\rm id})$ denotes the extension of $V \ot {\rm id}$ to the completed tensor product
  $\clh \ot \clq$ which exists as $V$ is an isometry.

Every  CQG $\clq$ contains   a canonical dense unital $\ast$-subalgebra
$\clq_0$ of $\clq$ on which linear maps $\kappa$ and
$\epsilon$ (called the antipode and the counit respectively) are defined making the above subalgebra a Hopf $\ast$ algebra. In fact, this is  the algebra generated by the `matrix coefficients' of
the (finite dimensional) irreducible non degenerate representations (see \cite{VanDaele} )
 of the CQG. The antipode is an anti-homomorphism and also satisfies $\kappa(a^{\ast})=(\kappa^{-1}(a))^{\ast}$ for $a \in \clq_{0}$.

 It is known that there is a unique state $h$ on a CQG $\clq$ (called the Haar
state) which is bi invariant in the sense that $({\rm id} \ot h)\circ
\Delta(a)=(h \ot {\rm id}) \circ \Delta(a)=h(a)1$ for all $a$. The Haar state
need not be faithful in general, though it is always faithful on $\clq_0$ at
least. Given the Hopf $\ast$-algebra $\clq_{0}$, there can be several CQG's
which have this $\ast$-algebra as the Hopf $\ast$-algebra generated by the
matrix elements of finite dimensional representations. We need two of such CQG's: the reduced and the universal one. By definition, the reduced
 CQG $\clq_r$ is the image of $\clq$ in the GNS representation of $h$, i.e. $\clq_r=\pi_r(\clq)$, $\pi_r: \clq \raro \clb(L^2(h))$ is the GNS representation.

 There also exists a
largest such CQG $\clq^{u}$, called the universal CQG corresponding to
$\clq_{0}$. It is obtained as the universal enveloping $C^{\ast}$-algebra of
$\clq_{0}$. We also say that a CQG $\clq$ is universal if $\clq=\clq^{u}$. Given two CQG's $(\clq_{1},\Delta_{1})$ and $(\clq_{2},\Delta_{2})$, a $\ast$ homomorphism $\pi:\clq_{1}\raro\clq_{2}$ is said to be a CQG morphism if $(\pi\ot\pi)\circ\Delta_{1}=\Delta_{2}\circ\pi$ on $\clq_{1}$. In case $\pi$ is surjective, $\clq_{2}$ is said to be a quantum subgroup of $\clq_{1}$  and denoted by $\clq_{2}\leq\clq_{1}$.

\bdfn
We say that a CQG $(\clq, \Delta)$ (co)-acts on a (unital) $C^*$-algebra $\clc$ if there is a unital $\ast$-homomorphism $\alpha: \clc \raro \clc \ot \clq$ such that
$(\alpha \ot {\rm id})\circ \alpha=({\rm id} \ot \Delta)\circ \alpha$, and the linear span of $\alpha(\clc)(1 \ot \clq)$ is norm-dense in $\clc \ot \clq$.
\edfn
 In Woronowicz theory, it is customary to drop `co', and call the above co-action simply `action' of the CQG on the $C^*$-algebra. Let us adopt this convention for the rest of the note.
    An action  $\alpha$ on $\clc$ is called faithful if the $\ast$-subalgebra generated by $\{ (\omega \ot {\rm id})(\alpha(b)) \}$, where $b \in \clc$ and $\omega$ varying over the
   set of bounded linear functionals on $\clc$, is dense in $\clq$.

   Given an action $\alpha$, we define $\alpha_r=({\rm id} \ot \pi_r)\circ \alpha$ and call it the reduced action. If the Haar state is faithful on $\clq$, we have $\alpha=\alpha_r$.
      \bdfn
 We call an action $\alpha$ of a CQG $\clq$ on a unital $C^{\ast}$-algebra $\clc$ to be implemented by a unitary representation $U$ of $\clq$ in $\clh$, say, if
  there is a faithful representation $\pi : \clc \raro \clb(\clh)$ such that $U (\pi(x) \ot 1)U^{\ast}=(\pi \ot {\rm id})(\alpha(x))$ for all $x \in \clc$.
  \edfn
  It is clear that if an action is implemented by a unitary representation then it is one-to-one. In fact, as $({\rm id} \ot \pi_r)(U)$ gives a unitary representation of $\clq_r$ in $\clh$
   and the `reduced action' $\alpha_r:=({\rm id} \ot \pi_r)\circ \alpha$ of $\clq_r$ is also implemented by a unitary representation, it follows that even
    $\alpha_r$ is one-to-one. We see below that this is actually equivalent to implementability by unitary representation:
    \blmma
    \label{unitary_impl}
    Given an action $\alpha$ of $\clq$ on a unital separable $C^{\ast}$-algebra $\clc$, the following are equivalent:\\
    (a) There is a faithful positive functional $\phi$ on $\clc$ which is invariant w.r.t. $\alpha$, i.e. $(\phi \ot {\rm id})(\alpha(x))=\phi(x)1_\clq$ for all $x \in \clc$.\\
    (b) The action is implemented by some unitary representation.\\
    (c) The reduced action $\alpha_r$ of $\clq_r$ is injective.
\elmma
{\it Proof:}\\
If (a) holds, we consider $\clh$ to be the GNS space of the faithful positive functional $\phi$. The GNS representation $\pi$ is faithful, and the linear map $V$ defined by
 $V(x):=\alpha(x)$ from $\clc \subset \clh=L^2(\clc,\phi)$ to $\clh \ot \clq$ is an isometry by the invariance of $\phi$. Thus $V$ extends to $\clh$ and it is easy to check that
  it induces a unitary representation $U$, given by $U(\xi \ot q)=V(\xi)q$, which implements $\alpha$.\\
   \indent We have already argued $(b) \Rightarrow (c)$, and finally, if (c) holds, we choose any faithful state say $\tau$ on the separable $C^{\ast}$ algebra $\clc$ and take $\phi(x)=(\tau \ot h)(\alpha_r(x))$,
   which is faithful as $h$ is faithful on $\clq_r$ and $\alpha_r$ is injective. It can easily be verified that $\phi$ is $\alpha$-invariant on the dense subalgebra $\clc_0$ mentioned before, and hence on the whole of
    $\clc$.\qed

   We'll also need the following facts about actions on commutative $C^{\ast}$-algebras.
   \bppsn
   \label{abc}
   If a CQG $\clq$ acts faithfully on $C(X)$ for some compact metrizable  space $X$, then $\clq$ is separable and it is also of Kac type, i.e. $\kappa$ is norm-bounded
    on $\clq_r$, $\kappa^2={\rm id}$ and the Haar state is tracial.
    \eppsn
    {\it Proof}\\
    Note that $X$ is second countable and hence $C(X)$ is separable. Choose a countable dense set of points $\{ x_i,~i=1,2,\ldots \}$ and a countable norm-dense subset $\{ f_n ,~n=1,2,\ldots\}$
     of $C(X)$. It follows from faithfulness of the action $\alpha$ (say) that $\clq$ is generated as a $C^{\ast}$-algebra by the countable set $\{ \alpha(f_n)(x_i),~i,n=1, 2,\ldots \}$, hence it is separable.\\
     \indent For the proof of the Kac conditions, see \cite{Huichi}.\qed
 \section{Smooth actions are injective}
Let $M$ be a compact smooth manifold. Let us recall the definition of smooth CQG action on it from \cite{gafa}.
\bdfn
We say that an action $\alpha$ of a CQG $\clq$ on $C(M)$ is smooth if
 $\alpha$ maps $C^\infty(M)$ into $C^\infty(M, \clq)$ and   $Sp \ \alpha(C^\infty(M))(1\ot \clq)$ is dense in $C^\infty(M, \clq)$ in the Frechet topology.\edfn

 \bthm
\label{smooth_bdd_counit}
If $\clq$ has a faithful smooth action $\alpha$ on $C^\infty(M)$, where $M$ is compact manifold, then for every fixed $x \in M$ there is a well-defined, $\ast$-homomorphic
 extension $\epsilon_x$ of the counit map
 $\epsilon$ of $\clq_0$  to the unital $\ast$-subalgebra $\clq^\infty_x:=\{ \alpha_r(f)(x):~f \in C^\infty(M)\}$ satisfying $\epsilon_x(\alpha_r(f)(x))=f(x)$, where $\alpha_r$ is
  the reduced action discussed earlier.\ethm
 {\it Proof:}\\
 Adapting the arguments of \cite{Podles} and \cite{free}, we can get a  the Fr\'echet dense subalgebra $\clc_0$ of $C^\infty(M)$ on which $\alpha$ restricts to an algebraic co-action of $\clq_0$.
 For example, $\clc_0$ may be chosen as the Peter-Weyl subalgebra in the sense of \cite{free}.
 Replacing $\clq$ by $\clq_r$ we can assume without loss of generality that $\clq$ has faithful Haar state and $\alpha=\alpha_r$.
 In this case $\clq$ will have bounded antipode $\kappa$ (by Proposition \ref{abc}). Let $\alpha_x: C^\infty(M) \raro \clq^\infty_x$ be the map defined by $\alpha_x(f):=\alpha(f)(x)$.
 It is clearly continuous w.r.t. the Fr\'echet topology of $C^\infty(M)$ and hence the kernel  $\cli_x$ (say)
  is a closed ideal, so that the quotient, which is isomorphic to $\clq^\infty_x$, is a nuclear space. Let us consider $\clq^\infty_x$ with this topology and then by nuclearity,
   the projective and injective tensor products with $\clq$ (viewed as a
 Banach space, which is separable by Proposition \ref{abc}) coincide.  The
multiplication map $m: \clq^\infty_x \ot_{\rm alg} \clq \raro \clq$ extends to a
continuous map (to be denoted by $m$ again) on $\clq_x^\infty \hat{\ot} \clq$.

          It follows from the relation $({\rm id}\ot\Delta) \circ \alpha=(\alpha \ot {\rm id})\circ \alpha$ that $\Delta(\alpha_x(f))=(\alpha_x \ot {\rm id})(\alpha(f))$, i.e. $\Delta$
           maps $\clq_x^\infty$ to the
           $(\alpha_x \ot {\rm id}_\clq)(C^\infty(M) \hat{\ot} \clq) \subseteq \clq_x^\infty \hat{\ot} \clq$.  Thus, the composite map
 $\beta:=m \circ ({\rm id} \ot \kappa) \circ \Delta: \clq^\infty_x \raro \clq$ is continuous.  Clearly, this map  coincides with
 $\epsilon(\cdot) 1_\clq$ on the Fr\'echet-dense subalgebra of $\clq^\infty_x$ spanned by elements of the form $\alpha(f)(x)$,
  with $f$ varying in the Fr\'echet-dense  subalgebra $\clc_0$  of $C^\infty(M)$. By continuity of $\beta$, it follows that the range of $\beta$ is $\IC 1_\clq$.
   Hence we can define $\epsilon_x$ by setting $\epsilon_x(\cdot)1_\clq=\beta(\cdot).$ This completes the proof of the lemma.\qed

      \bcrlre
  \label{automatic_inj}
  For any smooth action $\alpha$ on $C^\infty(M)$, the conditions of Theorem \ref{unitary_impl} are satisfied, hence the reduced  action is injective on $C(M)$.
  \ecrlre
  {\it Proof:}\\
  Replacing $\clq$ by the Woronowicz subalgebra generated by $\{ \alpha(f)(x), f \in C(M), x \in M\}$ we may assume that
   $\alpha$ is faithful. If $\alpha_r(f)=0$ for $f \in C^\infty(M)$  then by Lemma \ref{smooth_bdd_counit} applying the extended $\epsilon$ we conclude
    $f=0$. Now, consider any positive Borel measure $\mu$ of full support on $M$, with $\phi_\mu$ being the positive functional obtained by integration w.r.t $\mu$.
    Let $\psi:=(\phi_\mu \ot h) \circ \alpha_r$ be the positive functional which is clearly $\alpha_r$-invariant and faithful on $C^\infty(M)$,
    i.e. $\psi(f)=0, f \in C^\infty(M)$ and $f$ nonnegative  implies $f=0$. But  by Riesz Representation Theorem there is a positive Borel measure
     $\nu$ such that $\psi(f)=\int_M f d\nu$. We claim that $\nu$ has full support, hence $\psi$ is faithful also on $C(M)$.
     Indeed, for any nonempty open subset $U$ of $M$ there is a nonzero positive $f \in C^\infty(M)$, with $0 \leq f \leq 1,$ and support of $f$ is contained in $ U$.
     By faithfulness of $\psi$ on $C^\infty(M)$ we get $0 < \psi(f) =\int_U f d\nu \leq \nu(U).$\qed

\brmrk
In a recent work \cite{final}, Goswami has proved that any CQG which admits a faithful smooth action on a compact connected smooth manifold must be isomorphic to $C(G)$ for some
 group $G$ and the CQG action becomes $G$-action. Hence it is in particular injective. However, the result of the present note is applicable to a  possibly disconnected manifold. Moreover,
  the proof of injectivity given here is rather short and direct.
\ermrk

\section{Smooth action on Rieffel deformation}
Let us now consider CQG actions on noncommutative $C^{\ast}$-algebras.
Rieffel deformation (see \cite{rieffel}) is a well-known and very useful procedure to obtain interesting noncommutative $C^{\ast}$-algebras from the commutative ones. In particular, given  a smooth
 compact Riemannian manifold $M$ equipped with a toral subgroup $T \cong \IT^n \subseteq ISO(M)$, one can construct a family of (typically noncommutative) $C^{\ast}$-algebras
  $C(M)_\theta$ indexed by $n \times n$ skew symmetric matrices $\theta$. There is a similar procedure (see \cite{wang_def}) for deforming a CQG
    with some toral quantum subgroup of rank $n$ inducing an action of torus of rank $2n$ combining left and right action by the elements of the $n$-toral subgroup.
     In this case, one gets a CQG by retaining the same co-algebra structure as the original one but changing  the algebra structure. This will be called the Rieffel-Wang deformation of $G$.

