1 Introduction
Let R be a commutative Artin ring and let
$\Lambda $
be a not-necessarily commutative Artin R-algebra. Let
$\operatorname {\underline {mod}}(\Lambda )$
denote the stable category of
$\Lambda $
. The study of equivalences of stable categories of Artin R-algebras has a rich history; see [Reference Auslander, Reiten and Smalø2, Chapter X]. Auslander discovered that many concepts in representation theory of Artin algebras have natural analogs in the study of maximal Cohen-Macaulay ( = MCM) modules of a commutative Cohen-Macaulay local ring A. See [Reference Yoshino15] for a nice exposition of these ideas. By following Auslander’s idea, we investigate equivalences of the stable category of MCM modules over commutative Cohen-Macaulay local rings. If A is Gorenstein local, then
$\operatorname {\underline {CM}}(A)$
, the stable category of MCM A-modules has a triangulated structure, see [Reference Buchweitz4, 4.4.1]. So as a first iteration in this program, we may investigate triangle equivalences of stable categories of MCM modules over commutative Gorenstein local rings.
There is paucity of examples of Gorenstein local rings
$A, B$
such that
$\operatorname {\underline {CM}}(A)$
is triangle equivalent to
$\operatorname {\underline {CM}}(B)$
. There are two well-known examples of triangle equivalences. First is Knörrer periodicity [Reference Knörrer9] for hypersurfaces. Another gives an equivalence
$\operatorname {\underline {CM}}(A) \rightarrow \operatorname {\underline {CM}}(\widehat {A})$
, where A is a excellent, Henselian, Gorenstien isolated singularity, see [Reference Keller, Murfet and Van den Bergh8, A.6]. In this paper, we show existence of bountiful examples of Gorenstein local rings A and B such that there is a triangle equivalence between the stable categories
$\operatorname {\underline {CM}}(A), \operatorname {\underline {CM}}(B)$
.
1.1 The example
Let
$(A,\mathfrak {m} )$
be a Gorenstein local ring of dimension
$d \geq 1$
. Assume A is essentially of finite type over a field K and that A is an isolated singularity. Also assume that the completion of A is a domain. Further assume that the Grothendieck group of the completion of A is a finitely generated abelian group. Let
$\mathfrak {E} _P(A)$
be the set of isomorphism classes of pointed etale neighborhoods of A, see 3.1, 3.2. It is well known that
$\mathfrak {E} _P(A)$
is a directed set and
$A^h$
, the henselization of A, is
$$\begin{align*}A^h = \lim_{R \in \mathfrak{E} _P(A)}R. \end{align*}$$
By Remark 3.3, for any
$B \in \mathfrak {E} _P(A)$
there exists at least one infinite chain in
$\mathfrak {E} _P(A)$
starting at B.
We show
Theorem 1.1 (with hypotheses as in 1.1.)
-
(1) Let
$A_1 \rightarrow A_2 \rightarrow \cdots \rightarrow A_n \rightarrow \cdots $
be a chain in
$\mathfrak {E} _P(A)$
. Then there exists
$n_0$
such that
$$ \begin{align*}\operatorname{\underline{CM}}(A_n) \cong \operatorname{\underline{CM}}(A_{n+1}) \quad \text{for all }n \geq n_0.\end{align*} $$
-
(2) There exists a chain
$B_1 \rightarrow B_2 \rightarrow \cdots \rightarrow B_n \rightarrow \cdots $
in
$\mathfrak {E} _P(A)$
such that
$$ \begin{align*}\operatorname{\underline{CM}}(B_n) \cong \operatorname{\underline{CM}}(A^h) \quad \text{for all }n \geq 1.\end{align*} $$
Remark 1.2 In (2) note that
$B_n$
is NOT henselian, see A.1. Nevertheless
$\operatorname {\underline {CM}}(B_n) \cong \operatorname {\underline {CM}}(A^h)$
is Krull-Schmidt for all
$n \geq 1$
.
Theorem 1.1 is existensial. We show
Theorem 1.3 Let
$A = \mathbb {C}[X,Y, Z]_{(X, Y, Z)}/ (Z^2 - XY)$
. Let
$B \in \mathfrak {E} _P(A)$
. Then
$ \operatorname {\underline {CM}}(B) \cong \operatorname {\underline {CM}}(A^h)$
.
Remark 1.4 Let
$(A,\mathfrak {m} )$
be a Gorenstein local ring, essentially of finite type over a field K and that A is an isolated singularity. Then by [Reference Kamoi and Kurano7, 1.5] the natural map
$G(A) \rightarrow G(A^h)$
is injective. Assume
$\widehat {A}$
is also a domain (automatic if
$\dim A \geq 2$
). Our entire argument goes through if we just assume the quotient group
$G(\widehat {A})/G(A)$
is a finitely generated abelian group. Regrettably, we do not have a nice class of rings with this property (i.e., with
$G(\widehat {A})$
infinitely generated abelian group and
$G(\widehat {A})/G(A)$
finitely generated abelian group).
