arXiv:1211.3445v2 [] 9 Sep 2013

K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE
HENRIK HOLM
Abstract. For a local Cohen–Macaulay ring R of finite CM-type, Yoshino has
applied methods of Auslander and Reiten to compute the Grothendieck group
K0 of the category mod R of finitely generated R-modules. For the same type of
rings, we compute in this paper the first Quillen K-group K1 (mod R). We also
describe the group homomorphism R∗ → K1 (mod R) induced by the inclusion
functor proj R → mod R and illustrate our results with concrete examples.

1. Introduction
Throughout this introduction, R denotes a commutative noetherian local Cohen–
Macaulay ring. The lower K-groups of R are known: K0 (R) ∼
= Z and K1 (R) ∼
= R∗ .
For n ∈ {0, 1} the classical K-group Kn (R) of the ring coincides with Quillen’s Kgroup Kn (proj R) of the exact category of finitely generated projective R-modules;
and if R is regular, then Quillen’s resolution theorem shows that the inclusion
functor proj R → mod R induces an isomorphism Kn (proj R) ∼
= Kn (mod R). If R is
non-regular, then these groups are usually not isomorphic. The groups Kn (mod R)
are often denoted Gn (R) and they are classical objects of study called the G-theory
of R. A celebrated result of Quillen is that G-theory is well-behaved under (Laurent)
polynomial extensions: Gn (R[t]) ∼
= Gn (R) and Gn (R[t, t−1 ]) ∼
= Gn (R) ⊕ Gn−1 (R).
Auslander and Reiten [4] and Butler [9] computed K0 (mod Λ) for an Artin algebra Λ of finite representation type. Using similar techniques, Yoshino [32] computed
K0 (mod R) in the case where R has finite (as opposed to tame or wild) CM-type:
Theorem (Yoshino [32, thm. (13.7)]). Assume that R is henselian and that it
has a dualizing module. If R has finite CM-type, then there is a group isomorphism,
K0 (mod R) ∼
= Coker Υ ,
where Υ : Zt → Zt+1 is the Auslander–Reiten homomorphism from (2.3).
We mention that Yoshino’s result is as much a contribution to algebraic Ktheory as it is to the representation theory of the category MCM R of maximal
Cohen–Macaulay R-modules. Indeed, the inclusion functor MCM R → mod R induces an isomorphism Kn (MCM R) ∼
= Kn (mod R) for every n. The theory of maximal Cohen–Macaulay modules, which originates from algebraic geometry and integral representations of finite groups, is a highly active area of research.
In this paper, we build upon results and techniques of Auslander and Reiten [4],
Bass [7], Lam [20], Leuschke [21], Quillen [24], Vaserstein [28, 29], and Yoshino [32]
2010 Mathematics Subject Classification. 13C14, 13D15, 19B28.
Key words and phrases. Auslander–Reiten sequence; Bass’ universal determinant group; finite
Cohen–Macaulay type; maximal Cohen–Macaulay module; Quillen’s K-theory.
1

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HENRIK HOLM

to compute the group K1 (mod R) when R has finite CM-type. Our main result is
Theorem (2.12); it asserts that there is an isomorphism,
∼ AutR (M )ab /Ξ ,
K1 (mod R) =
where M is any representation generator of the category of maximal Cohen–Macaulay R-modules and AutR (M )ab is the abelianization of its automorphism group.
The subgroup Ξ is more complicated to describe; it is determined by the Auslander–
Reiten sequences and defined in (2.10). Observe that in contrast to K0 (mod R), the
group K1 (mod R) is usually not finitely generated.
We also prove that if one writes M = R ⊕ M ′ , then the group homomorphism
∗ ∼
R = K1 (proj R) → K1 (mod R) induced by the inclusion functor proj R → mod R
can be identified with the map


r1R
0
∗
λ : R −→ AutR (M )ab /Ξ
given by
r 7−→
.
0
1M ′
The paper is organized as follows: In Section 2 we formulate our main result,
Theorem (2.12). This theorem is not proved until Section 8, and the intermediate Sections 3 (on the Gersten–Sherman transformation), 4 (on Auslander’s and
Reiten’s theory for coherent pairs), 5 (on Vaserstein’s result for semilocal rings), 6
(on certain equivalences of categories), and 7 (on Yoshino’s results for the abelian
category Y) prepare the ground.
In Sections 9 and 10 we apply our main theorem to compute the group K1 (mod R)
and the homomorphism λ : R∗ → K1 (mod R) in some concrete examples. E.g. for
the simple curve singularity R = k[[T 2 , T 3 ]] we obtain K1 (mod R) ∼
= k[[T ]]∗ and show
2
3 ∗
∗
that the homomorphism λ : k[[T , T ]] → k[[T ]] is the inclusion. It is well-known
that if R is artinian with residue field k, then one has K1 (mod R) ∼
= k ∗ . We apply
Theorem (2.12) to confirm this isomorphism for the ring R = k[X]/(X 2 ) of dual
numbers and to show that the homomorphism λ : R∗ → k ∗ is given by a+ bX 7→ a2 .
We end this introduction by mentioning a related preprint [23] of Navkal. Although the present work and the paper of Navkal have been written completely
independently (this fact is also pointet out in the latest version of [23]), there is a
significant overlap between the two manuscripts: Navkal’s main result [23, thm. 1.2]
is the existence of a long exact sequence involving the G-theory of the rings R and
EndR (M )op (where M is a particular representation generator of the category of
maximal Cohen–Macaulay R-modules) and the K-theory of certain division rings.
In Section 5 in loc. cit., Navkal applies his main result to give some description of
the group K1 (mod R) for the ring R = k[[T 2 , T 2n+1 ]] where n > 1. We point out
that the techniques used in this paper and in Navkal’s work are quite different.
2. Formulation of the Main Theorem
Let R be a commutative noetherian local Cohen–Macaulay ring. By mod R we
denote the abelian category of finitely generated R-modules. The exact categories
of finitely generated projective modules and of maximal Cohen–Macaulay modules
over R are written proj R and MCM R, respectively. The goal of this section is to
state our main Theorem (2.12); its proof is postponed to Section 8.
(2.1) Setup. Throughout this paper, (R, m, k) is a commutative noetherian local
Cohen–Macaulay ring satisfying the following assumptions.
(1) R is henselian.

K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE

3

(2) R admits a dualizing module.
(3) R has finite CM-type, that is, up to isomorphism, there are only finitely many
non-isomorphic indecomposable maximal Cohen–Macaulay R-modules.
Note that (1) and (2) hold if R is m-adically complete. Since R is henselian, the
category mod R is Krull–Schmidt by [32, prop. (1.18)]; this fact will be important
a number of times in this paper.
Set M0 = R and let M1 , . . . , Mt be a set of representatives for the isomorphism
classes of non-free indecomposable maximal Cohen–Macaulay R-modules. Let M
be any representation generator of MCM R, that is, a finitely generated R-module
such that addR M = MCM R (where addR M denotes the category of R-modules
that are isomorphic to a direct summand of some finite direct sum of copies of M ).
For example, M could be the square-free module
(2.1.1)

M = M0 ⊕ M1 ⊕ · · · ⊕ Mt .

We denote by E = EndR (M ) the endomorphism ring of M .
It follows from [32, thm. (4.22)] that R is an isolated singularity, and hence by
loc. cit. thm. (3.2) the category MCM R admits Auslander–Reiten sequences. Let
(2.1.2)

0 −→ τ (Mj ) −→ Xj −→ Mj −→ 0

(1 6 j 6 t)

be the Auslander–Reiten sequence in MCM R ending in Mj , where τ is the Auslander–Reiten translation.
(2.2) Remark. The one-dimensional Cohen–Macaulay rings of finite CM-type are
classified by Cimen [10, 11], Drozd and Roı̆ter [12], Green and Reiner [18], and
Wiegand [30, 31]. The two-dimensional complete Cohen–Macaulay rings of finite
CM-type that contains the complex numbers are classified by Auslander [2], Esnault
[14], and Herzog [19]. They are the invariant rings R = C[[X, Y ]]G where G is a nontrivial finite subgroup of GL2 (C). In this case, M = C[[X, Y ]] is a representation
generator for MCM R which, unlike the one in (2.1.1), need not be square-free.
(2.3) Definition. For each Auslander–Reiten sequence (2.1.2) we have
n
n
n
Xj ∼
= M0 0j ⊕ M1 1j ⊕ · · · ⊕ Mt tj

for uniquely determined n0j , n1j , . . . , ntj > 0. Consider the element,
τ (Mj ) + Mj − n0j M0 − n1j M1 − · · · − ntj Mt ,
in the free abelian group ZM0 ⊕ ZM1 ⊕ · · · ⊕ ZMt , and write this element as,
y0j M0 + y1j M1 + · · · + ytj Mt ,
where y0j , y1j , . . . , ytj ∈ Z. Define the Auslander–Reiten matrix Υ as the (t + 1) × t
matrix with entries in Z whose j’th column is (y0j , y1j , . . . , ytj ). When Υ is viewed
as a homomorphism of abelian groups Υ : Zt → Zt+1 (elements in Zt and Zt+1 are
viewed as column vectors), we refer to it as the Auslander–Reiten homomorphism.
(2.4) Example. Let R = C[[X, Y, Z]]/(X 3 + Y 4 + Z 2 ). Besides M0 = R there are
exactly t = 6 non-isomorphic indecomposable maximal Cohen–Macaulay modules,