     Let $\cla_\theta$ be the noncommutative $n$-torus, which is the universal $C^{\ast}$-algebra
   generated by unitaries $U_1, \ldots U_n$ satisfying the commutation relations $U_j U_k={\rm exp}(2\pi i \theta_{jk})U_k U_j$, where $\theta=(( \theta_{jk} )).$
    Given a unital $C^{\ast}$-algebra $\clc$ with a $\IT^n$-action given by $\beta_z$ (say), the deformed $C^*$-algebra $\clc_\theta$ can be described in two alternative ways: either in the original picture of Rieffel where one defines a new, twisted multiplication on
     the spectral algebra for the toral action and then considers appropriate $C^{\ast}$-completion, or as   in \cite{connes_dubois}, identifying
  $\clc_\theta$  with the fixed point subalgebra $(\clc \ot \cla_\theta)^{\beta \ot v^{-1}}$  where
  $v_z$ denotes the canonical toral action on $\cla_\theta$ satisfying $v_z(U_i)=z_iU_i$ for all $i$. We have a `dual' $T$-action on $\clc_\theta$ which is the restriction of $({\rm id} \ot v)$ on $\clc \ot \cla_\theta$.

    Given a CQG $\clq$ and a quantum subgroup of $\clq$ isomorphic with $T=\IT^n$, with the corresponding
       surjective CQG morphism $\pi : \clq \raro C(T)$, we can define a left and a right $\IT^n$-action, say $\chi^l_z,\chi^r_z$ respectively, by setting
       $\chi^l_z=({\rm id} \ot ({\rm ev}_z \circ \pi))\circ \Delta$ and $\chi^r_z=(( {\rm ev}_z \circ \pi) \ot {\rm id})\circ \Delta.$ Using this, we have a $\IT^{2n}$-action
       $\chi_{z,w}=\chi^l_{z}\chi^r_{w}$ on $\clq$ and the corresponding deformed CQG   is the $C^{\ast}$-algebra $\clq_{\theta \oplus (-\theta)}.$

        We have the following from Theorem 3.11 of \cite{bh_gos}.
        \blmma
        \label{action}
        Let $\clc$ be a unital $C^{\ast}$-algebra equipped with a $\IT^n$-action given by $\ast$-automorphism $\beta_z$,  $\clq$ be a reduced CQG with an action $\alpha$ of $\clq$ on $\clc$ and a quantum subgroup of $\clq$ isomorphic with $\IT^n$ as above (with the corresponding
        morphism $\pi$) satisfying $\beta_z:=({\rm id}\ot ({\rm ev}_z \circ \pi))\circ \alpha$.  Then we have an action $\alpha_\theta$ of $\clq_{\tilde{\theta}}$ ($\tilde{\theta}=
        \theta \oplus (-\theta)$) on $\clc_\theta$. Here the deformation of $\clc$ is taken w.r.t. the action $\beta$.
        \elmma

   Consider now $\clc=C(M)$, $M$ being a compact smooth Riemannian manifold equipped with $\IT^n$ action, which also induces a $\IT^n$ action (say $\beta$) on $C(M)$.
    Let $\gamma=\beta \ot v^{-1}$ as before and let
     us call the subalgebra $C^\infty(M, \cla_\theta)^\gamma  \subset C(M, \cla_\theta)^\gamma\equiv
      (C(M) \ot \cla_\theta)^\gamma=C(M)_\theta$ the `smooth subalgebra' and call an action $\alpha$ on $C(M)_\theta$ by a CQG $\clq$
    to be
     smooth if it maps the above smooth subalgebra into $C^\infty(M, \cla_\theta \ot \clq)^{(\gamma \ot {\rm id})}$ and the linear span of  $\alpha(F)(1 \ot \clq)$, $F \in C^\infty(M,\cla_\theta)^\gamma$ is
      dense in $C^\infty(M,\cla_\theta \ot \clq)^{(\gamma \ot {\rm id})}$. Now, we can state and prove the following:
      \bthm
      Let $M$ be  as above and let  $\alpha$ be a  smooth action of a CQG on $C(M)_\theta$ in the above sense. Moreover, assume that there is a quantum subgroup of $\clq$ isomorphic with $T=\IT^n$, given by a
       surjective CQG morphism $\pi : \clq \raro C(T)$ such that $({\rm id}\ot ({\rm ev}_z \circ \pi))\circ \alpha$ coincides with the canonical `dual' $T$-action  on $C(M)_\theta$. Then the
        action (in fact, the reduced one) is injective.
      \ethm
{\it Proof:}\\ We only very briefly sketch the proof.  As before, assume without loss of generality that the CQG is reduced.
 It follows from the proof of Theorem 3.11 of \cite{bh_gos} that the action $\alpha_{-\theta}$ of $\clq_{\tilde{\theta}}$ on  $(C(M)_\theta)_{-\theta}
\cong C(M)$   is  smooth. We note that the word `smooth' in the statement of Theorem 3.11 of \cite{bh_gos} is used in a  sense weaker  than ours: it only means the
 invariance of the smooth algebra there. However, the dense  subalgebra (e.g. Peter-Weyl subalgebra)of  $C(M)_\theta$ for
  the action $\alpha$, on which $\clq_0$ (co)acts algebraically, can be identified as a vector space  with a dense subalgebra of
   $C^\infty(M)$ on which the deformed action (which is the same as $\alpha$ a linear  map on this space) $\alpha_{-\theta}$ is algebraic. From this,
    the Podles-type density condition follows, i.e. $\alpha_{-\theta}$ is smooth in our sense. Hence it is injective by Corollary \ref{automatic_inj}. Moreover, by that corollary and
     Theorem \ref{unitary_impl}, we get a unitary representation of $\clq_{\tilde{\theta}}$ which implements $\alpha_{-\theta}$. But by the generalities of Rieffel-Wang (or, more general cocycle-twisted)
      deformation of CQG as in the Chapter 7 of \cite{book}, we conclude that $\alpha=(\alpha_{-\theta})_\theta$ is unitarily implemented too, where the corresponding Hilbert space and unitary
       essentially remain the same. In particular, $\alpha$ is injective.\qed

{\bf Ackowledgement:}\\ The first author would like to recall his fond memories of interaction with Prof. Paul Baum on several occasions: ICTP (Trieste), TIFR (India) and also in the conference in
 the Fields Institute in 2016. Although there was no discussion on the particular topic of this note, the first author is grateful to him for encouragement and suggestions regarding Baum-Connes' conjecture
  for quantum groups and other problems related to quantum symmetry on operator algebras. The authors wish him a long, healthy and creative  life.


text:
\begin{document}
\title[Relations for Grothendieck groups and representation-finiteness]{Relations for Grothendieck groups and representation-finiteness}

\author[H. Enomoto]{Haruhisa Enomoto}

\address{Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya. 464-8602, Japan}
\email{m16009t@math.nagoya-u.ac.jp}
\subjclass[2010]{16E20, 18E10, 16G70, 16G30}
\keywords{Grothendieck group; exact category; Auslander-Reiten sequence}
\begin{abstract}
 For an exact category $\mathcal{E}$, we study the Butler's condition ``AR=Ex": the relation of the Grothendieck group of $\mathcal{E}$ is generated by Auslander-Reiten conflations. Under some assumptions, we show that AR=Ex is equivalent to that $\mathcal{E}$ has finitely many indecomposables. This can be applied to functorially finite torsion(free) classes and contravariantly finite resolving subcategories of the module category of an artin algebra, and the category of Cohen-Macaulay modules over an order which is Gorenstein or has finite global dimension. Also we showed that under some weaker assumption, AR=Ex implies that the category of syzygies in $\mathcal{E}$ has finitely many indecomposables.
\end{abstract}

\maketitle

\tableofcontents

\section{Introduction}
Let $\Lambda$ be a finite-dimensional $k$-algebra over a field $k$. To the abelian category $\mod\Lambda$ of finitely generated $\Lambda$-modules, we can associate an abelian group $\KKK_0(\mod\Lambda)$ called the \emph{Grothendieck group}. This is the quotient group $\KKK_0(\mod\Lambda,0) / \Ex (\mod\Lambda)$, where $\KKK_0(\mod\Lambda,0)$ denotes the free abelian group with the basis the set of isomorphism classes $[X]$ of indecomposable objects $X \in \mod\Lambda$, and $\Ex(\mod\Lambda)$  the subgroup generated by $[X]-[Y]+[Z]$ for every exact sequence $0 \to X \to Y \to Z\to 0$.

\emph{Auslander-Reiten sequences}, or \emph{AR sequences} for short, are special kind of short exact sequences in $\mod\Lambda$, which are ``minimal'' in some sense. We denote by $\AR(\mod\Lambda)$ the subgroup of $\Ex(\mod\Lambda)$ generated by AR sequences. Then Butler and Auslander proved the following result.
\begin{theorem}[\cite{but,relations}]
Let $\Lambda$ be a finite-dimensional algebra. Then $\AR(\mod\Lambda) = \Ex(\mod\Lambda)$ holds if and only if there are only finitely many indecomposable objects in $\mod\Lambda$ up to isomorphism.
\end{theorem}

Auslander conjectured that similar results hold for a more general class of categories other than $\mod\Lambda$. In this paper, we study this conjecture in the context of Quillen's \emph{exact categories}.
For a Krull-Schmidt exact category $\EE$, the Grothendieck group $\KKK_0(\EE)= \KKK_0(\EE,0)/\Ex(\EE)$ of $\EE$ is defined in the same way. Also the notion of AR sequences is defined over $\EE$, which we call \emph{AR conflations}. Thus we have the subgroup $\AR(\EE)$ of $\Ex(\EE)$, and the aim of this paper is to study the following question.
\begin{question}\label{q}
For a Krull-Schmidt exact category $\EE$, when are the following equivalent?
\begin{enumerate}
\item $\EE$ is of finite type, that is, there exist only finitely many indecomposable objects in $\EE$ up to isomorphism.
\item $\AR(\EE) = \Ex(\EE)$ holds.
\end{enumerate}
\end{question}

Let us mention some known results on this question. For the implication (1) $\Rightarrow$ (2), we have the following quite general result by the author:
\begin{theorem}[{\cite[Theorem 3.18]{en2}, Corollary \ref{finadm}}]
  Let $R$ be a complete noetherian local ring and $\EE$ a Krull-Schmidt exact $R$-category such that $\EE(X,Y)$ is a finitely generated $R$-module for each $X,Y \in \EE$. If $\EE$ has enough projectives and is of finite type, then $\AR(\EE) = \Ex(\EE)$ holds.
\end{theorem}
This can be applied to various situations in the representation theory of noetherian algebras. We provide a proof of this in Corollary \ref{finadm}.

On the other hand, the implication (2) $\Rightarrow$ (1) is more subtle. Indeed, some counter-examples are known, e.g. \cite[Section 3]{mmp}. However, there are affirmative results for this implication in some concrete situations: $\EE = \CM \Lambda$ for an $R$-order $\Lambda$ with finite global dimension \cite[Proposition 2.3]{ar1}, and the category of modules with good filtrations over standardly stratified algebras \cite{mmp,pr}. Also some partial results are known (\cite{hir,ko}).

In this paper, we first give a sufficient condition for (1) and (2) in Question \ref{q} to be equivalent, by using a functorial method. For a Krull-Schmidt exact category $\EE$, we consider the category $\Mod\EE$ of functors $\EE^{\op} \to \Ab$. There is an important subcategory $\eff\EE$ of $\Mod\EE$, the category of \emph{effaceable} functors (Definition \ref{effdef}), which nicely reflects the exact structure of $\EE$ and is closely related to the group $\Ex(\EE)$.
Then let us consider the following conditions:
\begin{enumerate}[label={\upshape(\alph*)}]
\item $\EE$ is of finite type.
\item $\EE$ is \emph{admissible} \cite{en2}, that is, every object in $\eff\EE$ has finite length in $\Mod\EE$.
\item $\AR(\EE)=\Ex(\EE)$ holds.
\end{enumerate}
Simple modules contained in $\eff\EE$ bijectively correspond to AR conflations in $\EE$ (Proposition \ref{exsimp}). Thus the condition (b) can be regarded as a functorial analogue of (c), hence it is somewhat closer to (c) than (a) is. The relation of these conditions are summarized in Figure \ref{fig}.