Here is an overview of the contents of this paper. In section two, we prove some general results on Grothendieck groups of triangulated categories. In section three, we discuss some preliminary facts on pointed étale neighborhoods of a local ring. In section four, we give proof of Theorem 1.1. In the next section, we give many examples of local rings with
$G(\widehat {A})$
finitely generated. In section six we prove Theorem 1.3. In the appendix, we discuss a fact which is crucial for us.
2 Some generalities on Grothendieck groups of triangulated categories
Throughout all triangulated categories considered will be skeletally small. Let
$\mathcal {C} , \mathcal {D} $
be triangulated categories.
2.1
A triangulated functor
$F\colon \mathcal {C} \rightarrow \mathcal {D} $
is called an equivalence up to direct summand’s if it is fully faithful and any object
$X \in \mathcal {D} $
is isomorphic to a direct summand of
$F(Y)$
for some
$Y \in \mathcal {C} $
.
2.2
We say
$\mathcal {C} $
has weak cancellation, if for
$U, V, W \in \mathcal {C} $
we have
$$ \begin{align*}U \oplus V \cong U \oplus W \implies V \cong W.\end{align*} $$
2.3
Let
$\mathcal {C} $
be a triangulated category. The Grothendieck group
$G(\mathcal {C} )$
is the quotient group of the free abelian group on the set of isomorphism classes of objects of
$\mathcal {C} $
by the Euler relations:
$[V] = [U] + [W]$
whenever there is an exact triangle in
$\mathcal {C} $
$$ \begin{align*}U \rightarrow V \rightarrow W \rightarrow U[1].\end{align*} $$
As
$[U[1]] = -[U]$
in
$G(\mathcal {C} )$
, it follows that any element of
$G(\mathcal {C} )$
is of the form
$[V]$
for some
$V \in \mathcal {C} $
.
We first show:
Theorem 2.1 Let
$\phi \colon \mathcal {C} \rightarrow \mathcal {D} $
be a triangulated functor which is an equivalence upto direct summands. Then the natural map
$G(\phi ) \colon G(\mathcal {C} ) \rightarrow G(\mathcal {D} )$
is injective.
Proof Suppose
$G(\phi )([U]) = [\phi (U)] = 0$
in
$G(\mathcal {D} )$
. So by [Reference Thomason13, 2.4] there exists
$X, Y, Z \in \mathcal {D} $
and triangles
$$ \begin{align*} X &\rightarrow Y\oplus \phi(U) \rightarrow Z \rightarrow X[1], \quad \text{and} \\ X &\rightarrow Y \rightarrow Z \rightarrow X[1]. \end{align*} $$
There exists
$X_1, Z_1 \in \mathcal {D} $
such that
$X \oplus X_1 = \phi (M)$
and
$Z \oplus Z_1 = \phi (N)$
. We add the triangles
$X_1 \xrightarrow {1} X_1 \rightarrow 0 \rightarrow X_1 \rightarrow 0 \rightarrow X_1[1]$
and
$0 \rightarrow Z_1 \xrightarrow {1} Z_1 \rightarrow 0$
to the above triangles.
The modified second triangle is
$$\begin{align*}\phi(M) \rightarrow Y^\prime \rightarrow \phi(N) \rightarrow \phi(M)[1]. \end{align*}$$
It follows that
$Y^\prime = \phi (L)$
for some
$L \in \mathcal {C} $
. The modified first triangle becomes
$$ \begin{align*}\phi(M) \rightarrow\phi(L)\oplus \phi(U) \rightarrow \phi(N) \rightarrow \phi(M)[1].\end{align*} $$
Note
$\phi (L) \oplus \phi (U) = \phi (L\oplus U)$
. We now use the fact that if
$W_1, W_2 \in \mathcal {C} $
then
$W_1 \cong W_2$
if and only if
$\phi (W_1) \cong \phi (W_2)$
. As
$\phi $
is fully faithful we get triangles in
$\mathcal {C} $
$$ \begin{align*} M &\rightarrow L \oplus U \rightarrow N \rightarrow M[1], \quad \text{and} \\ M &\rightarrow L \rightarrow N \rightarrow M[1]. \end{align*} $$
It follows that
$[U] = 0$
in
$G(\mathcal {C} )$
.