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HENRIK HOLM

and the Auslander–Reiten sequences have the following form,
0 −→ M1 −→ M2 −→ M1 −→ 0
0 −→ M2 −→ M1 ⊕ M3 −→ M2 −→ 0
0 −→ M3 −→ M2 ⊕ M4 ⊕ M6 −→ M3 −→ 0
0 −→ M4 −→ M3 ⊕ M5 −→ M4 −→ 0
0 −→ M5 −→ M4 −→ M5 −→ 0
0 −→ M6 −→ M0 ⊕ M3 −→ M6 −→ 0 ;
see [32, (13.9)]. The 7×6 Auslander–Reiten matrix Υ

0
0
0
0
0
 2 −1 0
0
0

−1 2 −1 0
0

Υ=
0
−1
2
−1
0

0
0 −1 2 −1

0
0
0 −1 2
0
0 −1 0
0

is therefore given by

−1
0

0

−1
.
0

0
2

In this case, the Auslander–Reiten homomorphism Υ : Z6 → Z7 is clearly injective.
One hypothesis in our main result, Theorem (2.12) below, is that the Auslander–
Reiten homomorpism Υ over the ring R in question is injective. We are not aware
of an example where Υ is not injective. The following lemma covers the situation of
the rational double points, that is, the invariant rings R = k[[X, Y ]]G , where k is an
algebraically closed field of characteristic 0 and G is a non-trivial finite subgroup
of SL2 (k); see [5].

(2.5) Lemma. Assume that R is complete, integrally closed, non-regular, Gorenstein, of Krull dimension 2, and that the residue field k is algebraically closed. Then
the Auslander–Reiten homomorphism Υ is injective.
Proof. Let 1 6 j 6 t be given and consider the expression
τ (Mj ) + Mj − n0j M0 − n1j M1 − · · · − ntj Mt = y0j M0 + y1j M1 + · · · + ytj Mt
in the free abelian group ZM0 ⊕ ZM1 ⊕ · · · ⊕ ZMt , see Definition (2.3). Let Γ be
the Auslander–Reiten quiver of MCM R. We recall from [5, thm. 1] that the arrows
/ ◦ , and that collapsing each pair to an undirected edge
in Γ occur in pairs ◦ o
˜ Moreover, removing the vertex corresponding
gives an extended Dynkin diagram ∆.
to M0 = R and any incident edges gives a Dynkin graph ∆.
Now, Xj has a direct summand Mk if and only if there is an arrow Mk → Mj in Γ.
Also, the Auslander–Reiten translation τ satisfies τ (Mj ) = Mj by [5, proof of thm.
1]. Combined with the structure of the Auslander–Reiten quiver, this means that

if k = j ,
 2
˜ ,
ykj =
−1 if there is an edge Mk
Mj in ∆

0
otherwise .
Hence the t × t matrix Υ0 with (y1j , . . . , ytj ) as j’th column, where 1 6 j 6 t, is the
Cartan matrix of the Dynkin graph ∆; cf. [8, def. 4.5.3]. This matrix is invertible
by [15, exer. (21.18)]. Deleting the first row (y01 , . . . , y0t ) in the Auslander–Reiten
matrix Υ, we get the invertible matrix Υ0 , and consequently, Υ : Zt → Zt+1 determines an injective homomorphism.


K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE

5

For a group G we denote by Gab its abelianization, i.e. Gab = G/[G, G], where
[G, G] is the commutator subgroup of G.
We refer to the following as the tilde construction. It associates to every automorphism α : X → X of a maximal Cohen–Macaulay module X an automorphism
α̃ : M q → M q of the smallest power q of the representation generator M such that
X is a direct summand of M q .
(2.6) Construction. The chosen representation generator M for MCM R has the
form M = M0m0 ⊕ · · · ⊕ Mtmt for uniquely determined integers m0 , . . . , mt > 0. For
any module X = M0n0 ⊕ · · · ⊕ Mtnt in MCM R, we define natural numbers,
q = q(X) = min{p ∈ N | pmj > nj for all 0 6 j 6 t} , and
vj = vj (X) = qmj − nj > 0 ,
∼
=

and a module Y = M0v0 ⊕ · · · ⊕ Mtvt in MCM R. Let ψ : X ⊕ Y −→ M q be the Risomorphism that maps an element
((x0 , . . . , xt ), (y 0 , . . . , y t )) ∈ X ⊕ Y = (M0n0 ⊕ · · · ⊕ Mtnt ) ⊕ (M0v0 ⊕ · · · ⊕ Mtvt ) ,
v

n

where xj ∈ Mj j and y j ∈ Mj j , to the element
((z 01 , . . . , z t1 ), . . . , (z 0q , . . . , z tq )) ∈ M q = (M0m0 ⊕ · · · ⊕ Mtmt )q ,
m

qm

n +v

where z j1 , . . . , z jq ∈ Mj j are given by (z j1 , . . . , z jq ) = (xj , y j ) ∈ Mj j = Mj j j .
Now, given α in AutR (X), we define α̃ to be the uniquely determined element
in AutR (M q ) that makes the following diagram commutative,
X ⊕Y
α⊕1Y

ψ
∼
=

∼
= α̃

∼
=


X ⊕Y

/ Mq

∼
=
ψ


/ Mq .

The automorphism α̃ of M q has the form α̃ = (α̃ij ) for uniquely determined endomorphisms α̃ij of M , that is, α̃ij ∈ E = EndR (M ). Hence α̃ = (α̃ij ) can naturally
be viewed as an invertible q × q matrix with entries in E.
(2.7) Example. Let M = M0 ⊕ · · · ⊕ Mt and X = Mj . Then q = 1 and
Y = M0 ⊕ · · · ⊕ Mj−1 ⊕ Mj+1 ⊕ · · · ⊕ Mt .
The isomorphism ψ : X ⊕ Y → M maps (xj , (x0 , . . . , xj−1 , xj+1 , . . . , xt )) in X ⊕ Y
to (x0 , . . . , xj−1 , xj , xj+1 , . . . , xt ) in M . Therefore, for α ∈ AutR (X) = AutR (Mj ),
Construction (2.6) yields the following automorphism of M ,
α̃ = ψ(α ⊕ 1Y )ψ −1 = 1M0 ⊕ · · · ⊕ 1Mj−1 ⊕ α ⊕ 1Mj+1 ⊕ · · · ⊕ 1Mt ,
which is an invertible 1 × 1 matrix with entry in E = EndR (M ).
The following result on Auslander–Reiten sequences is quite standard. We provide a few proof details along with the appropriate references.
(2.8) Proposition. Let there be given Auslander–Reiten sequences in MCM R,
0 −→ τ (M ) −→ X −→ M −→ 0

and

0 −→ τ (M ′ ) −→ X ′ −→ M ′ −→ 0 .

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HENRIK HOLM

If α : M → M ′ is a homomorphism, then there exist homomorphisms β and γ that
make the following diagram commutative,
0

0

/ τ (M )
✤
✤γ
✤
/ τ (M ′ )

/X
✤
✤β
✤

/ X′

/M

/0

α


/ M′

/ 0.

Furthermore, if α is an isomorphism, then so are β and γ.
Proof. Write ρ : X → M and ρ′ : X ′ → M ′ . It suffices to prove the existence of β
such that ρ′ β = αρ, because then the existence of γ follows from diagram chasing.
As 0 → τ (M ′ ) → X ′ → M ′ → 0 is an Auslander–Reiten sequence, it suffices by
[32, lem. (2.9)] to show that αρ : X → M ′ is not a split epimorphism. Suppose that
there do exist τ : M ′ → X with αρτ = 1M ′ . Hence α is a split epimorphism. As
M is indecomposable, α must be an isomorphism. Thus ρτ α = α−1 (αρτ )α = 1M ,
which contradicts the fact that ρ is not a split epimorphism.
The fact that β and γ are isomorphisms if α is so follows from [32, lem. (2.4)]. 
The choice requested in the following construction is possible by Proposition (2.8).
(2.9) Construction. Choose for each 1 6 j 6 t and every α ∈ AutR (Mj ) elements
βj,α ∈ AutR (Xj ) and γj,α ∈ AutR (τ (Mj )) that make the next diagram commute,
0
(2.9.1)

/ τ (Mj )
∼
= γj,α

0


/ τ (Mj )