Secondly, we apply this condition to concrete situations and obtain results on Question \ref{q}.
For an artin algebra, we show the following result, which extends the results \cite[Theorem 3]{mmp} and \cite[Theorem 3.6]{pr}.
\begin{theoremi}[{\upshape = Theorem \ref{artinmain}}]\label{cora}
Let $\Lambda$ be an artin algebra and $\EE$ a contravariantly finite resolving subcategory of $\mod\Lambda$. Then the following are equivalent.
\begin{enumerate}
\item $\EE$ is of finite type.
\item $\AR(\EE) = \Ex(\EE)$ holds.
\end{enumerate}
\end{theoremi}
When $R$ is not artinian, we prove the following result. This extends \cite{hir,ko} to the non-commutative case, and \cite[Proposition 2.3]{ar1} to Gorenstein orders. Also our method provides another proof of \cite[Proposition 2.3]{ar1}.
\begin{theoremi}[{\upshape = Theorem \ref{ordermain}, Corollary \ref{syzygymain}}]
Let $R$ be a complete Cohen-Macaulay local ring and $\Lambda$ an $R$-order with at most an isolated singularity. Suppose that $\Ex(\CM \Lambda) = \AR(\CM \Lambda)$ holds. Then the following hold.
\begin{enumerate}
\item $\mathsf{\Omega CM}\,\Lambda$, the category of syzygies of Cohen-Macaulay $\Lambda$-modules, is of finite type.
\item If $\Lambda$ has finite global dimension or $\Lambda$ is a Gorenstein order, then $\CM\Lambda$ is of finite type.
\end{enumerate}
\end{theoremi}

Actually, most of the results above are deduced from the following relations between (a), (b) and (c):
\begin{theoremi}[Proposition \ref{propab}, Theorem \ref{fcfl}]
  Let $\EE$ be a Krull-Schmidt exact $R$-category with a projective generator over a commutative noetherian ring $R$. Then the implications in Figure \ref{fig} hold.
\end{theoremi}
\begin{figure}[h]
\begin{tikzpicture}
 \node (a) at (-6,0)  {(a) $\EE$ is of finite type};
 \node (b) at (0,0)  {(b) $\EE$ is admissible};
 \node (c) at (5,0) {(c) $\AR(\EE) = \Ex(\EE)$};

 \draw [-implies,double equal sign distance] (a) to [bend left=10] node[auto] {\footnotesize $\un{\EE}$ is Hom-finite} (b) ;
 \draw [-implies,double equal sign distance] (b) to [bend left=10] node[auto] {\footnotesize $\un{\EE}$ has a weak cogenerator} (a) ;
 \draw [-implies,double equal sign distance] (b) to [bend left=10] (c) ;
 \draw [-implies,double equal sign distance] (c) to [bend left=10] node[auto] {\footnotesize $\EE$ satisfies (CF)} (b) ;
\end{tikzpicture}
\caption{Summary of Section \ref{sec3}}\label{fig}
\end{figure}
Here we introduce the condition \emph{(CF)} on exact categories concerning the Grothendieck groups (see Definition \ref{fcdef}).

The paper is organized as follows.
In Section \ref{sec2}, we introduce basic definitions and collect some results about Grothendieck group and effaceable functors.
In Section \ref{sec3}, we introduce some conditions on $\EE$ and prove the relations indicated in Figure \ref{fig}.
In Section \ref{sec4}, we verify that the conditions are satisfied in some concrete cases.
In Section \ref{sec5}, we study the finiteness of the category of syzygies.
In Appendix A, we provide some interesting properties of the category of effaceable functors.

\subsection{Notation and conventions}
Throughout this paper, we assume that \emph{all categories and functors are additive} and \emph{all subcategories are additive, full and closed under isomorphisms}. For an object $G$ of an additive category $\EE$, we denote by $\add G$ the category consisting of all direct summands of finite direct sums of $G$. We say that an additive category is \emph{of finite type} if there exists an object $G \in \EE$ such that $\EE = \add G$ holds.

For a Krull-Schmidt category $\EE$, we denote by $\ind\EE$ the set of isomorphism classes of indecomposable objects in $\EE$. We refer the reader to \cite{krause} for the basics of Krull-Schmidt categories.
We say that an additive category $\EE$ is an \emph{idempotent complete} if every idempotent morphism splits. It is well-known that a Krull-Schmidt category is idempotent complete.

As for exact categories, we use the terminologies \emph{inflations}, \emph{deflations} and \emph{conflations}. We refer the reader to \cite{buhler} for the basics of exact categories.
We say that an exact category $\EE$ has a \emph{projective generator $P$} if $\EE$ has enough projectives $\add P$.

\section{Preliminaries}\label{sec2}
First we introduce the basic definition about Auslander-Reiten theory in a Krull-Schmidt category. Let $\EE$ be a Krull-Schmidt category and $\JJ$ its Jacobson radical. A morphism $g:Y\to Z$ in $\EE$ is called \emph{right almost split} if $Z$ is indecomposable, $g$ is in $\JJ$ and any morphism $h:W \to Z$ in $\JJ$ factors through $g$. Dually we define \emph{left almost split}.

For a Krull-Schmidt exact category $\EE$, we say that a conflation $0 \to X \xrightarrow{f} Y \xrightarrow{g} Z \to 0$ in $\EE$ is an \emph{AR conflation} if $f$ is left almost split and $g$ is right almost split. We say that an indecomposable object $Z$ in $\EE$ \emph{admits a right AR conflation} if there exists an AR conflation ending at $Z$. We say that $\EE$ \emph{has right AR conflations} if every indecomposable non-projective object admits a right AR conflation. Left AR conflations are defined dually, and we say that $\EE$ \emph{has AR conflations} if it has both right and left AR conflations.

Next we introduce some notation concerning the Grothendieck group. For a Krull-Schmidt exact category $\EE$, let $\KKK_0(\EE,0)$ be a free abelian group $\bigoplus_{[X] \in \ind \EE} \mathbb{Z}\cdot[X]$ generated by the set $\ind \EE$ of isomorphism classes of indecomposable objects in $\EE$, which we identify with the Grothendieck group of $\EE$ with the split exact structure, that is, the exact structure where all conflations are split exact.
We denote by $\Ex(\EE)$ the subgroup of $\KKK_0(\EE,0)$ generated by
\[
\{ [X] - [Y] + [Z] \text{ $|$ there exists a conflation $X \infl Y \defl Z$ in $\EE$} \}.
\]
We call the quotient group $\KKK_0(\EE) :=\KKK_0(\EE,0) / \Ex(\EE)$ the \emph{Grothendieck group} of $\EE$.
We denote by $\AR(\EE)$ the subgroup of $\Ex(\EE)$ generated by
\[
\{ [X] - [Y] + [Z] \text{ $|$ there exists an AR conflation $X \infl Y \defl Z$ in $\EE$} \}.
\]

Throughout this paper, we make use of functor categorical arguments. Let us recall the related concepts. For an additive category $\EE$, a \emph{right $\EE$-module} $M$ is a contravariant additive functor $M: \EE^{\op} \to \Ab$ from $\EE$ to the category of abelian groups $\Ab$. We denote by $\Mod\EE$ the category of right $\EE$-modules, and morphisms are natural transformations between them.
Then the category $\Mod \EE$ is a Grothendieck abelian category with enough projectives, and projective objects are precisely direct summands of (possibly infinite) direct sums of representable functors.

We denote by $\mod \EE$ the category of finitely presented $\EE$-modules, that is, the modules $M$ such that there exists an exact sequence $\EE(-,X) \to \EE(-,Y) \to M \to 0$ for some $X,Y$ in $\EE$. This category $\mod\EE$ is not necessarily an abelian category, but $\mod\EE$ is closed under extensions in $\Mod\EE$ by the horseshoe lemma. Thereby we always regard $\mod\EE$ as an exact category, which has enough projectives in the sense of exact category. Moreover, if $\EE$ is idempotent complete, then projectives in $\mod\EE$ are precisely representable functors.

Now let us introduce the notion of \emph{effaceability} of $\EE$-modules, which plays an essential role throughout this paper. This notion was originally introduced by Grothendieck \cite[p. 141]{gro}.
\begin{definition}\label{effdef}
Let $\EE$ be an exact category. A right $\EE$-module $M$ is called \emph{effaceable} if there exists a deflation $f:Y \defl Z$ in $\EE$ such that $\EE(-,Y) \xrightarrow{\EE(-,f)} \EE(-,Z) \to M \to 0$ is exact. We denote by $\eff \EE$ the subcategory of $\mod\EE$ consisting of effaceable functors.
\end{definition}
The category $\eff\EE$ plays a quite essential role to study exact categories in a functorial method.
This definition for exact categories is due to \cite{fiorot}, and the category $\eff\EE$ plays an important role in \cite{en2} to classify exact structures on a given additive category. For the compatibility of Grothendieck's effaceability, see Proposition \ref{grodef}.

In the rest of this section, we collect basic properties about functor categories and effaceable functors which we need in the sequel.
\begin{proposition}\label{kssimp}
Let $\EE$ be a Krull-Schmidt category and $\JJ$ its Jacobson radical. Put $S_Z := \EE(-,Z)/\JJ(-,Z)$ for $Z \in \EE$. Then the following hold.
\begin{enumerate}
\item The map $Z \mapsto S_Z$ gives a bijection between $\ind\EE$ and the set of isomorphism classes of simple $\EE$-modules.
\item A morphism $g:Y \to Z$ in $\EE$ is right almost split if and only if $Z$ is indecomposable and $\EE(-,Y) \xrightarrow{\EE(-,g)} \EE(-,Z) \to S_Z \to 0$ is exact.
\end{enumerate}
\end{proposition}
\begin{proof}
It is well-known, see \cite{artin2} for example.
\end{proof}
In particular, simple $\EE$-modules contained in $\mod\EE$ bijectively correspond to indecomposable objects in $\EE$ which admit right almost split maps.
Moreover if $\EE$ is an exact category, then one can show more on $\eff\EE$ as follows.
\begin{proposition}\label{exsimp}
Let $\EE$ be a Krull-Schmidt exact category and $Z$ an indecomposable object in $\EE$. Then the following are equivalent.
\begin{enumerate}
\item $Z$ admits a right AR conflation.
\item $S_Z$ belongs to $\eff\EE$.
\item $S_Z$ is finitely presented and $Z$ is non-projective.
\end{enumerate}
In particular, simple $\EE$-modules contained in $\eff\EE$ bijectively correspond to indecomposables in $\EE$ which admit right AR conflations.
\end{proposition}
\begin{proof}
(1) $\Rightarrow$ (2):
We have a deflation $g:Y\defl Z$ which is right almost split. Thus $S_Z \iso \coker \EE(-,g)$ belongs to $\eff\EE$.

(2) $\Rightarrow$ (3):
Since $S_Z$ is in $\eff\EE$, it is finitely presented. By \cite[Propositions 2.11]{en2}, we have that $Z$ is non-projective since $S_Z(Z) \neq 0$ holds.

(3) $\Rightarrow$ (1):
This is precisely \cite[Proposition A.3]{en2}.
\end{proof}

The following says that the category $\eff\EE$ has nice properties and controls the exact structure of $\EE$. Here we say that an $\EE$-module $M$ is \emph{finitely generated} if there exists a surjection from a representable functor onto $M$.
\begin{proposition}\label{serre}
Let $\EE$ be an idempotent complete exact category. Then the following holds.
\begin{enumerate}
\item Suppose that there exists an exact sequence $0 \to M_1 \to M \to M_2 \to 0$ in $\Mod\EE$ and that $M_1$ is finitely generated. Then $M$ is in $\eff\EE$ if and only if both $M_1$ and $M_2$ are in $\eff\EE$.
\item A complex $X \xrightarrow{f} Y \xrightarrow{g} Z$ in $\EE$ is a conflation if and only if there exists an exact sequence
\[
0 \to \EE(-,X) \xrightarrow{\EE(-,f)} \EE(-,Y) \xrightarrow{\EE(-,g)} \EE(-,Z) \to M \to 0
\]
in $\Mod\EE$ with $M$ in $\eff\EE$.
\end{enumerate}
\end{proposition}
\begin{proof}
We refer the reader to \cite[Proposition 2.10]{en2} for the proof of (1), and \cite[Theorem 2.7]{en2} or the proof of Lemma \ref{grodef} for (2).
\end{proof}

The category $\eff\EE$ enjoys further nice properties, e.g. being abelian, and we refer the interested reader to Appendix A.

If $\EE$ has enough projectives, then one has a simpler description of the category $\eff\EE$. Let $\EE$ be an exact category with enough projectives and denote by $\PP$ the subcategory of $\EE$ consisting of projective objects. We define the \emph{stable category} $\un{\EE}$ by the ideal quotient $\EE/[\PP]$. Then $\eff\EE$ can be naturally identified with the module category of $\un{\EE}$, which we omit the proof.
\begin{lemma}[{\cite[Lemma 2.13]{en2}}]\label{effstable}
Let $\EE$ be an exact category with enough projectives $\PP$. Then the natural restriction functor $\Mod\un{\EE} \to \Mod\EE$ induces an equivalence $\mod\un{\EE} \equi \eff \EE$.
\end{lemma}

\section{General conditions for (a), (b) and (c) to be equivalent}\label{sec3}
In this section, we will provide some general results on the three conditions: (a) $\EE$ is of finite type, (b) $\EE$ is admissible and (c) $\AR(\EE) = \Ex(\EE)$. Here we say that $\EE$ is \emph{admissible} if every object in $\eff\EE$ has finite length in $\Mod\EE$ (\cite{en2}). See Proposition \ref{onaji} for the equivalent condition of the admissibility. Then our results in this section can be summarized in Figure \ref{fig} in the introduction.