A natural question is when is
$Y \in \mathcal {D} $
is isomorphic to
$\phi (X)$
for some
$X \in \mathcal {C} $
. A necessary condition is of course that
$[Y] \in \operatorname {image} G(\phi )$
. Some what surprisingly the converse also holds if
$\mathcal {D} $
satisfies weak cancellation.
Theorem 2.2 (with hypotheses as in Theorem 2.1). Also assume
$\mathcal {D} $
satisfies weak cancellation. If
$[V] \in \operatorname {image} G(\phi )$
then there exists
$W \in \mathcal {C} $
with
$\phi (W) \cong V$
.
We need the following:
Lemma 2.3 (with hypotheses as in Theorem 2.1). Let
$U, V \in \mathcal {C} $
. If
$\phi (U)$
is a direct summand of
$\phi (V)$
, then U is a direct summand of V.
Proof We have a diagram
with
$f = h \circ g$
,
$f^2 = f$
and
$g\circ h = 1_{\phi (U)}$
. As
$\phi $
is fully faithful, there exists
$f_1 \colon V \rightarrow V$
,
$g_1 \colon V \rightarrow U$
and
$h_1 \colon U \rightarrow V$
with
$\phi (f_1) = f$
,
$\phi (g_1) = g$
and
$\phi (h_1) = h$
. Again as
$\phi $
is fully faithful, we get
$h_1\circ g_1 = f_1$
,
$f_1^2 = f_1$
and
$g_1\circ h_1 = 1_{U}$
. The result follows.
We now give proof of Theorem 2.2.
Proof Say
$[V] = [\phi (U)]$
in
$G(\mathcal {D} )$
. By [Reference Thomason13, 2.4] we have triangles in
$\mathcal {D} $
$$ \begin{align*} X &\rightarrow Y\oplus \phi(U) \rightarrow Z \rightarrow X[1], \quad \text{and} \\ X &\rightarrow Y\oplus V \rightarrow Z \rightarrow X[1]. \end{align*} $$
There exists
$X_1, Z_1 \in \mathcal {D} $
such that
$X \oplus X_1 = \phi (M)$
and
$Z \oplus Z_1 = \phi (N)$
. We add the triangles
$X_1 \xrightarrow {1} X_1 \rightarrow 0 \rightarrow X_1 \rightarrow 0 \rightarrow X_1[1]$
and
$0 \rightarrow Z_1 \xrightarrow {1} Z_1 \rightarrow 0$
to the above triangles. Set
$Y^\prime = Y \oplus X_1 \oplus Z_1$
.
The (modified) first triangle yields
$Y^\prime \oplus \phi (U) \cong \phi (E)$
for some
$E \in \mathcal {C} $
. By 2.3 we have that U is a direct summand of E. Say
$E = U \oplus U_1$
. So
$Y^\prime \oplus \phi (U) \cong \phi (U) \oplus \phi (U_1)$
. As
$\mathcal {D} $
satisfies weak cancellation we get
$Y^\prime \cong \phi (U_1)$
.
The (modified) second triangle yields
$\phi (U_1) \oplus V \cong \phi (L)$
for some
$L \in \mathcal {C} $
. By 2.3 we have that
$U_1$
is a direct summand of L. Say
$L = U_1 \oplus U_2$
. So we obtain
$$ \begin{align*}\phi(U_1) \oplus V = \phi(U_1) \oplus \phi(U_2).\end{align*} $$
As
$\mathcal {D} $
satisfies weak cancellation we get
$V \cong \phi (U_2).$
The result follows.
The following consequence of Theorem 2.2 is significant.
Corollary 2.4 (with hypotheses as in Theorem 2.2). If
$G(\phi )$
is an isomorphism then
$\phi $
is an equivalence.
Proof By our assumption
$\phi $
is fully faithful. By Theorem 2.2 it follows that
$\phi $
is dense. So
$\phi $
is an equivalence.
Another surprising consequence of Theorem 2.2 is the following:
Corollary 2.5 (with hypotheses as in Theorem 2.2). Let
$Y \in \mathcal {D} $
. Then there exists
$X \in \mathcal {C} $
with
$\phi (X) = Y \oplus Y[1]$
.
Proof Notice in
$G(\mathcal {D} )$
we have
$$ \begin{align*}[Y \oplus Y[1]] = [Y] + [Y[1]] = [Y] - [Y] = 0.\end{align*} $$
The result follows from Theorem 2.2.
3 Pointed étale neighborhoods
In this section, we recall definition of pointed étale extensions and discuss a few of its properties.