/ Xj

/ Mj

∼
= βj,α

∼
= α


/ Xj


/ Mj

/0

/ 0;

here the row(s) is the j’th Auslander–Reiten sequence (2.1.2).
As shown in Lemma (5.1), the endomorphism ring E = EndR (M ) of the chosen
representation generator M is semilocal, that is, E/J(E) is semisimple. Thus, if
the ground ring R, and hence also the endomorphism ring E, is an algebra over the
residue field k and char(k) 6= 2, then a result by Vaserstein [29, thm. 2] yields that
∗
C
the canonical homomorphism θE : Eab
→ KC
1 (E) is an isomorphism. Here K1 (E)
is the classical K1 -group of the ring E; see (3.1). Its inverse,
−1
∗
θE
= detE : KC
1 (E) −→ Eab = AutR (M )ab ,

is called the generalized determinant map. The details are discussed in Section 5.
We are now in a position to define the subgroup Ξ of AutR (M )ab that appears in
our main Theorem (2.12) below.
(2.10) Definition. Let (R, m, k) be a ring satisfying the hypotheses in Setup (2.1).
Assume, in addition, that R is an algebra over k and that one has char(k) 6= 2.
Define a subgroup Ξ of AutR (M )ab as follows.
– Choose for each 1 6 j 6 t and each α ∈ AutR (Mj ) elements βj,α ∈ AutR (Xj )
and γj,α ∈ AutR (τ (Mj )) as in Construction (2.9).
– Let α̃, β̃j,α , and γ̃j,α be the invertible matrices with entries in E obtained by
applying the tilde construction (2.6) to α, βj,α , and γj,α .

K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE

7

Let Ξ be the subgroup of AutR (M )ab generated by the elements
(detE α̃)(detE β̃j,α )−1 (detE γ̃j,α ) ,
where j ranges over {1, . . . , t} and α over AutR (Mj ).
A priori the definition of the group Ξ involves certain choices. However, it follows
from Proposition (8.8) that Ξ is actually independent of the choices made.
(2.11) Remark. In specific examples it is convenient to consider the simplest possible representation generator M = M0 ⊕ M1 ⊕ · · · ⊕ Mt . In this case, Example (2.7)
shows that α̃ and γ̃j,α are 1×1 matrices with entries in E, that is, α̃, γ̃j,α ∈ E ∗ ,
∗
and consequently detE α̃ = α̃ and detE γ̃j,α = γ̃j,α as elements in Eab
.
We are now in a position to state our main result.
(2.12) Theorem. Let (R, m, k) be a ring satisfying the hypotheses in Setup (2.1).
Assume that R is an algebra over its residue field k with char(k) 6= 2, and that the
Auslander–Reiten homomorphism Υ : Zt → Zt+1 from Definition (2.3) is injective.
Let M be any representation generator of MCM R. There is an isomorphism,
K1 (mod R) ∼
= AutR (M )ab /Ξ ,
where Ξ is the subgroup of AutR (M )ab given in Definition (2.10).
Furthermore, if inc : proj R → mod R is the inclusion functor and M = R ⊕ M ′ ,
then K1 (inc) : K1 (proj R) → K1 (mod R) may be identified with the homomorphism,


r1R
0
λ : R∗ −→ AutR (M )ab /Ξ
given by
r 7−→
.
0
1M ′
As mentioned in the introduction, the proof of Theorem (2.12) spans Sections 3
to 8. Applications and examples are presented in Sections 9 and 10. The interested
reader could go ahead and read Sections 9–10 right away, since these sections are
practically independent of 3–8.
3. The Gersten–Sherman Transformation
To prove Theorem (2.12), we need to compare and/or identify various K-groups.
The relevant definitions and properties of these K-groups are recalled below. The
(so-called) Gersten–Sherman transformation is our most valuable tool for comparing
K-groups, and the main part of this section is devoted to this natural transformation. Readers who are familiar with K-theory may skip this section altogether.
In the following, the Grothendieck group functor is denoted by G.
(3.1) Let A be a unital ring.
The classical K0 -group of A is defined as KC
0 (A) = G(proj A), that is, the Grothendieck group of the category of finitely generated projective A-modules.
The classical K1 -group of A is defined as KC
1 (A) = GL(A)ab , that is, the abelianization of the infinite (or stable) general linear group; see e.g. Bass [7, chap. V].
(3.2) Let C be any category. Its loop category ΩC is the category whose objects are
pairs (C, α) with C ∈ C and α ∈ AutC (C). A morphism (C, α) → (C ′ , α′ ) in ΩC is

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HENRIK HOLM

a commutative diagram in C,
C

ψ

∼
= α′

α ∼
=


C

/ C′

ψ


/ C ′.

(3.3) Let C be a skeletally small exact category. Its loop category ΩC is also skeletally small, and it inherits a natural exact structure from C. Bass’ K1 -group (also
called Bass’ universal determinant group) of C, which we denote by KB
1 (C), is the
Grothendieck group of ΩC, that is G(ΩC), modulo the subgroup generated by all
elements of the form
(C, α) + (C, β) − (C, αβ) ,
where C ∈ C and α, β ∈ AutC (C); see the book of Bass [7, chap. VIII§1] or Rosenberg [25, def. 3.1.6]. For (C, α) in ΩC we denote by [C, α] its image in KB
1 (C).
(3.4) For every C in C one has [C, 1C ] + [C, 1C ] = [C, 1C 1C ] = [C, 1C ] in KB
1 (C).
Consequently, [C, 1C ] is the neutral element in KB
(C).
1
(3.5) For a unital ring A there is by [25, thm. 3.1.7] a natural isomorphism,
∼
=

B
ηA : KC
1 (A) −→ K1 (proj A) .

The isomorphism ηA maps ξ ∈ GLn (A), to the class [An , ξ] ∈ KB
1 (proj A). Here
ξ is viewed as an automorphism of the row space An (a free left A-module), that
is, ξ acts by multiplication from the right.
−1
The inverse map ηA
acts as follows. Let [P, α] be in KB
1 (proj A). Choose any Q
in proj A and any isomorphism ψ : P ⊕ Q → An with n ∈ N. In KB
1 (proj A) one has
[P, α] = [P, α] + [Q, 1Q ] = [P ⊕ Q, α ⊕ 1Q ] = [An , ψ(α ⊕ 1Q )ψ −1 ] .
The automorphism ψ(α ⊕ 1Q )ψ −1 of (the row space) An can be identified with a
−1
matrix in β ∈ GLn (A). The action of ηA
on [P, α] is now β’s image in KC
1 (A).
(3.6) Quillen defines in [24] functors KQ
n from the category of skeletally small exact
categories to the category of abelian groups. More precisely, KQ
n (C) = πn+1 (BQC, 0)
where Q is Quillen’s Q-construction and B denotes the classifying space.
The functor KQ
0 is naturally isomorphic to the Grothendieck group functor G; see
∼ C
[24, §2 thm. 1]. For a ring A there is a natural isomorphism KQ
1 (proj A) = K1 (A);
see for example Srinivas [27, cor. (2.6) and thm. (5.1)].
Gersten sketches in [17, §5] the construction of a natural transformation ζ : KB
1 →
of functors on the category of skeletally small exact categories. The details of
this construction were later given by Sherman [26, §3], and for this reason we refer to
ζ as the Gersten–Sherman transformation 1. Examples due to Gersten and Murthy
[17, prop. 5.1 and 5.2] show that for a general skeletally small exact category C,
Q
the homomorphism ζC : KB
1 (C) → K1 (C) is neither injective nor surjective. For the
Q
exact category proj A, where A is a ring, it is known that KB
1 (proj A) and K1 (proj A)
are isomorphic, indeed, they are both isomorphic to the classical K-group KC
1 (A);
see (3.5) and (3.6). Therefore, a natural question arises: is ζproj A an isomorphism?
Sherman answers this question affirmatively in [26, pp. 231–232]; in fact, in loc. cit.
KQ
1

1 In the papers by Gersten [17] and Sherman [26], the functor KB is denoted by Kdet .
1
1

K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE

9

Theorem 3.3 it is proved that ζC is an isomorphism for every semisimple exact category, that is, an exact category in which every short exact sequence splits. We
note these results of Gersten and Sherman for later use.
Q
(3.7) Theorem. There exists a natural transformation ζ : KB
1 → K1 , which we call
the Gersten–Sherman transformation, of functors on the category of skeletally small
Q
exact categories such that ζproj A : KB
1 (proj A) → K1 (proj A) is an isomorphism for
every ring A.