To begin with, let us introduce some terminologies on $R$-categories needed in this paper. Let $R$ be a commutative noetherian ring and $\EE$ an $R$-category. We say that an $R$-category $\EE$ is \emph{Hom-noetherian} (resp. \emph{Hom-finite}) if $\EE(X,Y)$ is finitely generated (resp. has finite length) as an $R$-module for each $X,Y$ in $\EE$.

To deal with admissibility, it is convenient to introduce the following criterion for deciding whether the functor has finite length or not.
Let $\EE$ be a Krull-Schmidt category and $M \in \Mod\EE$ a right $\EE$-module. Then the support $\supp M$ of $M$ is defined by
\[
\supp M := \{ X \in \ind\EE \ |\  M(X) \neq 0\}.
\]
The Hom-finiteness ensures that finiteness of modules can be checked only by considering their support:
\begin{lemma}\label{fl}
Let $\EE$ be a Krull-Schmidt category and $M \in \mod\EE$ an $\EE$-module. If $M$ has finite length in $\Mod\EE$, then $\supp M$ is a finite set. The converse holds if $\EE$ is Hom-finite $R$-category.
\end{lemma}
\begin{proof}
See, for example, \cite[Theorem 2.12]{artin2}.
\end{proof}

\subsection{(a) Representation-finiteness and (b) admissibility}\label{secab}
First, we consider the relation between: (a) $\EE$ is of finite type, and (b) $\EE$ is admissible.

\begin{definition}
Let $\CC$ be an additive category. We say that an object $C \in \CC$ is a \emph{weak cogenerator} if $\CC(X,C)=0$ implies that $X$ is a zero object for every $X \in \CC$.
\end{definition}
If we assume the existence of a weak cogenerator of the stable category, then (a) is almost equivalent to (b).
\begin{proposition}\label{propab}
Let $\EE$ be a Krull-Schmidt exact category with a projective generator. Consider the following conditions:
\begin{enumerate}
\item $\EE$ is admissible and $\un{\EE}$ has a weak cogenerator.
\item $\EE$ is of finite type.
\end{enumerate}
Then {\upshape (1)} $\Rightarrow$ {\upshape (2)} holds. The converse holds if $\un{\EE}$ is a Hom-finite $R$-category for a commutative noetherian ring $R$.
\end{proposition}
\begin{proof}
(1) $\Rightarrow$ (2):
Since $\EE$ has enough projectives, we may identify $\eff\EE$ with $\mod\un{\EE}$ by Lemma \ref{effstable}. Let $\un{X}$ be a weak cogenerator of $\un{\EE}$. By the assumption, we have that $\un{\EE}(-,\un{X})$ has finite length. This implies that $\supp \un{\EE}(-,\un{X})$ is finite by Lemma \ref{fl}. Since every indecomposable object $\un{W}$ in $\un{\EE}$ should satisfy  $\un{\EE}(\un{W},\un{X})\neq 0$, it follows that $\un{\EE}$ is of finite type. Now $\EE$ itself is of finite type because we have a natural identification $\ind\EE =\ind\un{\EE} \sqcup \ind (\add P)$.

(2) $\Rightarrow$ (1):
Suppose that $\EE$ is of finite type and $\un{\EE}$ is Hom-finite over $R$. Then obviously the direct sum of all indecomposables in $\un{\EE}$ is a weak cogenerator of $\un{\EE}$. Moreover, since $\un{\EE}$ is Hom-finite and $\ind\un{\EE}$ is finite, every object in $\eff\EE = \mod\un{\EE}$ has finite length by Lemma \ref{fl}.
\end{proof}

\subsection{(b) Admissibility and (c) AR=Ex}\label{secbc}

Now let us investigate the relation between the condition (b) and (c).
First we show that if (b) $\EE$ is admissible, then (c) $\AR(\EE) =\Ex(\EE)$ holds. Next we will introduce the \emph{condition (CF)} and show that the converse implication holds under it.

\begin{proposition}\label{part1}
Let $\EE$ be a Krull-Schmidt exact category. Suppose that $\EE$ is admissible. Then $\Ex(\EE) = \AR(\EE)$ holds.
\end{proposition}
\begin{proof}
This is proved in the similar way as \cite[Theorem 3.17]{en2}. We provide a proof here for the convenience of the reader.

Let $0 \to X_1 \xrightarrow{f_1} Y_1 \xrightarrow{g_1} Z_1 \to 0$ be a conflation in $\EE$ and we have to show $[X_1] - [Y_1] + [Z_1] \in \AR(\EE)$.
Put $M:= \coker \EE(-,g_1)$ in $\Mod\EE$, then $M$ is in $\eff\EE$ by the definition.
Suppose that there exists another conflation $0 \to X_2 \xrightarrow{f_2} Y_2 \xrightarrow{g_2} Z_2 \to 0$ in $\EE$ such that $M \iso \coker \EE(-,g_2)$. Then we have exact sequences
\[
0 \to {\EE(-,X_i)} \xrightarrow{\EE(-,f_i)} \EE(-,Y_i) \xrightarrow{\EE(-,g_i)} \EE(-,Z_i) \to M \to 0
\]
in $\Mod\EE$ for $i=1,2$. Thus Schanuel's lemma shows that $\EE(-,X_1) \oplus \EE(-,Y_2) \oplus \EE(-,Z_1) \iso \EE(-,X_2) \oplus \EE(-,Y_1) \oplus \EE(-,Z_2)$ in $\Mod\EE$, hence $X_1 \oplus Y_2 \oplus Z_1 \iso X_2 \oplus Y_1 \oplus Z_2$
 in $\EE$ by the Yoneda lemma. This implies that $[X_1] - [Y_1] + [Z_1] = [X_2] - [Y_2] + [Z_2]$ in $\KKK_0(\EE,0)$. Thus it suffices to show the following claim.

\emph{Claim: For any $M \in \eff \EE$, there exists at least one exact sequence in $\Mod\EE$
\[
0 \to \EE(-,X) \xrightarrow{\EE(-,f)} \EE(-,Y) \xrightarrow{\EE(-,g)} \EE(-,Z) \to M \to 0
\] with $[X]-[Y]+[Z] \in \AR(\EE)$.} Note that such a complex $0 \to X \to Y \to Z \to 0$ is automatically a conflation by Proposition \ref{serre}(2).

We will show this claim by induction on $l(M)$, the length of $M$ as an $\EE$-module.
Suppose that $l(M) = 1$, that is, $M$ is a simple $\EE$-module. We may assume that $M=S_Z$ for an indecomposable object $Z \in \EE$ by Proposition \ref{kssimp}, and there exists an AR conflation $0 \to X \xrightarrow{f} Y \xrightarrow{g} Z \to 0$ in $\EE$ by Proposition \ref{exsimp}. This gives the desired projective resolution $0 \to \EE(-,X) \xrightarrow{\EE(-,f)} \EE(-,Y) \xrightarrow{\EE(-,g)} \EE(-,Z) \to S_Z \to 0$ of $S_Z$, and $[X] - [Y] + [Z] \in \AR(\EE)$ holds.

Now suppose that $l(M) > 1$. Take a simple $\EE$-submodule $M_1$ of $M$. Then we have an exact sequence $0 \to M_1 \to M \to M_2 \to 0$ in $\Mod\EE$. Since $M_1$ is finitely generated, Proposition \ref{serre}(1) shows that $M_1$ and $M_2$ are both in $\eff\EE$. By induction hypothesis, we have an exact sequence $0 \to \EE(-,X_i) \xrightarrow{\EE(-,f_i)} \EE(-,Y_i) \xrightarrow{\EE(-,g_i)} \EE(-,Z_i) \to M_i \to 0$ with $[X_i] -[Y_i] + [Z_i] \in \AR(\EE)$ for each $i=1,2$.
By the horseshoe lemma, we obtain a projective resolution
\[
0 \to \EE(-,X_1 \oplus X_2) \xrightarrow{\EE(-,f)} \EE(-,Y_1\oplus Y_2) \xrightarrow{\EE(-,g)} \EE(-,Z_1 \oplus Z_2) \to M \to 0.
\]
Then we have
\[
[X_1\oplus X_2 ] -[Y_1\oplus Y_2] + [Z_1 \oplus Z_2]
=\sum_{i=1,2}([X_i] -[Y_i] + [Z_i]) \in \AR(\EE).
\]
Thus the claim follows.
\end{proof}

Next let us investigate when the converse of Proposition \ref{part1} holds. The author does not know whether this always holds, but we show that this is the case for particular cases. To this purpose, let us introduce the condition \emph{(CF)}, which stands for the \emph{Conservation of Finiteness}.

\begin{definition}\label{fcdef}
For an exact category $\EE$, we say that $\EE$ satisfies \emph{(CF)} if it satisfies the following:
\begin{itemize}
\item[(CF)] Let $0 \to X \xrightarrow{f_i} Y \xrightarrow{g_i} Z \to 0$ be two conflations in $\EE$ for $i=1,2$ and put $M_i := \coker \EE(-,g_i) \in \eff\EE$. If $M_1$ has finite length in $\Mod\EE$, then so does $M_2$.
\end{itemize}
\end{definition}
For later purposes, the following equivalent condition is more useful.
\begin{lemma}
For an exact category $\EE$, the condition {\upshape (CF)} is equivalent to the following condition.
\begin{itemize}
\item[\upshape (CF$'$)] Let $0 \to X_i \xrightarrow{f_i} Y_i \xrightarrow{g_i} Z_i \to 0$ be conflations in $\EE$ for $i=1,2$ and put $M_i := \coker \EE(-,g_i)$ in $\eff\EE$.
Suppose that $[X_1]-[Y_1]+[Z_1]=[X_2]-[Y_2]+[Z_2]$ holds in $\KKK_0(\EE,0)$. If $M_1$ has finite length in $\Mod\EE$, then so does $M_2$.
\end{itemize}
\end{lemma}
\begin{proof}
Obviously it suffices to show that (CF) implies (CF$'$). Since $[X_1]-[Y_1]+[Z_1]=[X_2]-[Y_2]+[Z_2]$ holds in $\KKK_0(\EE,0)$, we have that $X_2 \oplus Y_1 \oplus Z_2 \iso X_1 \oplus Y_2 \oplus Z_1$ holds in $\EE$ by the Krull-Schmidtness of $\EE$. Now consider the following two complexes:
\begin{align*}
0 \to X_1 \oplus X_2 \xrightarrow{
\begin{sbmatrix} 0 & 1 \\ f_1 & 0 \\ 0 & 0 \end{sbmatrix}
} X_2 \oplus Y_1 \oplus Z_2 \xrightarrow{
\begin{sbmatrix}
0 & g_1 & 0 \\
0 & 0 & 1
\end{sbmatrix}
} Z_1 \oplus Z_2 \to 0,\\
0 \to X_1 \oplus X_2 \xrightarrow{
\begin{sbmatrix} 1 & 0 \\ 0 & f_2 \\ 0 & 0 \end{sbmatrix}
} X_1 \oplus Y_2 \oplus Z_1 \xrightarrow{
\begin{sbmatrix}
0 & 0 & 1 \\
0 & g_2 & 0
\end{sbmatrix}
} Z_1 \oplus Z_2 \to 0.
\end{align*}
These are conflations since they are direct sums of conflations and split exact sequences. Applying (CF) to them, it is easy to check that (CF$'$) holds.
\end{proof}

Now we can state the following main result in this section.
\begin{theorem}\label{fcfl}
Let $\EE$ be a Krull-Schmidt exact category. Then the following are equivalent.
\begin{enumerate}
\item $\EE$ is admissible.
\item $\Ex(\EE) = \AR(\EE)$ holds and $\EE$ satisfies {\upshape (CF)}.
\item $\Ex(\EE)\otimes_\Z \Q = \AR(\EE) \otimes_\Z \Q$ holds and $\EE$ satisfies {\upshape (CF)}.
\end{enumerate}
\end{theorem}
\begin{proof}
(1) $\Rightarrow$ (2):
If every object in $\eff\EE$ has finite length, then we have $\Ex(\EE) = \AR(\EE)$ by Proposition \ref{part1}, and the condition (CF) is trivially satisfied.

(2) $\Rightarrow$ (3): Obvious.