3.1
A local homomorphism
$ \psi \colon (A,\mathfrak {m} ) \rightarrow (B,\mathfrak {n} ) $
of local rings is unramified provided B is essentially of finite type over A (that is, B is a localization of some finitely generated A-algebra) and the following properties hold. (i)
$\mathfrak {m} B = \mathfrak {n} $
, and (ii)
$B/\mathfrak {m} B$
is a finite separable field extension of
$A/\mathfrak {m} $
. If, in addition,
$\psi $
is flat, then we say
$\psi $
is étale. (We say also that B is an unramified, respectively, étale extension of A.) Finally, a pointed étale neighborhood of A is an étale extension
$(A,\mathfrak {m} ) \rightarrow (B, \mathfrak {n} )$
inducing an isomorphism on residue fields.
3.2
The isomorphism classes of pointed étale neighborhoods of a local ring
$(A,\mathfrak {m} )$
form a direct system (see [Reference Iversen6, Chapter 3] for details). This implies that if
$A \rightarrow B$
and
$A \rightarrow C$
are pointed étale neighborhoods, then there is at most one homomorphism
$B \rightarrow C$
making the obvious diagram commute. Let
$\mathfrak {E} _P(A)$
be the set of isomorphism classes of pointed étale neighborhoods of A. Then
$A^h$
, the henselization of A, is
$$\begin{align*}A^h = \lim_{R \in \mathfrak{E} _P(A)}R. \end{align*}$$
3.3
Assume A is Cohen–Macaulay and essentially of finite type over a field K. Also assume
$\widehat {A}$
is a domain. By A.1 it follows that
$A^h$
is NOT of essentially finite type over K. Thus,
$\mathfrak {E} _P(A)$
contains infinitely many terms. So there exists at least one infinite chain in
$\mathfrak {E} _P(A)$
. Note we may assume this infinite chain starts at A
Let
$B \in \mathfrak {E} _P(A)$
. Then
$B^h = A^h$
, and we may assume
$\mathfrak {E} _P(B) \subseteq \mathfrak {E} _P(A)$
. By argument as before there exists an infinite chain in
$\mathfrak {E} _P(B)$
(and so in
$\mathfrak {E} _P(A)$
) starting at B.
3.4
Let A be an excellent Gorenstein isolated singularity. Let
$B, C \in \mathfrak {E} _P(A)$
and let
$\psi \colon B \rightarrow C$
be a morphism in
$\mathfrak {E} _P(A)$
. Note
$B, C$
are also an excellent Gorenstein isolated singularity. Note
$\psi $
is flat, see [Reference Iversen6, Chapter 1, 2.7], and so induces a functor
$\psi \colon \operatorname {\underline {CM}}(B) \rightarrow \operatorname {\underline {CM}}(C)$
. As B is an isolated singularity we have
$\operatorname {\underline {Hom}}_B(M, N)$
has finite length for every
$M, N \in \operatorname {\underline {CM}}(B)$
. It follows that
$\psi $
is fully faithful. Furthermore, as C is étale over B, we get that every MCM C-module M is a direct summand of
$N \otimes _B C$
, where N is a MCM B-module, see [Reference Leuschke and Wiegand11, 10.5, 10.7]. By Theorem 2.1, the natural map
$G(\psi ) \colon G(\operatorname {\underline {CM}}(B)) \rightarrow G(\operatorname {\underline {CM}}(C))$
is injective.
Lemma 3.1 Let
$(A,\mathfrak {m} )$
be a Gorenstein local ring. Then,
$\operatorname {\underline {CM}}(A)$
has weak cancellation (see 2.2)
Proof Suppose
$M, N, L$
are MCM A-modules such that
$M\oplus N \cong M \oplus L$
in
$\operatorname {\underline {CM}}(A)$
. Then as A-modules
$M \oplus N \oplus A^r \cong M \oplus L \oplus A^s$
for some
$r, s$
. By [Reference Leuschke and Wiegand11, 1.16] it follows that
$N \oplus A^r \cong L \oplus A^s$
as A-modules. So
$N \cong L$
in
$\operatorname {\underline {CM}}(A)$
.
3.5
(with hypotheses as in 3.4) If
$G(\psi ) \colon G(\operatorname {\underline {CM}}(B)) \rightarrow G(\operatorname {\underline {CM}}(C))$
is an isomorphism then by 2.4 and Lemma 3.1, we get that
$\psi \colon \operatorname {\underline {CM}}(B) \rightarrow \operatorname {\underline {CM}}(C)$
is an equivalence.
3.6
Let
$(A,\mathfrak {m} )$
be an excellent Gorenstein isolated singularity. Then
$A^h$
is also an excellent Gorenstein isolated singularity, see [Reference Leuschke and Wiegand11, 10.7]. In particular we have an equivalence of categories
$\operatorname {\underline {CM}}(A^h) \cong \operatorname {\underline {CM}}(\widehat {A})$
, by [Reference Keller, Murfet and Van den Bergh8, A.6]. If
$G(\widehat {A})$
is finitely generated abelian group then
$G(\operatorname {\underline {CM}}(\widehat {A})) = G(A)/([A])$
is finitely generated abelian group. It follows that
$G(\operatorname {\underline {CM}}(A^h))$
is finitely generated abelian group.