We will also need the next result on the Gersten–Sherman transformation. Recall
that a length category is an abelian category in which every object has finite length.
(3.8) Theorem. If A is a skeletally small length category with only finitely many
Q
simple objects (up to isomorphism), then ζA : KB
1 (A) → K1 (A) is an isomorphism.
Proof. We begin with a general observation. Given skeletally small exact categories
C1 and C2 , there are exact projection functors pj : C1 × C2 → Cj (j = 1, 2). From the
“elementary properties” of Quillen’s K-groups listed in [24, §2], it follows that the
Q
Q
Q
Q
homomorphism (KQ
1 (p1 ), K1 (p2 )) : K1 (C1 × C2 ) → K1 (C1 ) ⊕ K1 (C2 ) is an isomorB
B
phism. A similar argument shows that (K1 (p1 ), K1 (p2 )) is an isomorphism. Since
Q
ζ : KB
1 → K1 is a natural transformation, it follows that ζC1 ×C2 is an isomorphism
if and only if ζC1 and ζC2 are isomorphisms.
Denote by Ass the full subcategory of A consisting of all semisimple objects. Note
that Ass is a Serre subcategory of A, and hence Ass is itself an abelian category.
Let i : Ass ֒→ A be the (exact) inclusion and consider the commutative diagram,
KB
1 (Ass )

KB
1 (i)
∼
=

ζAss


KQ
(A
ss )
1

/ KB
1 (A)
ζA

KQ
1 (i)
∼
=


/ KQ (A) .
1

Since A is a length category, Bass’ and Quillen’s devissage theorems [7, VIII§3 thm.
Q
(3.4)(a)] and [24, §5 thm. 4] show that KB
1 (i) and K1 (i) are isomorphisms. Hence,
it suffices to argue that ζAss is an isomorphism. By assumption there is a finite set
{S1 , . . . , Sn } of representatives of the isomorphism classes of simple objects in A.
Note that every object A in Ass has unique decomposition A = S1a1 ⊕· · ·⊕Snan where
a1 , . . . , an ∈ N0 ; we used here the assumption that A has finite length to conclude
that the cardinal numbers ai must be finite. Since one has HomA (Si , Sj ) = 0 for
i 6= j, it follows that there is an equivalence of abelian categories,
Ass ≃ (add S1 ) × · · · × (add Sn ) .
Consider the ring Di = EndA (Si )op . As Si is simple, Schur’s lemma gives that Di is
a division ring. It easy to see that the functor HomA (Si , −) : A → Mod Di induces
an equivalence add Si ≃ proj Di . By Theorem (3.7) the maps ζproj D1 , . . . , ζproj Dn are
isomorphisms, so it follows from the equivalence above, and the general observation

in the beginning af the proof, that ζAss is an isomorphism, as desired.
Note that in this section, superscripts “C” (for classical), “B” (for Bass), and
“Q” (for Quillen) have been used to distinguish between various K-groups. In the
rest of the paper, K-groups without superscripts refer to Quillen’s K-groups.

10

HENRIK HOLM

4. Coherent Pairs
We recall a few results and notions from the paper [4] by Auslander and Reiten
which are central in the proof of our main Theorem (2.12). Throughout this section,
A denotes a skeletally small additive category.
(4.1) Definition. A pseudo (or weak ) kernel of a morphism g : A → A′ in A is a
morphism f : A′′ → A in A such that gf = 0, and which satisfies that every diagram
in A as below can be completed (but not necessarily in a unique way).

′′

A

~⑤

⑤
f

⑤

B❇
❇❇
❇❇0
h
❇❇

/A
/ A′ .
g

⑤

We say that A has pseudo kernels if every morphism in A has a pseudo kernel.
(4.2) Observation. Let A be a full additive subcategory of an abelian category
M. An A-precover of an object M ∈ M is a morphism u : A → M with A ∈ A
with the property that for every morphism u′ : A′ → M with A′ ∈ A there exists
a (not necessarily unique) morphism v : A′ → A such that uv = u′ . Following [13,
def. 5.1.1] we say that A is precovering (or contravariantly finite) in M if every
object M ∈ M has an A-precover. In this case, A has pseudo kernels. Indeed, if
i : K → A is the kernel in M of g : A → A′ in A, and if f : A′′ → K is an A-precover
of K, then if : A′′ → A is a pseudo kernel of g.
(4.3) Definition. Let B be a full additive subcategory of A. Auslander and Reiten
[4] call (A, B) a coherent pair if A has pseudo kernels in the sense of Definition (4.1),
and B is precovering in A.
If (A, B) is a coherent pair then also B has pseudo kernels by [4, prop. 1.4(a)].
(4.4) Definition. Write Mod A for the abelian category of additive contravariant
functors A → Ab, where Ab is the category of abelian groups. Denote by mod A
the full subcategory of Mod A consisting of finitely presented functors.
(4.5) If the category A has pseudo kernels then mod A is abelian, and the inclusion
functor mod A → Mod A is exact, see [4, prop. 1.3].
If (A, B) is a coherent pair, see (4.3), then the exact restriction Mod A → Mod B
maps mod A to mod B by [4, prop. 1.4(b)]. In this case, there are functors,
(4.5.1)

i

r

Ker r −→ mod A −→ mod B ,

where r is the restriction and i the inclusion functor. The kernel of r, that is,
Ker r = {F ∈ mod A | F (B) = 0 for all B ∈ B } ,
is a Serre subcategory of the abelian category mod A. The quotient (mod A)/(Ker r),
in the sense of Gabriel [16], is equivalent to the category mod B, and the canonical
functor mod A → (mod A)/(Ker r) may be identified with r. These assertions are
proved in [4, prop. 1.5]. Therefore (4.5.1) induces by Quillen’s localization theorem

K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE

11

[24, §5 thm. 5] a long exact sequence of K-groups,

(4.5.2)

···

/ Kn (Ker r)

···

/ K0 (Ker r)

Kn (i)
K0 (i)

/ Kn (mod A)
/ K0 (mod A)

Kn (r)
K0 (r)

/ Kn (mod B)

/ ···

/ K0 (mod B)

/ 0.

5. Semilocal Rings
A ring A is semilocal if A/J(A) is semisimple. Here J(A) is the Jacobson radical
of A. If A is commutative then this definition is equivalent to A having only finitely
many maximal ideals; see Lam [20, prop. (20.2)].
(5.1) Lemma. Let R be a commutative noetherian semilocal ring, and let M 6= 0
be a finitely generated R-module. Then the ring EndR (M ) is semilocal.
Proof. As R is commutative and noetherian, EndR (M ) is a module-finite R-algebra.
Since R is semilocal, the assertion now follows from [20, prop. (20.6)].

(5.2) Denote by A∗ the group of units in a ring A, and let ϑA : A∗ → KC
1 (A) be the
composite of the group homomorphisms,
(5.2.1)

A∗ ∼
= GL1 (A) ֒→ GL(A) ։ GL(A)ab = KC
1 (A) .

Some authors refer to ϑA as the Whitehead determinant. If A is semilocal, then
ϑA is surjective by Bass [7, V§9 thm. (9.1)]. As the group KC
1 (A) is abelian one has
[A∗ , A∗ ] ⊆ Ker ϑA , and we write θA : A∗ab → KC
(A)
for
the
induced
homomorphism.
1
Vaserstein [28] showed that the inclusion [A∗ , A∗ ] ⊆ Ker ϑA is strict for the semilocal ring A = M2 (F2 ) where F2 is the field with two elements. In [28, thm. 3.6(a)]
it is shown that if A is semilocal, then Ker ϑA is the subgroup of A∗ generated by
elements of the form (1 + ab)(1 + ba)−1 where a, b ∈ A and 1 + ab ∈ A∗ .
If A is semilocal, that is, A/J(A) is semisimple, then by the Artin–Wedderburn
theorem there is an isomorphism of rings,
A/J(A) ∼
= Mn1 (D1 ) × · · · × Mnt (Dt ) ,
where D1 , . . . , Dt are division rings, and n1 , . . . , nt are natural numbers all of which
are uniquely determined by A. The next result is due to Vaserstein [29, thm. 2].
(5.3) Theorem. Let A be semilocal and write A/J(A) ∼
= Mn1 (D1 )× · · ·× Mnt (Dt ).
If none of the Mni (Di )’s is M2 (F2 ), and at most one of the Mni (Di )’s is M1 (F2 ) = F2
then one has Ker ϑA = [A∗ , A∗ ]. In particular, ϑA induces an isomorphism,


∼
=

θA : A∗ab −→ KC
1 (A) .

(5.4) Remark. Note that if A is a semilocal ring which is an algebra over a field k
with characteristic 6= 2, then the hypothesis in Theorem (5.3) is satisfied.
If A is a commutative semilocal ring, then Ker ϑA and the commutator subgroup
[A∗ , A∗ ] = {1} are identical, i.e. the surjective homomorphism ϑA = θA : A∗ → KC
1 (A)
is an isomorphism. Indeed, the determinant homomorphisms detn : GLn (A) → A∗
∗
induce a homomorphism detA : KC
1 (A) → A that evidently satisfies detA θA = 1A∗ .
−1
Since θA is surjective, it follows that θA is an isomorphism with θA
= detA .