(3) $\Rightarrow$ (1):
Let $M$ be in $\eff\EE$ and take a conflation $0 \to X \xrightarrow{f} Y \xrightarrow{g} Z \to 0$ in $\EE$ such that $0 \to \EE(-,X) \xrightarrow{\EE(-,f)} \EE(-,Y) \xrightarrow{\EE(-,g)} \EE(-,Z) \to M \to 0$ is exact.
By $\AR(\EE) \otimes_\Z \Q = \Ex(\EE) \otimes_\Z \Q$, there exist a positive integer $a$, integer $b_i$ and an AR conflation $0 \to X_i \to Y_i \xrightarrow{g_i} Z_i \to 0$ for each $i$ such that we have an equality in $\KKK_0(\EE,0)$:
\[
a([X]-[Y]+[Z]) = \sum_{i=1}^n b_i ([X_i] - [Y_i] + [Z_i]).
\]
We may assume that $b_i > 0$ for $i\leq m$ and $-c_i:= b_i <0$ for $i > m$. Thus we have an equality
\begin{equation}\label{eq}
a([X]-[Y]+[Z]) + \sum_{i=m+1}^n c_i([X_i]-[Y_i]+[Z_i]) = \sum_{i=1}^m b_i ([X_i] - [Y_i] + [Z_i])
\end{equation}
in $\KKK_0(\EE,0)$ such that all the coefficients are positive integers. Thus each side of the equation (\ref{eq}) comes from the single conflation in $\EE$.
Put $S_i:= \coker \EE(-,g_i) \in \mod\EE$ for each $i$, which is simple $\EE$-module contained in $\eff\EE$. Then the left hand side in (\ref{eq}) corresponds to $M^a \oplus \bigoplus_{i=m+1}^n S_i^{c_i}$ in $\eff\EE$ and the right hand side corresponds to $\bigoplus_{i=1}^m S_i^{b_i}$ in $\eff\EE$.
Since the latter has finite length, the condition (CF$'$) implies that so does the former. Thus $M$ has finite length.
\end{proof}

\section{Applications}\label{sec4}
In Section \ref{sec3}, we have shown the equivalence of (a), (b) and (c) under three technical assumptions: Hom-finiteness of $\un{\EE}$, the existence of a weak cogenerator of $\un{\EE}$, and the condition (CF) (see Figure \ref{fig} for a summary of these results). In this section, we show that these assumptions are satisfied for some concrete cases.

Main applications we have in mind are the representation theory of noetherian algebras, where the base ring is not necessarily artinian. For the convenience of the reader, we recall some related definitions.
\begin{definition}
Let $R$ be a noetherian local ring and $\Lambda$ an $R$-algebra.
\begin{enumerate}
\item $\Lambda$ is called a \emph{noetherian $R$-algebra} if $\Lambda$ is finitely generated as an $R$-module. If in addition $R$ is artinian, then we say that $\Lambda$ is an \emph{artin $R$-algebra}.
\item Suppose that $R$ is Cohen-Macaulay. For a noetherian $R$-algebra, we denote by $\CM\Lambda$ the category of finitely generated $\Lambda$-modules which are maximal Cohen-Macaulay as $R$-modules.
\item A noetherian $R$-algebra $\Lambda$ is called an \emph{$R$-order} if $\Lambda \in \CM\Lambda$ holds.
\item We say that an $R$-order $\Lambda$ \emph{has at most an isolated singularity} if $\gl \Lambda_p = \height p$ holds for every non-maximal prime $p \in \Spec R$.
\item An $R$-order is called a \emph{Gorenstein} order if $\CM\Lambda$ is a Frobenius exact category.
\end{enumerate}
\end{definition}
Let $R$ be a complete Cohen-Macaulay local ring and $\Lambda$ an $R$-order. Then $\CM\Lambda$ is closed under extensions in $\mod\Lambda$, thus is an exact category with a projective generator $\Lambda$. Moreover, $\CM\Lambda$ has AR conflations if and only if $\Lambda$ has at most an isolated singularity \cite{isolated}.

\subsection{Hom-finiteness of the stable category}
First we consider the Hom-finiteness of $\un{\EE}$. Recall that (a) $\Rightarrow$ (b) holds if $\un{\EE}$ is Hom-finite (Proposition \ref{propab}).
This condition is trivial if $R$ is artinian, hence we obtain the following.
\begin{corollary}
Let $\EE$ be a Hom-finite idempotent complete exact $R$-category over a commutative artinian ring $R$. Suppose that $\EE$ is of finite type. Then $\EE$ is admissible and $\AR(\EE) = \Ex(\EE)$ holds.
\end{corollary}
For an artin algebra $\Lambda$, we can apply this result to any subcategory $\EE$ of $\mod\Lambda$ which is closed under extensions and direct summands.

If $R$ is not artinian, we can use the following fact on Hom-finiteness.
\begin{lemma}[{\cite[Proposition A.5]{en2}}]\label{homfin}
Let $R$ be a complete noetherian local ring and $\EE$ a Hom-noetherian idempotent complete exact $R$-category. Suppose that $\EE$ has enough projectives and consider the following conditions:
\begin{enumerate}
\item $\EE$ is of finite type.
\item $\EE$ has AR conflations.
\item $\un{\EE}$ is Hom-finite over $R$.
\end{enumerate}
Then the implication {\upshape (1)} $\Rightarrow$ {\upshape (2)} $\Rightarrow$ {\upshape (3)} holds.
\end{lemma}
Thus we immediately obtain the following.
\begin{corollary}[{\cite[Corollary 3.18]{en2}}]\label{finadm}
Let $R$ be a complete noetherian local ring and $\EE$ a Hom-noetherian idempotent complete exact $R$-category. Suppose that $\EE$ has enough projectives and $\EE$ is of finite type. Then $\EE$ is admissible and $\AR(\EE) = \Ex(\EE)$ holds.
\end{corollary}
Note that the assumption on $\EE$ here is rather mild. In particular, this corollary can be applied to $\EE:= \CM\Lambda$ for an $R$-order $\Lambda$ which is \emph{CM-finite}, that is, $\CM\Lambda$ is of finite type. Thus this provides a further generalization of \cite[Proposition 2.2]{relations}.

\subsection{Existence of a weak cogenerator of the stable category}
For the implication (b) $\Rightarrow$ (a), we need a weak cogenerator of $\un{\EE}$ (Proposition \ref{propab}). To this purpose, let us recall the notions of functorially finiteness and resolving subcategories.
\begin{definition}\label{functfin}
Let $\AA$ be an additive category and $\EE$ an additive subcategory of $\AA$.
\begin{enumerate}
 \item A morphism $f:E_X \to X$ in $\AA$ is said to be a \emph{right $\EE$-approximation} if $E_X$ is in $\EE$ and every morphism $E \to X$ with $E\in\DD$ factors through $f$.
 \item $\EE$ is said to be \emph{contravariantly finite} if every object in $\AA$ has a right $\EE$-approximation.
\end{enumerate}
A \emph{left $\EE$-approximation} and \emph{covariantly finiteness} are defined dually. We say that $\EE$ is \emph{functorially finite} if it is both contravariantly and covariantly finite.
\end{definition}

\begin{definition}
Let $\Lambda$ be a noetherian ring and $\EE$ a subcategory of $\mod\Lambda$. We say that $\EE$ is a \emph{resolving} subcategory of $\mod\Lambda$ if it satisfies the following conditions:
\begin{enumerate}
 \item $\EE$ contains $\Lambda$.
 \item $\EE$ is closed under extensions, that is, for each exact sequence $0 \to X \to Y \to Z \to 0$, if both $X$ and $Z$ are in $\EE$, then so is $Y$.
 \item $\EE$ is closed under kernels of surjections, that is, for each exact sequence $0 \to X \to Y \to Z \to 0$, if both $Y$ and $Z$ are in $\EE$, then so is $X$.
 \item $\EE$ is closed under summands.
 \end{enumerate}
\end{definition}
Since each resolving subcategory is an extension-closed subcategory of an abelian category, it has the natural exact structure. Thereby we always regard it as an exact category.

The following result provides a rich source of exact categories such that its stable category has a weak cogenerator.
\begin{proposition}\label{weakcogen}
Let $\Lambda$ be a semiperfect noetherian ring and $\EE$ a contravariantly finite resolving subcategory of $\mod\Lambda$. Then $\un{\EE}$ has a weak cogenerator, which can be chosen as a right $\EE$-approximation of $\Lambda / \rad \Lambda$.
\end{proposition}
\begin{proof}
Since $\EE$ is resolving in $\mod\Lambda$, it is straightforward to see that $\EE$ has a projective generator $\Lambda$.
Let us construct a weak cogenerator $\un{X}$ of $\un{\EE}$ (cf. \cite[Lemma 2.4]{ar1}).
Since $\Lambda$ is semiperfect, $\Lambda / \rad \Lambda$ is a direct sum of all simple $\Lambda$-modules up to multiplicity. Take a right $\EE$-approximation $f:X \to \Lambda / \rad \Lambda$ of $\Lambda / \rad \Lambda$, and we claim that $\un{X}$ is a weak cogenerator of $\un{\EE}$.

Suppose that $\un{\EE}(\un{W},\un{X})=0$. Then it follows from this that every morphism from $W$ to finitely generated semisimple modules factors through some projective module. We will show that $W$ is projective, that is, $\un{W}=0$ in $\un{\EE}$.

For a Jacobson radical $\rad W$ of $W$, let $\pi:W \defl W/\rad W$ be a natural projection and take a projective cover $p:P \defl W/\rad W$ (this is possible since $\Lambda$ is semiperfect). It follows that there exists a projective cover $\varphi:P \defl W$ such that the following diagram commutes:

\[
\begin{tikzcd}
P \arrow[r,"\varphi", two heads] \arrow[rd, "p"', two heads] & W \arrow[d,"\pi", two heads]\arrow["\psi", r, dashed] & P \arrow[ld, "p", two heads] \\
 & W/\rad W
\end{tikzcd}
\]
Since $W/\rad W$ is semisimple, $\pi$ must factor through some projective module. Thus there exists $\psi:W \to P$ which makes the above diagram commute. On the other hand, since $p$ is right minimal, $\psi \circ \varphi$ must be an isomorphism. It follows that $\varphi$ is an isomorphism, thus $W$ is projective.
\end{proof}

Now one immediately obtains the following result about the equivalence of (a) and (b) for a contravariantly finite resolving subcategories over noetherian algebras. We will treat the case of artin algebras later (see Theorem \ref{artinmain}) since one can prove more.
\begin{corollary}\label{cmab}
Let $R$ be a complete noetherian local ring, $\Lambda$ a noetherian $R$-algebra and $\EE$ a contravariantly finite resolving subcategory of $\mod\Lambda$. Then $\EE$ is of finite type if and only if $\EE$ is admissible.
\end{corollary}
\begin{proof}
There exists a weak cogenerator of $\un{\EE}$ by Proposition \ref{weakcogen} since $\Lambda$ is semiperfect. Thus $\EE$ is of finite type if $\EE$ is admissible by Proposition \ref{propab}.
The other implication follows from Corollary \ref{finadm}.
\end{proof}

\begin{example}\label{cmcontfin}
Let $R$ be a complete Cohen-Macaulay local ring and $\Lambda$ an $R$-order. Then it is well-known that $\CM\Lambda$ is a contravariantly finite resolving subcategory of $\mod\Lambda$ (e.g. \cite{ab}). In particular, $\un{\CM}\,\Lambda$ has a weak cogenerator by Proposition \ref{weakcogen} and Corollary \ref{cmab} applies to $\CM\Lambda$.
Thus Corollary \ref{cmab} generalizes the result \cite[Lemma 2.4 (b)]{relations} on the category $\CM\Lambda$.
\end{example}

\subsection{On the condition (CF)}
Recall that in Section \ref{secbc}, we have investigated the relation between (b) admissibility and (c) AR=Ex, and Theorem \ref{fcfl} shows that the condition (CF) ensures that (b) and (c) are equivalent. Now let us give some actual examples where $\EE$ satisfies (CF).
\emph{In this subsection, we fix a commutative noetherian ring $R$}.

First we deal with exact categories which are Hom-finite.
\begin{proposition}\label{homfinfc}
Let $\EE$ be a Hom-finite exact $R$-category. Then $\EE$ satisfies {\upshape (CF)}.
\end{proposition}
\begin{proof}
Let $0 \to X \xrightarrow{f_i} Y \xrightarrow{g_i} Z \to 0$ be conflations in $\EE$ for $i=1,2$, and put $M_i := \coker \EE(-,g_i) \in \eff\EE$. Then we have that $[M_1]= [\EE(-,X)] - [\EE(-,Y)] + [\EE(-,Z)] = [M_2]$ holds in $\KKK_0(\mod\EE)$. We will show that if $M_1$ has finite length, then so does $M_2$. It suffices to show that $\supp M_2$ is finite by Lemma \ref{fl}.