3.7
Let A be an excellent Gorenstein isolated singularity. Let
$B \in \mathfrak {E} _P(A) $
. Then the map
$f \colon B \rightarrow A^h$
is flat. It is readily verified that
$f \colon \operatorname {\underline {CM}}(B) \rightarrow \operatorname {\underline {CM}}(A^h)$
is fully faithful (use
$B^h = A^h$
). Also as
$f \colon B \rightarrow A^h$
is a direct limit of étale-extensions every MCM
$A^h$
-module M is a direct summand of
$N \otimes _B A^h$
where N is a MCM B-module, see [Reference Leuschke and Wiegand11, 10.5, 10.7]. By Theorem 2.1 the natural map
$f_B \colon G(\operatorname {\underline {CM}}(B)) \rightarrow G(\operatorname {\underline {CM}}(A^h))$
is injective. If
$f_B$
is an isomorphism then by 2.4 and Lemma 3.1, we get that
$f \colon \operatorname {\underline {CM}}(B) \rightarrow \operatorname {\underline {CM}}(A^h)$
is an equivalence.
4 Proof of Theorem 1.1
In this section, we give:
Proof of Theorem 1.1
Let
$(A,\mathfrak {m} )$
be a Gorenstein local ring of dimension
$d \geq 1$
. By 3.6 it follows that
$G(\operatorname {\underline {CM}}(A^h))$
is a finitely generated abelian group.
1. Let
$A_1 \rightarrow A_2 \rightarrow \cdots \rightarrow A_n \rightarrow \cdots $
be a chain in
$\mathfrak {E} _P(A)$
.
Claim There exists
$n_0$
such that
$$ \begin{align*}\operatorname{\underline{CM}}(A_n) \cong \operatorname{\underline{CM}}(A_{n+1}) \quad \text{for all }n \geq n_0.\end{align*} $$
Proof of Claim
Let
$R, S \in \mathfrak {E} _P(A)$
and let
$R \xrightarrow {\psi } S$
be a homomorphism in
$ \mathfrak {E} _P(A)$
. Note as
$\psi , f_R \colon R \rightarrow A^h$
and
$f_S \colon S \rightarrow A^h $
are flat we get a commutative diagram
Here
$G(\psi ), f_R, f_S$
are injective by 3.4 and 3.7. By 3.5 it follows that if
$G(\psi )$
is an isomorphism then
$\psi \colon \operatorname {\underline {CM}}(R) \rightarrow \operatorname {\underline {CM}}(S)$
is an equivalence. We have
$\operatorname {image} f_R \subseteq \operatorname {image} f_S$
. We note that if
$A_1 \rightarrow A_2 \rightarrow \cdots \rightarrow A_n \rightarrow \cdots $
is a chain in
$\mathfrak {E} _P(A)$
then we have an ascending chain of subgroups of
$G(\operatorname {\underline {CM}}(A^h))$
$$ \begin{align*}\operatorname{image} f_{A_1} \subseteq \operatorname{image} f_{A_2} \subseteq \cdots \subseteq \operatorname{image} f_{A_n} \subseteq \cdots\end{align*} $$
As
$G(\operatorname {\underline {CM}}(A^h)) $
is a finitely generated abelian group it follows that there exists m such that
$\operatorname {image} f_{A_n} = \operatorname {image} f_{A_m}$
for all
$n \geq m$
. Thus, the map
$G(\operatorname {\underline {CM}}(A_n)) \rightarrow G(\operatorname {\underline {CM}}(A_{n+1}))$
is an isomorphism for all
$n \geq m$
. It follows that
$$ \begin{align*}\operatorname{\underline{CM}}(A_n) \cong \operatorname{\underline{CM}}(A_{n+1}) \quad \text{for all }n \geq m.\end{align*} $$
2. We have
$G(\operatorname {\underline {CM}}(A^h))$
is a finitely generated abelian group, we may assume
$[N_1], \ldots , [N_s]$
generate it as an abelian group. By proof of [Reference Leuschke and Wiegand11, Theorem 10.7] we may assume there exists
$A_i \in \mathfrak {E} _P(A)$
and finitely generated
$A_i$
module
$N_i$
such that
$N_i = A^h\otimes _{A_i} M_i$
. As
$\theta _i \colon A_i \rightarrow A^h$
is flat (with zero-dimensional fiber) it follows from [Reference Matsumura12, 23.3] that
$N_i$
is a MCM
$A_i$
-module. As
$\mathfrak {E} _P(A)$
is directed there exists
$B \in \mathfrak {E} _P(A)$
such that there exists maps
$\gamma _i \colon A_i \rightarrow B$
in
$\mathfrak {E} _P(A)$
for all i. Set
$E_i = B \otimes _{A_i} N_i$
. Then as
$\gamma _i$
is flat (with zero-dimensional fiber),
$E_i$
is MCM B-module. Furthermore,
$E_i \otimes _B A^h \cong N_i$
. It follows that
$f_B \colon G(\operatorname {\underline {CM}}(B)) \rightarrow G(\operatorname {\underline {CM}}(A^h))$
is surjective. As it is injective, see 3.7, it follows that
$f_B$
is an isomorphism. So again by 3.7 we get
$\operatorname {\underline {CM}}(B) \cong \operatorname {\underline {CM}}(A^h)$
.