12

HENRIK HOLM

(5.5) Definition. Let A be a ring for which the homomorphism θA : A∗ab → KC
1 (A)
from (5.2) is an isomorphism; for example, A could be a commutative semilocal ring
or a noncommutative semilocal ring satisfying the assumptions in Theorem (5.3).
−1
The inverse θA
is denoted by detA , and we call it the generalized determinant.
(5.6) Remark. Let ξ be an m × n and let χ be an n × p matrix with entries in a
ring A. Denote by “·” the product Mm×n (Aop ) × Mn×p (Aop ) → Mm×p (Aop ). Then
(ξ · χ)T = χT ξ T ,
where χT ξ T is computed using the product Mp×n (A) × Mn×m (A) → Mp×m (A).
Thus, transposition (−)T : GLn (Aop ) → GLn (A) is an anti-isomorphism (this is also
op
C
noted in [7, V§7]), which induces an isomorphism (−)T : KC
1 (A ) → K1 (A).
−1
(5.7) Lemma. Let A be a ring for which the generalized determinant detA = θA
exists; cf. Definition (5.5). For every invertible matrix ξ with entries in A one has
an equality detAop (ξ T ) = detA (ξ) in the abelian group (Aop )∗ab = A∗ab .

Proof. Clearly, there is a commutative diagram,
(Aop )∗ab

A∗ab

∼
= θAop

θA ∼
=


KC
(A)
1

∼
=
(−)T


op
/ KC
(A
),
1

−1
−1
T
It follows that one has θA
= θA
, that is, detAop ◦ (−)T = detA .
op ◦ (−)



6. Some Useful Functors
Throughout this section, A is a ring and M is a fixed left A-module. We denote by
E = EndA (M ) the endomorphism ring of M . Note that M = A,E M has a natural
left-A-left-E–bimodule structure.
(6.1) There is a pair of adjoint functors,
HomA (M,−)

/

Mod A o

Mod(E op ) .

−⊗E M

It is easily seen that they restrict to a pair of quasi-inverse equivalences,
HomA (M,−)

addA M o

≃
−⊗E M

/

proj(E op ) .

Auslander referred to this phenomenon as projectivization; see [6, I§2].
Let F ∈ Mod(addA M ), that is, F : addA M → Ab is a contravariant additive functor, see Definition (4.4). The compatible E-module structure on the given A-module
M induces an E op -module structure on the abelian group F M which is given by
zα = (F α)(z) for α ∈ E and z ∈ F M .
(6.2) Proposition. There are quasi-inverse equivalences of abelian categories,
Mod(addA M ) o

eM
≃
fM

/

Mod(E op ) ,

K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE

13

where eM (evaluation) and fM (functorfication) are defined as follows,
eM (F ) = F M

and

fM (Z) = Z ⊗E HomA (−, M )|addA M ,

for F in Mod(addA M ) and Z in Mod(E op ). They restrict to quasi-inverse equivalences between categories of finitely presented objects,
mod(addA M ) o

eM
≃
fM

/

mod(E op ) .

Proof. For Z in Mod(E op ) the canonical isomorphism
∼
=

Z −→ Z ⊗E E = Z ⊗E HomA (M, M ) = eM fM (Z)
is natural in Z. Thus, the functors idMod(E op ) and eM fM are naturally isomorphic.
For F in Mod(addA M ) there is a natural transformation,
(6.2.1)

δ

fM eM (F ) = F M ⊗E HomA (−, M )|addA M −→ F ;

for X in addA M the homomorphism δX : F M ⊗E HomA (X, M ) → F X is given by
z ⊗ ψ 7→ (F ψ)(z). Note that δM is an isomorphism
as it may be identified with
∼
=
the canonical isomorphism F M ⊗E E E −→ F M in Ab. As the functors in (6.2.1)
are additive, it follows that δX is an isomorphism for every X ∈ addA M , that is,
δ is a natural isomorphism. Since (6.2.1) is natural in F , the functors fM eM and
idMod(addA M) are naturally isomorphic.
It is straightforward to verify that the functors eM and fM map finitely presented
objects to finitely presented objects.

(6.3) Observation. In the case M = A one has E = EndA (M ) = Aop , and therefore Proposition (6.2) yields an equivalence fA : mod A → mod(proj A) given by
X 7−→ X ⊗Aop HomA (−, A)|proj A .
It is easily seen that the functor fA is naturally isomorphic to the functor given by
X 7−→ HomA (−, X)|proj A .
We will usually identify fA with this functor.
(6.4) Definition. The functor yM : addA M → mod(addA M ) which for X ∈ addA M
is given by yM (X) = HomA (−, X)|addA M is called the Yoneda functor.
Let A be a full additive subcategory of an abelian category M. If A is closed
under extensions in M, then A has a natural induced exact structure. However,
one can always equip A with the trivial exact structure. In this structure, the
“exact sequences” (somtimes called conflations) are only the split exact ones. When
viewing A as an exact category with the trivial exact structure, we denote it A0 .
(6.5) Lemma. Assume that A is commutative and noetherian and let M ∈ mod A.
Set E = EndA (M ) and assume that E op has finite global dimension. For the exact Yoneda functor yM : (addA M )0 → mod(addA M ), see (6.4), the homomorphisms
Kn (yM ), where n > 0, and KB
1 (yM ) are isomorphisms.

14

HENRIK HOLM

Proof. By application of Kn to the commutative diagram,
(addA M )0

HomA (M,−)
≃

/ proj(E op )

≃
eM


/ mod(E op ) ,

yM

inc


mod(addA M )

it follows that Kn (yM ) is an isomorphism if and only if Kn (inc) is an isomorphism.
The latter holds by Quillen’s resolution theorem [24, §4 thm. 3], since E op has finite
global dimension. A similar argument shows that KB
1 (yM ) is an isomorphism. This
time one needs to apply Bass’ resolution theorem; see [7, VIII§4 thm. (4.6)].

Since K0 may be identified with the Grothendieck group functor, cf. (3.6), the
following result is well-known. In any case, it is straightforward to verify.
(6.6) Lemma. Assume that mod A is Krull–Schmidt. Let N = N1n1 ⊕ · · · ⊕ Nsns
be a finitely generated A-module, where N1 , . . . , Ns are non-isomorphic indecomposable A-modules and n1 , . . . , ns > 0. The homomorphism of abelian groups,
ψN : ZN1 ⊕ · · · ⊕ ZNs −→ K0 ((addA N )0 ) ,
given by Nj 7→ [Nj ], is an isomorphism.



7. The Abelian Category Y
By the assumptions in Setup (2.1), the ground ring R has a dualizing module.
It follows from Auslander and Buchweitz [3, thm. A] that MCM R is precovering
in mod R. Actually, in our case MCM R equals addR M for some finitely generated
R-module M (a representation generator), and it is easily seen that every category
of this form is precovering in mod R. By Observation (4.2) we have a coherent pair
(MCM R, proj R), which by (4.5) yields a Gabriel localization sequence,
(7.0.1)

i

r

Y = Ker r −→ mod(MCM R) −→ mod(proj R) .

Here r is the restriction functor, Y = Ker r, and i is the inclusion. Since an additive
functor vanishes on proj R if and only if it vanishes on R, one has
Y = {F ∈ mod(MCM R) | F (R) = 0} .
The following two results about the abelian category Y are due to Yoshino. The first
result is [32, (13.7.4)]; the second is (proofs of) [32, lem. (4.12) and prop. (4.13)].
(7.1) Theorem. Every object in Y has finite length, i.e. Y is a length category. 
(7.2) Theorem. Consider for 1 6 j 6 t the Auslander–Reiten sequence (2.1.2) ending in Mj . The functor Fj , defined by the following exact sequence in mod(MCM R),
0 −→ HomR (−, τ (Mj )) −→ HomR (−, Xj ) −→ HomR (−, Mj ) −→ Fj −→ 0 ,
is a simple object in Y. Conversely, every simple functor in Y is naturally isomorphic
to Fj for some 1 6 j 6 t.

(7.3) Proposition. Let i : Y → mod(MCM R) be the inclusion functor from (7.0.1)
and let Υ : Zt → Zt+1 be the Auslander–Reiten homomorphism; see Definition (2.3).
The homomorphisms K0 (i) and Υ are isomorphic.

K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE

15

Proof. We claim that the following diagram of abelian groups is commutative,
ZM1 ⊕ · · · ⊕ ZMt

Υ

/ ZM0 ⊕ ZM1 ⊕ · · · ⊕ ZMt
∼
= ψM


K0 ((MCM R)0 )

ϕ ∼
=

∼
= K0 (yM )


K0 (Y)


/ K0 (mod(MCM R)) .

K0 (i)

The homomorphism ϕ is defined by Mj 7→ [Fj ] where Fj ∈ Y is described in (7.2).
From Theorems (7.1) and (7.2) and the proof of Rosenberg [25, thm. 3.1.8(1)] (or
the proof of Theorem (3.8)), it follows that ϕ is an isomorphism. The module M
is a representation generator for MCM R, see (2.1), and ψM is the isomorphism
given in Lemma (6.6). Finally, yM is the Yoneda functor from Definition (6.4).
By Leuschke [21, thm. 6] the ring E op , where E = EndR (M ), has finite global
dimension, and thus Lemma (6.5) implies that K0 (yM ) is an isomorphism.
From the definitions of the relevant homomorphims, it is straightforward to see
that the diagram is commutative; indeed, both K0 (i)ϕ and K0 (yM )ψM Υ map a
generator Mj to the element [Fj ] ∈ K0 (mod(MCM R)).