Let $X$ be an indecomposable object in $\EE$. For each $F$ in $\mod\EE$, the assignment $F \mapsto \length_R F(X)$ makes sense since $\EE$ is Hom-finite. It clearly extends to the group homomorphism $\chi_X: \KKK_0(\mod\EE) \to \Z$. Now  we have that $X\in\supp M_1 \Leftrightarrow \chi_X [M_1] \neq 0 \Leftrightarrow \chi_X [M_2] \neq 0 \Leftrightarrow X\in\supp M_2$ by $[M_1] = [M_2]$. Thus $\supp M_1 = \supp M_2$ holds, and this is finite since $M_1$ has finite length.
\end{proof}

Now we immediately obtain the following general result on artin algebras.
\begin{theorem}\label{artinmain}
Let $\Lambda$ be an artin algebra and $\EE$ a contravariantly finite resolving subcategory of $\mod\Lambda$. Then the following are equivalent.
\begin{enumerate}
\item $\EE$ is of finite type.
\item $\EE$ is admissible.
\item $\AR(\EE) = \Ex(\EE)$ holds.
\item $\AR(\EE) \otimes_\Z \Q = \Ex(\EE) \otimes_\Z \Q$ holds.
\end{enumerate}
\end{theorem}
\begin{proof}
Since $\Lambda$ is artin algebra, $\Lambda$ is semiperfect and $\un{\EE}$ is Hom-finite. Thus Propositions \ref{propab} and \ref{weakcogen} imply that (1) and (2) are equivalent. Furthermore, since $\EE$ is Hom-finite, $\EE$ satisfies (CF) by Proposition \ref{homfinfc}. Thus the other conditions are also equivalent by Theorem \ref{fcfl}.
\end{proof}

Contravariantly finite resolving subcategories are closely related to so called \emph{cotilting modules} by the famous result of \cite{applications}. For the convenience of the reader, we explain their result here.
\begin{definition}
Let $\Lambda$ be an artin algebra and $U$ a $\Lambda$-module in $\mod\Lambda$. Then $U$ is called a \emph{cotilting $\Lambda$-module} if it satisfies the following conditions:
\begin{enumerate}
\item the injective dimension of $U$ is finite.
\item $\Ext_\Lambda^{>0}(U,U)=0$ holds.
\item There exists an exact sequence $0 \to U_n \to \cdots \to U_0 \to D \Lambda \to 0$ of $\Lambda$-modules with $U_i \in \add U$ for each $i$, where $D$ denotes the standard duality $D: \mod \Lambda^{\op} \to \mod\Lambda$.
\end{enumerate}
\end{definition}
For a module $U$ in $\mod\Lambda$, we denote by $^\perp U$ the full subcategory of $\mod\Lambda$ consisting of modules $X$ satisfying $\Ext^{>0}_\Lambda(X,U)=0$.
\begin{proposition}[{\cite[Theorem 5.5]{applications}}]\label{cotiltcont}
Let $\Lambda$ be an artin algebra and $U$ a cotilting $\Lambda$-module. Then $^\perp U$ is a contravariantly finite resolving subcategory of $\mod\Lambda$.
\end{proposition}
Thus we can apply Theorem \ref{artinmain} to $\EE:= {}^\perp U$ for a cotilting module $U$ over an artin algebra. This provides a rich source of examples:
\begin{itemize}
\item Let $\EE$ be a functorially finite torsionfree class of $\mod\Lambda$ for an artin algebra $\Lambda$ (see \cite{ASS}). Then by factoring out the annihilator of $\EE$, we may assume that $\EE$ is a faithful functorially finite torsionfree class. It is well-known that such $\EE$ is of the form $^\perp U$ for a cotilting $\Lambda$-module with $\id U \leq 1$ (see e.g. \cite[Theorem]{smalo}), so Theorem \ref{artinmain} applies to $\EE$. By duality, every functorially finite torsion class over artin algebras is also an example.
\item An artin algebra $\Lambda$ is called \emph{Iwanaga-Gorenstein} if both $\id(\Lambda_\Lambda)$ and $\id({}_\Lambda \Lambda)$ are finite. In this case, we write $\GP\Lambda := {}^\perp \Lambda$ and call it the category of \emph{Gorenstein-projective} modules. It is immediate that $\Lambda$ itself is a cotilting $\Lambda$-module, so we can apply Theorem \ref{artinmain} to $\EE := \GP\Lambda$.
\end{itemize}

Next we will consider the case $\dim R > 0$.
Although $\EE$ is rarely Hom-finite, we can conclude (CF) by using the Hom-finiteness of the stable category $\un{\EE}$ in some cases. Let us introduce some terminologies to state our result.

Let $\EE$ be an exact category with enough projectives. In this case, we have the syzygy functor $\Omega: \un{\EE} \to \un{\EE}$. We denote by $\un{X}$ in $\un{\EE}$ for the image of $X$ in $\EE$ under the natural projection $\EE \defl \un{\EE}$. For an object $X$ in $\EE$, we say that $X$ has \emph{finite projective dimension} if there exists some integer $n\geq 0$ such that $\Omega^n \un{X} = 0$ holds in $\un{\EE}$.

Recall that an exact category $\EE$ is called \emph{Frobenius} if $\EE$ has enough projectives and enough injectives, and the classes of projectives and injectives coincide.

\begin{proposition}\label{glfc}
Let $R$ be a complete noetherian local ring and $\EE$ a Hom-noetherian idempotent complete exact $R$-category with AR conflations. Suppose that $\EE$ satisfies either one of the following conditions:
\begin{enumerate}
\item $\EE$ has enough projectives and every object in $\EE$ has finite projective dimension, or
\item $\EE$ is Frobenius.
\end{enumerate}
 Then $\EE$ satisfies {\upshape (CF)}.
\end{proposition}
\begin{proof}
Let $0 \to X \xrightarrow{f_i} Y \xrightarrow{g_i} Z \to 0$ be conflations in $\EE$ for $i=1,2$. Put $M_i := \coker \EE(-,g_i) \in \eff\EE$ for each $i$ and suppose that $M_1$ has finite length. We will show that so does $M_2$. First note that Lemma \ref{effstable} applies to this situation since $\EE$ has enough projectives, so we identify $\eff\EE$ with $\mod\un{\EE}$ naturally. Then $M_i$ has finite length as an $\EE$-module if and only if it does so as an $\un{\EE}$-module. Moreover, since $\un{\EE}$ is Hom-finite by Lemma \ref{homfin}, this occurs if and only if $\supp M_i$ is finite, where the support is considered inside $\ind \un{\EE}$.

(1)
First we prove that $[M_1]=[M_2]$ holds in $\KKK_0(\mod\un{\EE})$. Choose an integer $n \geq 0$ such that $\Omega^n \un{Z} = 0$ holds in $\un{\EE}$. Then it is classical that we have an exact sequence in $\mod\un{\EE}$ for each $i$:
\begin{align*}
0 &\to \un{\EE}(-,\Omega^{n-1}\un{X}) \to \un{\EE}(-,\Omega^{n-1}\un{Y}) \to \un{\EE}(-,\Omega^{n-1}\un{Z}) \to \cdots \\
&\to \un{\EE}(-,\Omega \un{X}) \to \un{\EE}(-,\Omega \un{Y}) \to \un{\EE}(-,\Omega \un{Z}) \to \un{\EE}(-,\un{X}) \to \un{\EE}(-,\un{Y}) \to \un{\EE}(-,\un{Z}) \to M_i \to 0.
\end{align*}
Therefore it immediately follows that $[M_1] = [M_2]$ holds in $\KKK_0(\mod\un{\EE})$.

Let $\un{X}$ be an indecomposable object of $\un{\EE}$. Since $\un{\EE}$ is a Hom-finite $R$-category by Lemma \ref{homfin}, we have the characteristic map $\chi_{\un{X}}:\KKK_0(\mod\un{\EE}) \to \Z$ which sends $F$ to $\length_R F(\un{X})$. Then we have $\un{X} \in \supp M_1 \Leftrightarrow \chi_{\un{X}}[M_1] \neq 0 \Leftrightarrow \chi_{\un{X}}[M_2] \neq 0 \Leftrightarrow \un{X} \in \supp M_2$, where the support is considered inside $\ind \un{\EE}$. Thus Proposition \ref{fl} implies that $M_2$ has finite length.

(2)
Put $N_i := \ker \un{\EE}(-,\un{f_i})$ for each $i=1,2$. Then we have exact sequences in $\mod\un{\EE}$ as in (1):
\begin{align*}
0 \to N_i \to \un{\EE}(-,\un{X}) \xrightarrow{\un{\EE}(-,\un{f_i})} \un{\EE}(-,\un{Y}) &\xrightarrow{\un{\EE}(-,\un{g_i})} \un{\EE}(-,\un{Z}) \to M_i \to 0, \\
\un{\EE}(-,\Omega\un{Y}) &\xrightarrow{\un{\EE}(-,\Omega\un{g_i})} \un{\EE}(-,\Omega\un{Z}) \to N_i \to 0.
\end{align*}
It follows from the first exact sequence that $[M_1\oplus N_1] = [M_1] + [N_1] = [M_2] + [N_2] = [M_2\oplus N_2]$ holds in $\KKK_0(\mod\un{\EE})$. We will show that $N_1$ has finite length. By the same argument as in (1), we have that $\supp (M_1\oplus N_1) = \supp (M_2 \oplus N_2)$ holds in $\ind\un{\EE}$.
Now $\Omega : \un{\EE} \to \un{\EE}$ is an equivalence since $\EE$ is Frobenius, and let us denote by $\Omega^-:\un{\EE} \to \un{\EE}$ the quasi-inverse of $\Omega$. Then it is easily checked that $\un{X} \in \supp N_1 \Leftrightarrow \Omega^- \un{X} \in \supp M_1 \Leftrightarrow \un{X} \in \Omega (\supp M_1)$ holds for an indecomposable object $\un{X} \in \un{\EE}$, hence $\supp N_1 = \Omega (\supp M_1)$. Therefore $\supp N_1$ is finite, and so is $\supp (M_1\oplus N_1) = \supp (M_2 \oplus N_2)$. Hence $\supp M_2$ is finite, which implies that $M_2$ has finite length.
\end{proof}

For an $R$-order $\Lambda$ with at most an isolated singularity, Proposition \ref{glfc} shows that $\CM\Lambda$ satisfies (CF) if either $\Lambda$ has finite global dimension or $\Lambda$ is a Gorenstein order. Thus we obtain the following result on the category $\CM\Lambda$.
\begin{theorem}\label{ordermain}
Let $R$ be a complete Cohen-Macaulay local ring and $\Lambda$ an $R$-order with at most an isolated singularity. Put $\EE := \CM\Lambda$. Then the following are equivalent.
\begin{enumerate}
\item $\EE$ is of finite type.
\item $\EE$ is admissible.
\end{enumerate}
Assume in addition that either $\Lambda$ has finite global dimension or $\Lambda$ is a Gorenstein order. Then the following are also equivalent.
\begin{enumerate}[resume]
\item $\AR(\EE) = \Ex(\EE)$ holds.
\item $\AR(\EE) \otimes_\Z \Q = \Ex(\EE) \otimes_\Z \Q$ holds.
\end{enumerate}
\end{theorem}
\begin{proof}
This follows from Example \ref{cmcontfin} and Proposition \ref{glfc}.
\end{proof}
\begin{remark}
Theorem \ref{ordermain} for the case $\Lambda$ has finite global dimension was shown in \cite[Proposition 2.3]{ar1}, but their proof relies on higher algebraic K-theory. Also Theorem \ref{ordermain} generalizes \cite{hir} where $\Lambda$ is assumed to be commutative and Gorenstein.
The author does not know whether all the conditions above are equivalent without any assumption on an $R$-order $\Lambda$, even if $\Lambda$ is commutative.
\end{remark}

\section{Finiteness of syzygies}\label{sec5}
Let $R$ be a complete Cohen-Macaulay local ring and suppose that $\AR(\CM R) = \Ex(\CM R)$ holds. Although we do not know whether $\CM R$ is of finite type, it is shown in \cite{ko} that $\mathsf{\Omega CM}\,R$, the category of syzygies of $\CM R$ is so. In this section, we will extend this result to a non-commutative order.

Let $\EE$ be an exact category with enough projectives. In this section, we denote by $\Omega\EE$ the subcategory of $\EE$ consisting of objects $X$ such that there exists an inflation $X \infl P$ in $\EE$ for some projective object $P$. If $\EE$ is Krull-Schmidt, then so is $\Omega \EE$ since $\Omega\EE$ is closed under direct sums and summands. The essential image of $\Omega\EE$ under the natural projection $\EE \defl \un{\EE}$ coincides with $\Omega \un{\EE}$, the essential image of the syzygy functor $\Omega:\un{\EE} \to \un{\EE}$.
We begin with the following property about $\Omega\EE$, which is of interest in itself.

\begin{lemma}\label{omega-}
Let $R$ be a commutative noetherian ring, and let $\EE$ be a Hom-noetherian idempotent complete exact $R$-category with a projective generator $P$. Then the syzygy functor $\Omega:\un{\EE} \to \Omega\un{\EE}$ has a fully faithful left adjoint $\Omega^-: \Omega\un{\EE}\to \un{\EE}$.
\end{lemma}
\begin{proof}
For each object $W\in\Omega \EE$, take a left $(\add P)$-approximation $f:W \to P^W$ (this is possible since $\EE$ is Hom-noetherian). Since $W$ is in $\Omega\EE$, there exists an inflation $f':W \infl P'$ with $P'\in \add P$. Thus $f'$ factors through $f$, which implies that $f$ is an inflation by the idempotent completeness of $\EE$ (\cite[Proposition 7.6]{buhler}).
Now we define $\Omega^- W\in\EE$ by the following conflation:
\[
0 \to W \xrightarrow{f} P^W \xrightarrow{g} \Omega^- W \to 0.
\]

Next suppose that we have a map $\varphi:W_1 \to W_2$ in $\Omega\EE$. This induces the commutative diagram
\[
\begin{tikzcd}
0 \arrow[r] & W_1 \arrow[r,"f_1"] \arrow[d,"\varphi"] & P^{W_1} \arrow[r,"g_1"] \arrow[d,"\psi"] & \Omega^- W_1 \arrow[r]\arrow[d,"\Omega^- \varphi"] & 0\\
0 \arrow[r] & W_2 \arrow[r,"f_2"] & P^{W_2} \arrow[r,"g_2"] & \Omega^- W_2 \arrow[r]& 0
\end{tikzcd}
\]
in $\EE$ where both rows are conflations, since $f_2 \circ \varphi$ must factor through $f_1$. A simple diagram chase shows that this defines a well-defined isomorphism $\Omega^-:\un{\EE}(W_1,W_2) \to \un{\EE}(\Omega^- W_1, \Omega^- W_2)$, with its inverse induced by the syzygy functor $\Omega:\un{\EE} \to \Omega \un{\EE}$. Thus we obtain a functor $\Omega^-: \Omega\un{\EE} \to \un{\EE}$, which is fully faithful and satisfies $\Omega \Omega^- \iso \id_{\Omega \un{\EE}}$.