Now let
$B = B_1 \rightarrow B_2 \rightarrow \cdots \rightarrow B_n \rightarrow \cdots $
be an infinite chain in
$\mathfrak {E} _P(A)$
, see 3.3. By the commutative diagram above we get
$f_{B_n} \colon G(\operatorname {\underline {CM}}(B_n)) \rightarrow G(\operatorname {\underline {CM}}(A^h))$
is an isomorphism for all
$n \geq 1$
. So by 3.7 we get
$\operatorname {\underline {CM}}(B_n) \cong \operatorname {\underline {CM}}(A^h)$
for all
$n \geq 1$
.
5 Rings with
$G(\widehat {A})$
finitely generated abelian group
In this section we give many examples of Gorenstein rings with
$G(\widehat {A})$
finitely generated abelain group (and
$\widehat {A}$
a domain).
5.1
Let
$R = k[t^{a_1}, \cdots , t^{a_m}]$
be a symmetric numerical semi-group ring. Then R is Gorenstein, see [Reference Bruns and Herzog5, 4.4.8]. Let
$$ \begin{align*}A = R_{(t^{a_1}, \cdots, t^{a_m})}.\end{align*} $$
The completion of A is
$k[[t^{a_1}, \cdots , t^{a_m}]]$
which is a domain. It is well known that for any Noetherian ring T the Grothendieck group of T is generated by elements of the form
$[T/P]$
where P is a prime in T. As
$\widehat {A}$
is one-dimensional we get that
$G(\widehat {A})$
is a finitely generated abelian group.
5.2
Let
$R = k[X_1, \ldots , X_n]^G$
where G is a finite subgroup of
$SL_n(k)$
with
$|G|$
nonzero in k (and
$n \geq 2$
). Then R is Gorenstein, [Reference Watanabe14, Theorem 1]. Let A be the localization of R at its irrelevant maximal ideal. Note the completion of A is
$k[[X_1, \ldots , X_n]]^G$
which is a domain. By [Reference Auslander and Reiten1, Chapters 3, 3.4] we get that
$G(\widehat {A})$
is a finitely generated abelian group. Also see [Reference Auslander and Reiten1, Chapters 3, 5.4, and 5,5] for sufficient conditions which ensure A is an isolated singularity.
5.3
Let k be an algebraically closed field of characteristic not equal to
$2,3,5$
. Let
$S = k[X, Y, Z_1, \ldots , Z_{m}]_{(X, Y, Z_1, \ldots , Z_{m})}$
. Let
$A = S/(f)$
where f is one of the equations defining a simple singularity (see [Reference Yoshino15, 8.8]). Then
$\widehat {A}$
is a simple singularity. They are isolated singularities (and so are domains if
$\dim A \geq 2$
). The Grothendieck groups of all simple singularities are computed and they are finitely generated, see [Reference Yoshino15, 13.10].
5.4
Let k denote a field of characteristic not equal to
$2$
. Let
$R = k[X_1, \ldots , X_n]_{(X_1, \ldots , X_n)}$
. Let
$f = \sum _{i=1}^{n} a_i X_i^2$
be a quadratic form with
$a_i \neq 0$
for all i. Set
$A = R/(f)$
. Note
$\widehat {A} = \widehat {R}/(f)$
is an isolated singularity (and so a domain if
$n \geq 3$
), see [Reference Yoshino15, 14.2]. Thus A is also an isolated singularity. By [Reference Buchweitz, Eisenbud and Herzog3],
$\widehat {A}$
has finite representation type (i.e., there exists only finitely many indecomposable MCM
$\widehat {A}$
-modules up-to isomorphism); also see [Reference Yoshino15, 14.10]. It follows that
$G(\widehat {A})$
is a finitely generated abelian group, see [Reference Yoshino15, 13.2].
6 Proof of Theorem 1.3
In this section we give a proof of Theorem 1.3. We need a few preliminary results.