8. Proof of the Main Theorem
Throughout this section, we fix the notation in Setup (2.1). Thus, R is a commutative noetherian local Cohen–Macaulay ring satisfying conditions (2.1)(1)–(3),
M is any representation generator of MCM R, and E is its endomorphism ring.
We shall frequently make use of the Gabriel localization sequence (7.0.1), and i
and r always denote the inclusion and the restriction functor in this sequence.
(8.1) Remark. Let C be an exact category. As in the paragraph preceding Lemma (6.5), we denote by C0 the category C equipped with the trivial exact structure.
Note that the identity functor idC : C0 → C is exact and the induced homomorphism

B
B
B
KB
1 (idC ) : K1 (C0 ) → K1 (C) is surjective, indeed, one has K1 (idC ) [C, α] = [C, α].

(8.2) Lemma. Consider the restriction functor r : mod(MCM R) → mod(proj R) and
identity functor idMCM R : (MCM R)0 → MCM R. The homomorphisms KB
1 (r) and
B
(id
)
are
isomorphic,
in
particular,
K
(r)
is
surjective
by
Remark
(8.1).
KB
MCM R
1
1

Proof. Consider the commutative diagram of exact categories and exact functors,
(MCM R)0

idMCM R

/ MCM R
j


mod R

yM

≃ fR


mod(MCM R)

r


/ mod(proj R) ,

where yM is the Yoneda functor from Definition (6.4), j is the inclusion, and fR is
the equivalence from Observation (6.3). We will prove the lemma by arguing that
the vertical functors induce isomorphisms on the level of KB
1.
The ring E op has finite global dimension by Leuschke [21, thm. 6], and hence
Lemma (6.5) gives that that KB
1 (yM ) is an isomorphism. Since fR is an equivalence,

16

HENRIK HOLM

B
KB
1 (fR ) is obviously an isomorphism. To argue that K1 (j) is an isomorphism,
we apply Bass’ resolution theorem [25, thm. 3.1.14]. We must check that the
subcategory MCM R of mod R satisfies conditions (1)–(3) in loc. cit. Condition (1)
follows as MCM R is precovering in mod R. As R is Cohen–Macaulay, every module
in mod R has a resolution of finite length by modules in MCM R, see [32, prop. (1.4)];
thus condition (2) holds. Condition (3) requires that MCM R is closed under kernels
of epimorphisms; this is well-known from e.g. [32, prop. (1.3)].


Next we show some results on the Gersten–Sherman transformation; see Sect. 3.
(8.3) Lemma. ζC : KB
1 (C) → K1 (C) is an isomorphism for C = mod(MCM R).
Proof. As ζ is a natural transformation, there is a commutative diagram,
op
KB
1 (proj(E ))

KB
1 (inc)

ζproj(E op )

op
/ KB
1 (mod(E ))

KB
1 (fM )

/ KB
1 (mod(MCM R))

ζmod(E op )


K1 (proj(E op ))

K1 (inc)


/ K1 (mod(E op ))

ζmod(MCM R)
K1 (fM )


/ K1 (mod(MCM R)) ,

where fM : mod(E op ) → mod(MCM R) is the equivalence from Proposition (6.2)
and inc is the inclusion of proj(E op ) into mod(E op ).
From Leuschke [21, thm. 6], the noetherian ring E op has finite global dimension.
Hence Bass’ and Quillen’s resolution theorems, [7, VIII§4 thm. (4.6)] (see also Rosenberg [25, thm. 3.1.14]) and [24, §4 thm. 3], imply that KB
1 (inc) and K1 (inc) are
isomorphisms. Since fM is an equivalence, KB
(f
)
and
K
(f
M
1 M ) are isomorphisms
1
as well. Consequently, ζmod(MCM R) is an isomorphism if and only if ζproj(E op ) is an
isomorphism, and the latter holds by Theorem (3.7).

The goal is to compute Quillen’s K-group K1 (mod R) for the ring R in question.
For our proof of Theorem (2.12), it is crucial that this group can be naturally identified with Bass’ K-group KB
1 (mod R). To put Proposition (8.4) in perspective, we
remind the reader that the Gersten–Sherman transformation ζmod A is not surjective
for the ring A = ZC2 ; see [17, prop. 5.1].
(8.4) Proposition. If the Auslander–Reiten homomorphism from Definition (2.3)
is injective, then the following assertions hold:
(a) The homomorphism ζmod R : KB
1 (mod R) → K1 (mod R) is an isomorphism.
(b) There is an exact sequence,
KB
1 (Y)

KB
1 (i)

/ KB (mod(MCM R))
1

KB
1 (r)

/ KB (mod(proj R))
1

/0.

Proof. The Gabriel localization sequence (7.0.1) induces by (4.5) a long exact sequence of Quillen K-groups,
···

/ K1 (Y) K1 (i)/ K1 (mod(MCM R)) K1 (r)/ K1 (mod(proj R))

/ K0 (Y) K0 (i)/ · · ·

By Proposition (7.3), we may identify K0 (i) with the Auslander–Reiten homomorphism, which is assumed to be injective. Therefore, the bottom row in the following

K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE

17

commutative diagram of abelian groups is exact,
KB
1 (Y)

KB
1 (i)

/ KB (mod(proj R))
1

∼
= ζmod(MCM R)

∼
= ζY


K1 (Y)

KB
1 (r)

/ KB (mod(MCM R))
1

K1 (i)

/0

ζmod(proj R)


/ K1 (mod(MCM R))

K1 (r)


/ K1 (mod(proj R))

/ 0.

The vertical homomorphisms are given by the Gersten–Sherman transformation;
see Section 3. It follows from Theorems (7.1) and (7.2) that Y is a length category
with only finitely many simple objects; thus ζY is an isomorphism by Theorem (3.8).
And ζmod(MCM R) is an isomorphism by Lemma (8.3). Since ri = 0, it follows that
B
B
B
KB
1 (r)K1 (i) = 0 holds, and a diagram chase now shows that Im K1 (i) = Ker K1 (r).
B
Furthermore K1 (r) is surjective by Lemma (8.2). This proves part (b).
The Five Lemma now implies that ζmod(proj R) is an isomorphism. Since the
category mod(proj R) is equivalent to mod R, see Observation (6.3), it follows that
ζmod R is an isomorphism as well. This proves (a).

We will also need the following classical notion.
(8.5) Definition. Let M be an abelian category, and let M be an object in M. A
projective cover of M is an epimorphism ε : P ։ M in M, where P is projective,
such that every endomorphism α : P → P satisfying εα = ε is an automorphism.
(8.6) Lemma. Let there be given a commutative diagram,
P

ε

// M

ε


// M

ϕ

α


P

in an abelian category M, where ε : P ։ M is a projective cover of M . If ϕ is an
automorphism, then α is an automorphism.
Proof. As P is projective and ε is an epimorphism, there exists β : P → P such
that εβ = ϕ−1 ε. By assumption one has εα = ϕε. Hence εαβ = ϕεβ = ϕϕ−1 ε = ε,
and similarly, εβα = ε. As ε is a projective cover, we conclude that αβ and βα are
automorphisms of P , and thus α must be an automorphism.

The following lemma explains the point of the tilde construction (2.6).
op
B
op
(8.7) Lemma. Consider the isomorphism ηE op : KC
1 (E ) → K1 (proj(E )) in (3.5).
Let X ∈ MCM R and α ∈ AutR (X) be given, and α̃ be the invertible matrix with
entries in E obtained by applying Construction (2.6) to α. There is an equality,


ηE op (α̃T ) = HomR (M, X), HomR (M, α) .
∼
=

Proof. Write (M, −) for HomR (M, −), and let ψ : X ⊕ Y −→ M q be as in Construction (2.6). The R-module isomorphism ψ induces an isomorphism of E op -modules,
(M, X) ⊕ (M, Y ) = (M, X ⊕ Y )

(M,ψ)
∼
=

/ (M, M q ) ∼
= Eq .

Consider the automorphism of the free E op -module E q given by

(M, ψ) (M, α) ⊕ 1(M,Y ) (M, ψ)−1 = (M, ψ(α ⊕ 1Y )ψ −1 ) = (M, α̃) .