Finally we show that $\Omega^-:\Omega\un{\EE} \to \un{\EE}$ is a left adjoint of $\Omega:\un{\EE} \to \Omega\un{\EE}$. Let $W$ be in $\Omega\EE$ and $X$ in $\EE$. Then the syzygy functor induces a morphism $\un{\EE}(\Omega^- W,X) \to \un{\EE}(W,\Omega X)$ by $\Omega \Omega^- W \iso W$ in $\un{\EE}$. It is easy to show that this map is bijective, by considering the following diagram
\[
\begin{tikzcd}
0 \arrow[r] & W \arrow[r,"f"] \arrow[d] & P^W \arrow[r,"g"] \arrow[d] & \Omega^- W \arrow[r]\arrow[d] & 0\\
0 \arrow[r] & \Omega X \arrow[r]& P_X \arrow[r]& X \arrow[r]& 0
\end{tikzcd}
\]
where $P_X$ is projective and $f$ is a left $\add P$-approximation. The details are left to the reader.
\end{proof}

\begin{theorem}\label{omega}
Let $R$ be a complete noetherian local ring and $\EE$ a Hom-noetherian idempotent complete exact $R$-category with a projective generator $P$ and AR conflations. Suppose that $\AR(\EE)\otimes_\Z \Q = \Ex(\EE) \otimes_\Z \Q$ holds. Then the following holds.
\begin{enumerate}
\item For every object $M$ in $\eff\EE$, we have that $\supp M \cap \ind (\Omega\un{\EE})$ is finite, where the support is considered inside $\ind\un{\EE}$.
\item Assume in addition that $\un{\EE}$ has a weak cogenerator. Then $\Omega \EE$ is of finite type.
\end{enumerate}
\end{theorem}

\begin{proof}
Throughout this proof, supports of functors are always considered inside $\ind\un{\EE}$.

(1)
First recall that the stable category $\un{\EE}$ is Hom-finite over $R$ by Lemma \ref{homfin}.
By  the similar argument as in Proposition \ref{fcfl}, it suffices to prove the following weaker version of (CF).

\begin{itemize}
\item[(CF)$_\Omega$] Let $0 \to X \xrightarrow{f_i} Y \xrightarrow{g_i} Z \to 0$ be conflations in $\EE$ for $i=1,2$ and put $M_i := \coker \EE(-,g_i) \in \eff\EE$. If $\supp M_1$ is finite, then $\supp M_2 \cap \ind(\Omega\un{\EE})$ is finite.
\end{itemize}
Assume the above situation and put $N_i := \ker \un{\EE}(-,\un{f_i})$ for each $i=1,2$. Then we have exact sequences in $\mod\un{\EE}$ for $i=1,2$, as in the proof of Proposition \ref{glfc}(2):
\begin{align*}
0 \to N_i \to \un{\EE}(-,\un{X}) \xrightarrow{\un{\EE}(-,\un{f_i})} \un{\EE}(-,\un{Y}) &\xrightarrow{\un{\EE}(-,\un{g_i})} \un{\EE}(-,\un{Z}) \to M_i \to 0, \\
\un{\EE}(-,\Omega\un{Y}) &\xrightarrow{\un{\EE}(-,\Omega \un{g_i})} \un{\EE}(-,\Omega\un{Z}) \to N_i \to 0.
\end{align*}
As in the proof of Proposition \ref{glfc}(2), it is enough to show that $\supp N_1 \cap \ind(\Omega\un{\EE})$ is finite. Actually we will show that every element of $\supp N_1 \cap \ind(\Omega\un{\EE})$ is isomorphic to $\Omega\un{A}$ for some $\un{A} \in \supp M_1$, which obviously implies the desired claim.

Let $\un{W}$ be an element of $\supp N_1 \cap \ind(\Omega\un{\EE})$, so $\un{\EE}(\un{W},\Omega \un{g_1}): \un{\EE}(\un{W},\Omega \un{Y_1}) \to \un{\EE}(\un{W},\Omega \un{Z_1})$ is not surjective. By Lemma \ref{omega-}, this is equivalent to that $\un{\EE}(\Omega^- \un{W},\un{g_1}) : \un{\EE}(\Omega^-\un{W},\un{Y_1}) \to \un{\EE}(\Omega^- \un{W}, \un{Z_1})$ is not surjective. On the other hand, since $\Omega^- :\Omega\EE \to \EE$ is fully faithful by Lemma \ref{omega-}, we have that $\Omega^- \un{W}$ is indecomposable. Thus $\Omega^- \un{W}$ belongs to $\supp M_1$. Now $\un{W} \iso \Omega \Omega^- \un{W}$ in $\un{\EE}$ holds, which completes the proof.

(2)
Let $\un{X}$ be a weak cogenerator of $\un{\EE}$. By (1), we have that that $\ind (\Omega\un{\EE}) \cap \supp \un{\EE}(-,\un{X})$ is finite. Since every indecomposable object $\un{W}$ in $\un{\EE}$ should satisfy  $\un{\EE}(\un{W},\un{X})\neq 0$, it follows that $\Omega\un{\EE}$ is of finite type. Now $\Omega\EE$ itself is of finite type because we have a natural identification $\ind(\Omega\EE) = \ind(\Omega \un{\EE}) \sqcup \ind (\add P)$.
\end{proof}
Now we apply this theorem to the category of Cohen-Macaulay modules. The obtained result extends \cite{hir,ko}, where $\Lambda$ was assumed to be commutative.
\begin{corollary}\label{syzygymain}
Let $R$ be a complete Cohen-Macaulay local ring and $\Lambda$ an $R$-order with at most an isolated singularity. If $\AR(\CM \Lambda)\otimes_\Z \Q = \Ex(\CM \Lambda) \otimes_\Z \Q$ holds, then $\mathsf{\Omega CM}\,\Lambda$ is of finite type.
\end{corollary}
\begin{proof}
We can apply Theorem \ref{omega} to $\EE:= \CM\Lambda$ since $\un{\EE}$ has a weak cogenerator by Example \ref{cmcontfin}.
\end{proof}

\appendix
\section{Some properties of the category of effaceable functors}
First we show that our notion of effaceability is equivalent to that of Grothendieck (see \cite[p. 148]{gro} or \cite[Exercise 2.4.5]{weibel} for the standard definition).

\begin{proposition}\label{grodef}
Let $\EE$ be an idempotent complete exact category and $M \in \Mod\EE$ a right $\EE$-module. Then the following are equivalent.
\begin{enumerate}
\item $M$ is in $\eff \EE$.
\item $M$ is finitely presented, and for every $W \in \EE$ and $w\in M(W)$, there exists a deflation $\psi: E \defl W$ in $\EE$ such that $(M\psi )(w)=0$.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) $\Rightarrow$ (2):
Suppose that $M$ is in $\eff\EE$ and take a conflation $0 \to X \to Y \xrightarrow{g} Z \to 0$ in $\EE$ such that $M \iso \coker \EE(-,g)$. Denote by $\pi:\EE(-,Z) \defl M$ the natural projection. Clearly $M$ is finitely presented. Let $W\in\EE$ and $w \in M(W)$. By the Yoneda lemma, we have a morphism $w_* :\EE(-,W) \to M$. By the projectivity of $\EE(-,W)$ and the Yoneda lemma, we have a map $\varphi : W \to Z$ such that $\pi \circ \EE(-,\varphi) = w_*$.
Now let us take the pullback of $g$ along $\varphi$:
\[
\begin{tikzcd}
0 \arrow[r]& X \arrow[d,equal] \arrow[r]& E \arrow[d] \arrow[r,"\psi"] & W \arrow[r] \arrow[d,"\varphi"] & 0 \\
0 \arrow[r]& X \arrow[r]& Y \arrow[r,"g"] & Z \arrow[r]& 0
\end{tikzcd}
\]
Note that $\psi$ is a deflation. This yields the following commutative diagram in $\Mod\EE$:
\[
\begin{tikzcd}
&  & \EE(-,E) \arrow[r,"{\EE(-,\psi)}"]\arrow[d] & \EE(-,W) \arrow[d,"{\EE(-,\varphi)}"'] \arrow[dr,"w_*"] \\
0 \arrow[r]& \EE(-,X) \arrow[r]& \EE(-,Y) \arrow[r,"{\EE(-,g)}"']& \EE(-,Z) \arrow[r,"\pi"']& M \arrow[r]& 0
\end{tikzcd}
\]
By applying these functors to $E$, it is straightforward to check that $(M\psi )(w)=0$.

(2) $\Rightarrow$ (1):
Suppose that $M$ satisfies the condition of (2). Since $M$ is finitely presented, we have a morphism $g:Y\to Z$ in $\EE$ such that $\EE(-,Y) \to \EE(-,Z) \to M \to 0$ is exact. We have an element $z \in M(Z)$ which corresponds to $\EE(-,Z) \to M$ by the Yoneda lemma. By the assumption, there exists a deflation $\psi: E \defl Z$ such that $(M \psi)(z)=0$. The Yoneda lemma implies that the composition $\EE(-,E) \to \EE(-,Z) \to M $ is a zero map, hence the map $\EE(-,\psi)$ lifts to $\EE(-,Y)$. Thus we have a map $h:E \to Y$ such that $g \circ h = \psi$. Since $\psi$ is a deflation, we can conclude that so is $g$ by the idempotent completeness of $\EE$. Therefore $M$ is in $\eff\EE$.
\end{proof}

Next we prove that $\eff\EE$ is an abelian subcategory of $\Mod\EE$.
\begin{theorem}\label{effabelian}
Let $\EE$ be an exact category. Then $\eff\EE$ is closed under kernels and cokernels in $\Mod\EE$. In particular, $\eff\EE$ is an abelian category such that the inclusion $\eff\EE \to \Mod\EE$ is exact. Moreover, $\eff\EE$ is closed under extensions in $\Mod\EE$.
\end{theorem}
\begin{proof}
Let $\varphi:M_1 \to M_2$ be a morphism in $\eff\EE$. For each $i=1,2$, we have the corresponding conflations $0 \to X_i \xrightarrow{f_i} Y_i \xrightarrow{g_i} Z_i \to 0$ in $\EE$ satisfying $M_i = \coker \EE(-,g_i)$.
We will show that the image, kernel and cokernel of $\varphi$ in $\Mod\EE$ are contained in $\eff\EE$. Sine $\varphi$ induces a morphism between the projective resolutions of $M_1$ and $M_2$, we have the following commutative diagram in $\EE$ by the Yoneda lemma:
\[
\begin{tikzcd}
0 \arrow[r]& X_1 \arrow[r,"f_1"] \arrow[d,"a"] & Y_1 \arrow[r,"g_1"] \arrow[d,"b"] & Z_1 \arrow[r] \arrow[d,"c"] & 0 \\
0 \arrow[r]& X_2 \arrow[r,"f_2"] & Y_2 \arrow[r,"g_2"] & Z_2 \arrow[r] & 0
\end{tikzcd}
\]
Take the pushout $E$ of the top conflation along $a:X_1 \to X_2$. By the universal property, we obtain the following commutative diagram
\begin{equation}\label{kore}
\begin{tikzcd}
0 \arrow[r]& X_1 \arrow[d,equal] \arrow[r, "{^t [-a,f_1]}"] & X_2 \oplus Y_1 \arrow[r,"{[f,b_1]}"] \arrow[d,"{[0,1]}"] & E \arrow[r]\arrow[d,"g"] & 0\\
0 \arrow[r]& X_1 \arrow[r,"f_1"] \arrow[d,"a"] & Y_1 \arrow[r,"g_1"] \arrow[d,"b_1"] & Z_1 \arrow[r] \arrow[d,equal] & 0 \\
0 \arrow[r]& X_2 \arrow[r,"f"] \arrow[d,equal] & E \arrow[r,"g"] \arrow[d,"b_2"] & Z_1 \arrow[r]\arrow[d,"c"] & 0 \\
0 \arrow[r]& X_2 \arrow[r,"f_2"]\arrow[d,"f"] & Y_2 \arrow[r,"g_2"]\arrow[d,"{^t [1,0]}"] & Z_2 \arrow[r] \arrow[d,equal] & 0\\
0 \arrow[r]& E \arrow[r,"{^t [b_2,g]}"] & Y_2 \oplus Z_1 \arrow[r,"{[g_2,-c]}"] & Z_2 \arrow[r]& 0
\end{tikzcd}
\end{equation}
in $\EE$ such that $b_2 \circ b_1 = b$ holds. It is straightforward to see that diagram (\ref{kore}) commutes. Moreover, all the rows are conflations, and the top-right square and the bottom-left square are pullback-pushout squares, see \cite[Proposition 2.12]{buhler}

By the Yoneda embedding, we obtain the following commutative diagram in $\Mod\EE$ corresponding to (\ref{kore}), where all the rows are exact.

\[
\begin{tikzcd}
0 \arrow[r] & \EE(-,X_1) \arrow[r]\arrow[equal,d]& \EE(-,X_2 \oplus Y_1) \arrow[r] \arrow[d] & \EE(-,E) \arrow[r] \arrow[d] & K \arrow[r]\arrow[d,"\iota"] & 0 \\
0 \arrow[r]& \EE(-,X_1) \arrow[r] \arrow[d] & \EE(-,Y_1) \arrow[r]\arrow[d] & \EE(-,Z_1) \arrow[r] \arrow[equal,d] & M_1 \arrow[r]\arrow[d,"\varphi_1"] & 0 \\
0 \arrow[r]& \EE(-,X_2) \arrow[r] \arrow[equal,d] & \EE(-,E) \arrow[r]\arrow[d] & \EE(-,Z_1) \arrow[r] \arrow[d] & M \arrow[r] \arrow[d,"\varphi_2"] & 0 \\
0 \arrow[r]& \EE(-,X_2) \arrow[r]\arrow[d] & \EE(-,Y_2) \arrow[r]\arrow[d] & \EE(-,Z_2) \arrow[r]\arrow[equal,d]  & M_2 \arrow[r]\arrow[d,"\pi"] & 0 \\
0 \arrow[r]& \EE(-,E) \arrow[r]& \EE(-,Y_2 \oplus Z_1) \arrow[r]& \EE(-,Z_2) \arrow[r]& C \arrow[r]& 0
\end{tikzcd}
\]
We have that $K$, $M$ and $N$ belong to $\eff\EE$. We can check that $\varphi_1$ is surjective and $\varphi_2$ is injective, hence $M$ is a image of $\varphi$. Moreover, one can show that $\iota$ is a kernel of $\varphi_1$ (or equivalently, $\varphi$) and $\pi$ is a cokernel of $\varphi_2$ (or equivalently $\varphi$). We leave the details to the reader.