6.1
Assume A is a domain. Then we have an obvious surjective map
$\operatorname {rk} \colon G(A) \rightarrow \mathbb {Z} $
define by
$\operatorname {rk}(M) = \operatorname {rank} M$
. Note
$\operatorname {rk} [A] = 1$
Therefore, it splits. We have
$\ker \operatorname {rk} = G(A)/([A])$
. In particular if A is Gorenstein then
$G(\operatorname {\underline {CM}}(A)) = G(A)/([A]) = \ker \operatorname {rk}$
The first result we need
Proposition 6.1 Let
$(A, \mathfrak {m} )$
be an excellent Gorenstein local ring of dimension
$d \geq 2$
with
$k = A/\mathfrak {m} $
algebraically closed. Assume A is an isolated singularity. Then
$[k]$
in the Grothendieck group
$G(A)$
is zero.
Proof This is known when A is Henselian, see [Reference Yoshino15, 13.4]. Let X be the MCM approximation of k, i.e., we have an exact sequence
$ 0 \rightarrow Y \rightarrow X \rightarrow k \rightarrow 0$
where
$\operatorname {projdim}_A Y $
is finite. Then note that
$ 0 \rightarrow Y\otimes _A A^h \rightarrow X\otimes _A A^h \rightarrow k \rightarrow 0$
is a MCM approximation of k in
$A^h$
. It follows that
$[X\otimes _A A^h] = s[A^h]$
for some s. Therefore,
$[X\otimes _A A^h] = 0$
in
$G(\operatorname {\underline {CM}}(A^h))$
. As the map
$G(\operatorname {\underline {CM}}(A)) \rightarrow G(\operatorname {\underline {CM}}(A^h))$
is injective it follows that
$[X] = 0$
in
$G(\operatorname {\underline {CM}}(A))$
. So
$[X] = r[A]$
in
$G(A)$
. So
$[k] = m[A]$
in
$G(A)$
.
Take a one-dimensional CM module M over A. Let
$x \in \mathfrak {m} $
be M-regular. The exact sequence
$0 \rightarrow M \xrightarrow {x} M \rightarrow M/xM \rightarrow 0$
yields that the class
$[M/xM] = 0$
in
$G(A)$
. As
$M/xM$
has finite length we get
$[M/xM] = l[k]$
where l is the length of
$M/xM$
.
We have
$ml[A] = 0$
. So
$\operatorname {rk}(ml[A]) = 0$
. But
$\operatorname {rk}(ml[A]) = ml$
and l is nonzero. So
$m = 0$
. The result follows.
The next result is [Reference Yoshino15, 13.3]. He needlessly assumed that A is Henselian.
Theorem 6.2 Let
$(A,\mathfrak {m} , k)$
be a excellent normal local domain of dimension
$2$
, and suppose the class
$[k]$
of the residue field is zero in
$G(A)$
. Then
$G(A)$
is isomorphic to
$\mathbb {Z} \oplus Cl(A)$
where
$Cl(A)$
denotes the divisor class group of A. In particular,
$\ker \operatorname {rk} = Cl(A)$
.
Next we give proof of Theorem 1.3.
Proof Let P be the prime ideal
$(X, Z)$
. We note that P is a height one prime in A and
$P\widehat {A}$
is also prime. Note it is well known that
$P\widehat {A}$
is not principal. So P is not principal in A. Thus
$Cl(A) \neq 0$
. We have an injection
$\psi _A \colon G(\operatorname {\underline {CM}}(A)) \rightarrow G(\operatorname {\underline {CM}}(A^h))$
. We note that by 6.1, 6.1 and 6.2 we have
$G(\operatorname {\underline {CM}}(A)) = Cl(A) \neq 0$
. It is well known that
$Cl(A^h) = \mathbb {Z} /2$
. So by 6.2 we have
$G(\operatorname {\underline {CM}}(A^h)) = \mathbb {Z} /2$
. As
$Cl(A) \neq 0$
it follows that
$\psi _A $
is an isomorphism.
Let
$B \in \mathfrak {E} _P(A)$
and let
$\psi _B \colon G(\operatorname {\underline {CM}}(B)) \rightarrow G(\operatorname {\underline {CM}}(A^h))$
. Let
$f \colon G(\operatorname {\underline {CM}}(A)) \rightarrow G(\operatorname {\underline {CM}}(B))$
. Then as argued before we have
$\psi _A = \psi _B \circ f$
. In particular,
$\psi _B$
is surjective. As it is injective it follows that
$\psi _B $
is an isomorphism. So by 3.7 we get that
$\operatorname {\underline {CM}}(B) \rightarrow \operatorname {\underline {CM}}(A^h)$
is an equivalence.