18

HENRIK HOLM

We view elements in the R-module M q as columns and elements in E q as rows. The
isomorphism E q ∼
= (M, M q ) identifies a row vector β = (β1 , . . . , βq ) ∈ E q with the
T
R-linear map β : M → M q whose coordinate functions are β1 , . . . , βq . The coordinate functions of (M, α̃)(β T ) = α̃ ◦ β T are the entries in the column α̃β T , where
the matrix product used is Mq×q (E) × Mq×1 (E) → Mq×1 (E). Thus, the action of
(M, α̃) on a row β ∈ E q is the row (α̃β T )T ∈ E q . In view of Remark (5.6) one has
(α̃β T )T = β · α̃T , where “·” is the product M1×q (E op ) × Mq×q (E op ) → M1×q (E op ).
Consequently, over the ring E op , the automorphism (M, α̃) of the E op -module E q
acts on row vectors by multiplication with α̃T from the right. These arguments
−1
T
show that ηE

op applied to [(M, X), (M, α)] is α̃ ; see (3.5).
(8.8) Proposition. Suppose, in addition to the blanket assumptions for this section, that R is an algebra over its residue field k and that char(k) 6= 2. Then there
is a group isomorphism,
∼
=

σ : AutR (M )ab −→ KB
1 (mod(MCM R)) ,
given by



α 7−→ HomR (−, M )|MCM R , HomR (−, α)|MCM R .

Furthermore, there is an equality,

σ(Ξ) = Im KB
1 (i) .
Here Ξ is the subgroup of AutR (M )ab given in (2.10), and i : Y → mod(MCM R) is
the inclusion functor from the Gabriel localization sequence (7.0.1).
Proof. We define σ to be the composite of the following isomorphisms,
∗
AutR (M )ab = Eab
= (E op )∗ab

(8.8.1)

θE op
∼
=
ηE op
∼
=
KB
1 ()
∼
=
KB
1 (fM )
∼
=

/ KC (E op )
1
/ KB (proj(E op ))
1
op
/ KB
1 (mod(E ))

/ KB
1 (mod(MCM R)) .

The ring E, and hence also its opposite ring E op , is semilocal by Lemma (5.1).
By assumption, R is a k-algebra, and hence so is E op . Thus, in view of Remark (5.4)
and the assumption char(k) 6= 2, we get the isomorphism θE op from Theorem (5.3).
op
It maps α ∈ AutR (M )ab to the image of the 1×1 matrix (α) ∈ GL(E op ) in KC
1 (E ).
op
The isomorphism ηE op is described in (3.5); it maps ξ ∈ GLn (E ) to the class
op
[(EE )n , ξ] ∈ KB
1 (proj(E )).
The third map in (8.8.1) is induced by the inclusion  : proj(E op ) → mod(E op ).
By Leuschke [21, thm. 6] the noetherian ring E op has finite global dimension and
hence Bass’ resolution theorem [7, VIII§4 thm. (4.6)], or Rosenberg [25, thm. 3.1.14],
B
op
implies that KB
1 () is an isomorphism. It maps an element [P, α] ∈ K1 (proj(E ))
B
op
to [P, α] ∈ K1 (mod(E )).
The fourth and last isomorphism KB
1 (fM ) in (8.8.1) is induced by the equivalence
fM : mod(E op ) → mod(MCM R) from Proposition (6.2).
Thus, σ is an isomorphism that maps an element α ∈ AutR (M )ab to the class


EE ⊗E HomR (−, M )|MCM R , (α ·) ⊗E HomR (−, M )|MCM R ,

K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE

19

which is evidently the same as the class


HomR (−, M )|MCM R , HomR (−, α)|MCM R .

It remains to show the equality σ(Ξ) = Im KB
1 (i). By the definition (8.8.1) of σ
B
−1
op
(Im KB
θ
this is tantamount to showing that KB
()η
E op (Ξ) = K1 (fM )
E
1 (i)). As
1
B
eM is a quasi-inverse of fM , see Proposition (6.2), we have K1 (fM )−1 = KB
1 (eM ),
and hence we need to show the equality
(8.8.2)

B
B
KB
1 ()ηE op θE op (Ξ) = K1 (eM )(Im K1 (i)) .

By Definition (2.10), the group Ξ is generated by all elements of the form
∗
ξj,α := (detE α̃)(detE β̃j,α )−1 (detE γ̃j,α ) ∈ Eab

for j ∈ {1, . . . , t} and α ∈ AutR (Mj ); here βj,α ∈ AutR (Xj ) and γj,α ∈ AutR (τ (Mj ))
are choices of automorphisms such that the diagram (2.9.1) is commutative. It
follows from Lemma (5.7) that
T
T −1
) ∈ (E op )∗ab .
) (detE op γ̃j,α
ξj,α = (detE op α̃T )(detE op β̃j,α

By Definition (5.5) the homomorphism detE op is the inverse of θE op , and consequently the group θE op (Ξ) is generated by the elements
T −1 T
op
′
) γ̃j,α ∈ KC
ξj,α
:= θE op (ξj,α ) = α̃T (β̃j,α
1 (E ) .
op
′
′′
) ∈ KB
:= ηE op (ξj,α
Thus ηE op θE op (Ξ) is generated by the elements ξj,α
1 (proj(E )),
and it follows from Lemma (8.7) that

 

′′
ξj,α
= HomR (M, Mj ), HomR (M, α) − HomR (M, Xj ), HomR (M, βj,α )


+ HomR (M, τ (Mj )), HomR (M, γj,α ) .

Thus, the group KB
1 ()ηE op θE op (Ξ) on the left-hand side in (8.8.2) is generated by
′′
B
′′
′′
the elements KB
1 ()(ξj,α ). Note that K1 ()(ξj,α ) is nothing but ξj,α viewed as an
B
op
element in K1 (mod(E )). We have reached the following conclusion:
′′
The group KB
1 ()ηE op θE op (Ξ) is generated by the elements ξj,α ,
where j ranges over {1, . . . , t} and α over all automorphisms of Mj .

B
To give a useful set of generators of the group KB
1 (eM )(Im K1 (i)) on the righthand side in (8.8.2), recall from Theorems (7.1) and (7.2) that every element in Y
has finite length and that the simple objects in Y are, up to isomorphism, exactly
the functors F1 , . . . , Ft . Thus, by [25, (proof of) thm. 3.1.8(2)] the group KB
1 (Y) is
generated by all elements of the form [Fj , ϕ], where j ∈ {1, . . . , t} and ϕ is an automorphism of Fj . It follows that the group Im KB
1 (i) is generated by the elements
B
KB
(i)([F
,
ϕ]).
Note
that
K
(i)([F
,
ϕ])
is
nothing
but [Fj , ϕ] viewed as an element
j
j
1
1
(mod(MCM
R)).
By
definition
of
the
functor
eM , see Prop. (6.2), one has
in KB
1

λj,ϕ := KB
1 (eM )([Fj , ϕ]) = [Fj M, ϕM ] .
We have reached the following conclusion:
B
The group KB
1 (eM )(Im K1 (i)) is generated by the elements λj,ϕ ,
where j ranges over {1, . . . , t} and ϕ over all automorphisms of Fj .
′′
With the descriptions of the generators ξj,α
and λj,ϕ at hand, we are now in a
position to prove the identity (8.8.2).

20

HENRIK HOLM

′′
Consider an arbitrary generator ξj,α
in the group KB
1 ()ηE op θE op (Ξ). Recall from
Theorem (7.2) that there is an exact sequence in mod(MCM R),

0 −→ HomR (−, τ (Mj )) −→ HomR (−, Xj ) −→ HomR (−, Mj ) −→ Fj −→ 0 .
Thus, the commutative diagram (2.9.1) in MCM R induces a commutative diagram
in mod(MCM R) with exact row(s),
0

/ HomR (−, τ (Mj ))

/ HomR (−, Xj )

∼
= HomR (−,γj,α )

0

/ HomR (−, Mj )

∼
= HomR (−,βj,α )


/ HomR (−, τ (Mj ))


/ HomR (−, Xj )

∼
= HomR (−,α)


/ HomR (−, Mj )

/ Fj
✤
✤
∼
= ϕ
✤
/ Fj

/0

/0,

where ϕ is the uniquely determined natural endotransformation of Fj that makes
this diagram commutative. Note that ϕ is an automorphism by the Five Lemma,
and thus [Fj , ϕ] is a well-defined element in KB
1 (mod(MCM R)). The diagram above
is an exact sequence in the loop category Ω(mod(MCM R)), see (3.2) and (3.3), so
in the group KB
1 (mod(MCM R)) there is an equality:

 

[Fj , ϕ] = HomR (−, Mj ), HomR (−, α) − HomR (−, Xj ), HomR (−, βj,α )


+ HomR (−, τ (Mj )), HomR (−, γj,α ) .

′′
Applying the homomorphism KB
1 (eM ) to this equality, we get λj,ϕ = ξj,α . These
′′
arguments show that every generator ξj,α has the form λj,ϕ for some ϕ, and hence
the inclusion ”⊆” in (8.8.2) is established.
B
Conversely, consider an arbitrary generator λj,ϕ in the group KB
1 (eM )(Im K1 (i)).
As the category MCM R is a Krull–Schmidt variety in the sense of Auslander [1, II,
§2], it follows by [1, II, prop. 2.1(b,c)] and [1, I, prop. 4.7] that HomR (−, Mj ) ։ Fj
is a projective cover in mod(MCM R) in the sense of Definition (8.5). In particular,
ϕ lifts to a natural transformation ψ of HomR (−, Mj ), which must be an automorphism by Lemma (8.6). Thus we have a commutative diagram in mod(MCM R),

HomR (−, Mj )
✤
✤=
ψ ∼
✤

HomR (−, Mj )

/ / Fj
∼
= ϕ


/ / Fj .