For the extension-closedness of $\eff\EE$, we refer the reader to \cite[Proposition 2.10]{en2}.
\end{proof}

In this paper, we often deal with the finiteness of length of effaceable $\EE$-modules. Since $\eff\EE$ and $\Mod\EE$ are both abelian, the length of effaceable modules seems to depend on the ambient category we adopt. This is not the case by the following.
\begin{proposition}\label{onaji}
Let $\EE$ be an exact category and $M$ an object in $\eff\EE$.
\begin{enumerate}
\item $M$ has finite length in $\eff\EE$ if and only if it does so in $\Mod\EE$.
\item Suppose that $\mod\EE$ is abelian. Then $M$ has finite length in $\Mod\EE$ if and only if it does so in $\mod\EE$.
\end{enumerate}
Moreover, the length and the composition factors of $M$ in $\Mod\EE$ coincide with those in $\eff\EE$ and in $\mod\EE$.
\end{proposition}

This follows from the following elementary observation.
\begin{lemma}\label{elementary}
Let $\EE$ be an additive category and $\AA$ an abelian subcategory of $\Mod\EE$ such that the inclusion $\AA \to \Mod\EE$ is exact. Suppose that $\AA$ is closed under finitely generated submodules. Then the following are equivalent for every object $M$ in $\AA$.
\begin{enumerate}
\item $M$ has finite length in $\Mod\EE$.
\item $M$ has finite length in $\AA$.
\end{enumerate}
Moreover, the length and the composition factors of $M$ in $\Mod\EE$ coincides with those in $\AA$.\end{lemma}
\begin{proof}
(1) $\Rightarrow$ (2):
This is clear since the lattice of subobjects of $M$ in $\AA$ is a subposet of that in $\Mod\EE$.

(2) $\Rightarrow$ (1):
It suffices to show that every simple object in $\AA$ is also simple in $\Mod\EE$. Let $M$ be a simple object in $\AA$ and $N$ a non-zero submodule of $M$ in $\Mod\EE$. By using the Yoneda lemma, it is easily checked that there is a non-zero finitely generated submodule $N'$ of $N$ in $\Mod\EE$. Since $N'$ is a finitely generated submodule of $M$, we have that $N'$ belongs to $\AA$ by the assumption. Thus $N'=M$ holds by $N' \leq M$ in $\AA$ and $N'\neq 0$.

The remaining assertions are clear from the above proof and the Jordan-H\"older theorem.
\end{proof}

\begin{proof}[Proof of Proposition \ref{onaji}]
(1)
The category $\eff\EE$ satisfies the conditions in Lemma \ref{elementary} by Propositions \ref{effabelian} and \ref{serre}(1).

(2)
Suppose that $\mod\EE$ is abelian. Then it is well-known that the embedding $\mod\EE \to \Mod\EE$ is exact and that $\mod\EE$ is closed under finitely generated submodules, see \cite{aus66}. Thus Lemma \ref{elementary} applies.
\end{proof}

\medskip\noindent
{\bf Acknowledgement.}
The author would like to express his deep gratitude to his supervisor Osamu Iyama for his support and many helpful comments. This work is supported by JSPS KAKENHI Grant Number JP18J21556.

config: arxiv


TITLE: Help understanding the axiom of regularity
QUESTION [1 upvotes]: If 0={}, 1={0}, 2={0,1} and so forth, the axiom of regularity holds for every natural number, since for each natural number n, there is an element x in n which is disjoint from n.
However, if we define 0={1}, 1={2}, 2={3} and so on (infinite descending sequence), regularity does no hold, given that there is not a single element of the set S = { all n : n is a natural number }  which is disjoint of S. Is that correct?

REPLY [1 votes]: As people have noted in the comments your definition doesn’t make a lot of sense (even if sets don’t need to be well founded, recursive definitions need to be). And more generally, you will never succeed at defining an infinite descending membership sequence in ZF minus regularity, because it is consistent that no such sequence exists.
Now, it is also consistent with ZF minus regularity that there is a countable sequence of distinct sets $x_n$ such that $x_n =\{x_{n+1}\}$ for all $n$. But even still, you can’t just define the natural numbers as $n=x_n$ unless you actually add antifoundation axioms to the theory that will allow you not only to prove the existence of such a sequence but to define a canonical one.
But this whole thing about the natural numbers feels like a non sequitur. It seems like the content of your question is really whether the existence of an infinite sequence such that $x_n=\{x_{n+1}\}$  implies that regularity fails. The answer to that is yes: in ZF minus regularity we can define the set $\{x_n:n\in\omega\}$ from
the sequence, and this set violates regularity. This follows more generally from any infinite descending sequence where $x_{n+1}\in x_n.$ If we also assume choice in the base theory (or even just dependent choice) then the implication goes the other way and the failure of the axiom of regularity implies the existence of an infinite descending sequence. (Thanks to Hanul Jeon in the comments for pointing out that the converse requires choice.)


TITLE: Index Manipulation and Angular Momentum Commutator Relations
QUESTION [4 upvotes]: I have been trying for hours and cannot figure it out. I am not asking anyone to do it for me, but to understand how to proceed.
We have the relations
$$[L_i,p_j] ~=~ i\hbar\;  \epsilon_{ijk}p_k,$$
$$[L_i,r_j] ~=~ i\hbar\;  \epsilon_{ijk}r_k,$$
$$[L_i,L_j] ~=~ i\hbar\;  \epsilon_{ijk}L_k.$$
Now I am trying to calculate
$$[L_i,(p\times L)_j].$$
I do not know how to reduce it. When I try I am left with too many dummy indices that I don't know what to do. For example, using
$$[AB,C] = A[B,C]+[A,C]B,$$
and expanding the cross product in terms of Levi-Civita symbols
$$[L_i,\epsilon_{jmn}p_m L_n],$$
but I don't know how to proceed correctly from here. For example, I tried,
$$ =~ \epsilon_{jmn}(\;p_m[L_i,L_n]+[L_i,p_m]L_n ).$$
Is this correct? If so, in the next step I used the known commutation relations
$$ =~i\hbar\; \epsilon_{jmn}(\; \epsilon_{ink}p_mL_k+\epsilon_{imk}p_kL_n).$$
Once again, I am stuck and do not know how to evaluate this further. Could someone tell me what I am doing wrong, or if not, how to proceed?

REPLY [3 votes]: It is possible to continue OP's calculation as follows
$$ =~i\hbar\; \epsilon_{jmn}(\epsilon_{ink}p_m L_k+\epsilon_{imk}p_k L_n)
~=~-i\hbar( \epsilon_{jkm} \epsilon_{min}+\epsilon_{kim} \epsilon_{mjn})p_k L_n $$
$$ \stackrel{\rm Jac.Id.}{=}~i\hbar\; \epsilon_{ijm} \epsilon_{mkn}p_k L_n
~=~i\hbar\; \epsilon_{ijm}(p\times L)_m ,$$
where one uses the Jacobi identity
$$\sum_{{\rm cycl.}~i,j,k} \epsilon_{ijm} \epsilon_{mkn}~=~0 $$
for the Levi-Civita symbols.


TITLE: Is Algebraic Number Theory still an active research field?
QUESTION [17 upvotes]: I've took an introductory course on Algebraic Number Theory during my Master's, which I really enjoyed. Now that I'm beginning to think about my PhD area, I wonder if I shouldn't go for something in that direction.
However, as in practically all courses I've taken, I was too busy trying to understand definitions and theorems, having little time left (or not enough knowledge) to understand the historical development or the research perspectives on the subject.
Is Algebraic Number Theory still an active area of research? I ask this because almost everything I read or hear about number theory from recent years seems to involve analytic methods (harmonic analysis, probability, ergodic theory etc.), which are not really my cup of tea.
Since I'm a more algebra-driven guy (meaning I'm instinctly attracted to things like Commutative Algebra, Algebraic Geometry, Galois Theory and, of course, Algebraic Number Theory), I wish there were still active lines of research in Number Theory using actual algebraic methods, at least for the most part.
I know this question is rather vague, but that's how well I'm able to articulate it right now, so any advice, insights, reading suggestions etc. would be greatly appreciated.

REPLY [25 votes]: Nowadays, the distinction between algebraic and analytic number theory is not in the proofs, but in the questions you are trying to answer.  Analytic number theory asks questions like "how are the primes distributed on the number line?"  Algebraic number theory asks questions like "how do primes split in a given extension of number fields?"
Many questions in algebraic number theory are hard to answer just by using algebra.  There has been an enormous amount of insight gained by bringing in analytic techniques.  For example, there is no known proof just using algebra that there are no nontrivial unramified extensions of $\mathbb Q$.  But the Minkowski bound for the discriminant shows that it is never trivial.
There are some deep questions about integer solutions to polynomial equations which have only been answered by connecting them to modular forms.  More generally, the representations and associated L-functions of reductive groups are expected to yield considerable arithmetic insight once the Langlangs Program is complete.  This seems to be the most promising direction for the future of algebraic number theory.
If you love algebraic number theory, I would recommend embracing the analytic techniques with the algebraic ones.  When you're really doing this kind of math, you won't be able to distinguish whether you are doing algebra or analysis.  If you truly dislike analysis, you might be better off doing something like commutative algebra.


TITLE: Calculate the probabilities of the same coulor appearing in a row
QUESTION [1 upvotes]: I am doing some experiments using the probabilities on a roulette.
I found this link: http://www.roulettestar.com/probability.php
Where you see "Same even money bet result in a row probability." the page reports the probabilities in percentage of the same coulor appearing in a row.
I am really not getting how is that calculated. Does anyone know how that site calculated that percentages? What is the math formula behind?
Thanks for any help

REPLY [0 votes]: The probability for a color is $18/37 =\sim 48.6\%$. To get the same (given) color $n$ times consecutively is simply $(18/37)^n$.


TITLE: If an extension of ZF is not consistent with $V=HOD$, does this imply that it must prove existence of large cardinals?
QUESTION [0 upvotes]: Suppose we have a first order theory $\text{T}$ such that

$\text{T} \supset \text{ZF}$
$\forall M [(M\models \text{T}) \to \neg (M\models \forall x (x \in HOD))]$

Does that mean that $\text{T}$ must prove existence of a large cardinal?
If Yes, then among the list of the known large cardinal proeprties , which one is the least that $\text{T}$ must prove its existence?

REPLY [4 votes]: Let me interpret your question in a slightly different way, which may be what you thought about.
$T$ extends $\sf ZF$, and if $M$ is a model of $T$, then there is no model extending it with the same ordinals which satisfies $V=\rm HOD$.
In this sense it's a sensible question, since every model of $\sf ZFC$ extends to a model of $\rm HOD$.
The answer is no, though. Simply because there are models of $\sf ZF$ that cannot be extended to models of $\sf ZFC$ without adding ordinals. The obvious example would be Gitik's model where all the ordinals have cofinality $\omega$. But of course that we can prove there are very large cardinals in some inner models. So it doesn't help.
However, D. B. Morris proved in his thesis, and later I rediscovered a similar proof, that it is consistent that there is an extension of every model of $\sf ZFC$ to a model where for every infinite ordinal $\alpha$, there is a set $A_\alpha$ which is a countable union of countable sets, and the power set of $A_\alpha$ can be mapped onto $\alpha$. It follows that this model cannot be extended to a model of $\sf ZFC$ without adding ordinals.
In particular, it cannot be extended to a model of $V=\rm HOD$.