A Appendix
In this section we prove the following result
Theorem A.1 Let
$(A,\mathfrak {m} )$
be a Cohen-Macaulay local ring of dimension
$d \geq 1$
. Assume A is of essentially of finite type over a field K. Also assume
$\widehat {A}$
, the completion of A is a domain. Then the henselization
$A^h$
of A is NOT of essentially finite type over K.
We believe Theorem A.1 is already known. However, we are unable to find a reference. As this result is crucial for us, we give a proof.
Lemma A.2 Let
$(A,\mathfrak {m} )$
be a Cohen-Macaulay local ring of dimension
$d \geq 1$
. Assume A is of essentially of finite type over a field K. Also assume that A is a domain. Then
$A = T_P$
for some affine domain T over K and P a prime in T.
Proof By assumption
$A = W^{-1}R$
where
$R = K[X_1, \ldots , X_n]/I$
and W is a multiplicatively closed ideal in R. The maximal ideal
$\mathfrak {m} $
of A is of the form
$W^{-1}P$
where P is a prime in R. It follows that
$A = R_P$
. Let
$P_1$
be a minimal prime of I such that
$\operatorname {height}(P/P_1) = \dim A$
. We have a a surjective map
$R \rightarrow T = R/P_1$
, which induces a surjection
$ \epsilon \colon A\rightarrow T_P$
. As
$\dim A = \dim T_P$
and A is a domain it follows that
$\epsilon $
is an isomorphism. The result follows.
We now give proof of Theorem A.1.
Proof By Lemma A.2 we may assume that
$A = T_P$
for some affine domain T and a prime P in T. Let
$S = K[X_1, \ldots ,X_n]$
be a Noether normalization of T. We can assume that
$P\cap T = (X_1, \ldots , X_d)$
(see [Reference Kunz10, Chapter 2, Theorem 3.1]). Let
$B = S_{(X_1, \ldots , X_d)}$
. Then we note that there is a local map
$B \rightarrow A$
with zero-dimensional fiber. This induces a local map
$B^h \rightarrow A^h$
(again with zero-dimensional fiber). We note that
$B^h$
is regular and
$A^h$
is Cohen-Macaulay. So by [Reference Matsumura12, 23.1]
$A^h$
is flat over
$B^h$
. As
$A^h$
is a subring of
$\widehat {A}$
it follows that
$A^h$
is a domain. Let
$L( E)$
be a quotient field of
$A^h(B^h))$
, respectively. Note we may consider
$B^h$
to be a subring of
$A^h$
, see [Reference Matsumura12, 7.5]. Then L contains E. Also E contains
$k(X_1,\ldots , X_n)$
the quotient field of B.
Let the characteristic of K be either zero or
$p> 0$
.
Claim For every prime
$q \neq p$
the field E contains
$\sqrt [q]{1 + X_1}$
.
Let
$f_q(t) = t^q - (1+X_1)$
. We note that
$X_1$
is in the maximal ideal
$\mathfrak {n} $
of
$B^h$
. Modulo
$\mathfrak {n} $
we get that
$$ \begin{align*}\overline{f_q(t)} = t^q - 1 = (t-1)(t^{q-1} + \cdots + t+ 1).\end{align*} $$
The polynomials
$t-1$
,
$t^{q-1} + \cdots + t+ 1$
are co-prime in
$(B^h/\mathfrak {n} )[t]$
. As
$B^h$
is Henselian, we get that there exists
$F(t), G(t)$
monic such that
$f_q(t) = F(t)G(t)$
with
$$ \begin{align*}\overline{F(t)} = t - 1 \quad \text{and} \quad \overline{G(t)} = t^{q-1} + \cdots + t+ 1.\end{align*} $$
Say
$F(t) = t - u$
. Then
$u \in B^h$
and
$u^q = 1+X_1$
. The claim follows.
Suppose if possible
$A^h$
is of finite type over K. Then by A.2 it follows that the quotient field L of
$A^h$
is a finite extension of
$K(Y_1, \ldots , Y_m)$
. Say
$Y_1, \ldots , Y_c$
is algebraically independent over
$k(X_1,\ldots , X_n)$
and
$Y_{c+1}, \ldots , Y_m$
are algebraic over
$V = k(X_1,\ldots , X_n, Y_1, \ldots , Y_c)$
. In particular, L is a finite extension of V. This is a contradiction as
$f_q(t) = t^q - (1+X_1)$
is irreducible over V for all
$q \neq p$
(use Eisenstein’s criterion and Gauss-Lemma).
Acknowledgement
I thank the referee for many pertinent comments.
Conflict of interest
The author have no conflict of interest to declare that are relevant to this article.
Data availability statement
My manuscript has no associated data.