As the Yoneda functor yM : MCM R → mod(MCM R) is fully faithful, see [32, lem.
(4.3)], there exists a unique automorphism α of Mj such that ψ = HomR (−, α). For
′′
this particular α, the arguments above show that λj,ϕ = ξj,α
. Thus every generator
′′
λj,ϕ has the form ξj,α for some α, and hence the inclusion ”⊇” in (8.8.2) holds. 
(8.9) Observation.∼ For any commutative noetherian local ring R, there is an iso=
morphism ρR : R∗ −→ KB
1 (proj R) given by the composite of
ηR
θR
/ KB
/ KC
R∗
1 (R)
1 (proj R) .
∼
=

∼
=

The first map is described in (5.2); it is an isomorphism by Srinivas [27, exa. (1.6)].
The second isomorphism is discussed in (3.5). Thus, ρR maps r ∈ R∗ to [R, r1R ].
We are finally in a position to prove the main result.
Proof of Theorem (2.12). By Proposition (8.4) we can identify K1 (mod R) with the
group KB
1 (mod R). Recall that i and r denote the inclusion and restriction functors
from the localization sequence (7.0.1). By the relations that define KB
1 (mod R), see

K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE

21

(3.3), there is a homomorphism π0 : AutR (M ) → KB
1 (mod R) given by α 7→ [M, α].
Since KB
(mod
R)
is
abelian,
π
induces
a
homomorphism
π, which is displayed as
0
1
the upper horizontal map in the following diagram,
π

AutR (M )ab
(8.9.1)

/ KB
1 (mod R)
∼
= KB
1 (fR )

σ ∼
=


KB
1 (mod(MCM R))

KB
1 (r)


/ KB (mod(proj R)) .
1

Here σ is the isomorphism from Proposition (8.8), and the isomorphism KB
1 (fR )
is induced by the equivalence fR from Observation (6.3). The diagram (8.9.1) is
B
commutative, indeed, KB
1 (r)σ and K1 (fR )π both map α ∈ AutR (M )ab to the class


HomR (−, M )|proj R , HomR (−, α)|proj R .

By Lemma (8.2) the homomorphism KB
1 (r) is surjective, and hence so is π. Exactness of the sequence in Proposition (8.4)(b) and commutativity of the diagram
(8.9.1) show that Ker π = σ −1 (Im KB
1 (i)). Therefore Proposition (8.8) implies that
there is an equality Ker π = Ξ, and it follows that π induces an isomorphism,
∼
=

π
b : AutR (M )ab /Ξ −→ KB
1 (mod R) .

This proves the first assertion in Theorem (2.12).
To prove the second assertion, let inc : proj R → mod R denote the inclusion functor. Note that the Gersten–Sherman transformation identifies the homomorphisms
K1 (inc) and KB
1 (inc); indeed ζproj R is an isomorphism by Theorem (3.7) and ζmod R
is an isomorphism by Proposition (8.4)(a). Thus, we must show that KB
1 (inc) can be
identified with the homomorphism λ : R∗ → AutR (M )ab /Ξ given by r 7→ r1R ⊕ 1M ′
(recall that we have written M = R ⊕ M ′ ). To this end, consider the isomorphism
ρR : R ∗ → K B
1 (proj R) from Observation (8.9) given by r 7→ [R, r1R ]. The fact that
KB
1 (inc) and λ are isomorphic maps now follows from the diagram,
R∗

λ

ρR ∼
=


KB
1 (proj R)

KB
1 (inc)

/ AutR (M )ab /Ξ
∼
= π
b


/ KB
1 (mod R) ,

which is commutative. Indeed, for r ∈ R∗ one has
(b
π λ)(r) = [M, r1R ⊕ 1M ′ ] = [R, r1R ] + [M ′ , 1M ′ ] = [R, r1R ] = (KB
1 (inc)ρR )(r) ,
where the penultimate equality is by (3.4).



9. Abelianization of Automorphism Groups
To apply Theorem (2.12), one must compute AutR (M )ab , i.e. the abelianization
of the automorphism group of the representation generator M . In Proposition (9.6)
we compute AutR (M )ab for the R-module M = R ⊕ m, which is a representation
generator for MCM R if m happens to be the only non-free indecomposable maximal
Cohen–Macaulay module over R. Specific examples of rings for which this is the
case will be studied in Section 10. Throughout this section, A denotes any ring.

22

HENRIK HOLM

(9.1) Definition. Let N1 , . . . , Ns be A-modules, and set N = N1 ⊕ · · · ⊕ Ns . We
view elements in N as column vectors.
For ϕ ∈ AutA (Ni ) we denote by di (ϕ) the automorphism of N which has as its
diagonal 1N1 , . . . , 1Ni−1 , ϕ, 1Ni+1 , . . . , 1Ns and 0 in all other entries.
For i 6= j and µ ∈ HomA (Nj , Ni ) we denote by eij (µ) the automorphism of N
with diagonal 1N1 , . . . , 1Ns , and whose only non-trivial off-diagonal entry is µ in
position (i, j).
(9.2) Lemma. Let N1 , . . . , Ns be A-modules and set N = N1 ⊕ · · · ⊕ Ns . If 2 ∈ A
is a unit, if i 6= j, and if µ ∈ HomA (Nj , Ni ) then eij (µ) is a commutator in AutA (N ).
Proof. The commutator of ϕ and ψ in AutA (N ) is [ϕ, ψ] = ϕψϕ−1 ψ −1 . It is easily
verified that eij (µ) = [eij ( µ2 ), dj (−1Nj )] if i 6= j.

The idea in the proof above is certainly not new. It appears, for example, already
in Litoff [22, proof of thm. 2] in the case s = 2. Of course, if s > 3 then eij (µ) is a
commutator even without the assumption that 2 is a unit; see e.g. [25, lem. 2.1.2(c)].
(9.3) Lemma. Let X and Y be non-isomorphic A-modules with local endomorphism rings. Let ϕ, ψ ∈ EndA (X) and assume that ψ factors through Y . Then one
has ψ ∈
/ AutA (X). Futhermore, ϕ ∈ AutA (X) if and only if ϕ + ψ ∈ AutA (X).
Proof. Write ψ = ψ ′′ ψ ′ with ψ ′ : X → Y and ψ ′′ : Y → X. If ψ is an automorphism,
then ψ ′′ is a split epimorphism and hence an isomorphism as Y is indecomposable.
This contradicts the assumption that X and Y are not isomorphic. The second
assertion now follows as AutA (X) is the set of units in the local ring EndA (X). 
(9.4) Proposition. Let N1 , . . . , Ns be pairwise non-isomorphic A-modules with
local endomorphism rings. An endomorphism
α = (αij ) ∈ EndA (N1 ⊕ · · · ⊕ Ns ) with αij ∈ HomA (Nj , Ni )
is an automorphism if and only if α11 , α22 , . . . , αss are automorphisms.
Furthermore, every α in AutA (N ) can be written as a product of automorphisms
of the form di (·) and eij (·), cf. Definition (9.1).
Proof. “Only if”: Assume that α = (αij ) is an automorphism with inverse
Ps β = (βij )
and let i = 1, . . . , s be given. In the local ring EndA (Ni ) one has 1Ni = j=1 αij βji ,
and hence one of the terms αij βji must be an automorphism. As αij βji is not an
automorphism for j 6= i, see Lemma (9.3), it follows that αii βii is an automorphism.
In particular, αii has a right inverse and βii has a left inverse, and since the ring
EndA (Ni ) is local this means that αii and βii are both automorphisms.
“If”: By induction on s > 1. The assertion is trivial for s = 1. Now let s > 1.
Assume that α11 , α22 , . . . , αss are automorphisms. Recall the notation from (9.1).
−1
−1
By composing α with es1 (−αs1 α−1
11 ) · · · e31 (−α31 α11 )e21 (−α21 α11 ) from the left
−1
−1
and with e12 (−α−1
11 α12 )e13 (−α11 α13 ) · · · e1s (−α11 α1s ) from the right, one gets an
endomorphism of the form




1N1 0
α11 0
= d1 (α11 )
,
α′ =
0 β
0
β
where β ∈ EndA (N2 ⊕ · · · ⊕ Ns ) is an (s − 1) × (s − 1) matrix with diagonal entries
given by αjj − αj1 α−1
11 α1j for j = 2, . . . , s. By applying Lemma (9.3) to the situa−1
tion ϕ = αjj − αj1 α11
α1j and ψ = αj1 α−1
11 α1j , it follows that the diagonal entries in

K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE

23

β are all automorphisms. By the induction hypothesis, β is now an automorphism
and can be written as a product of automorphisms of the form di (·) and eij (·).
Consequently, the same is true for α′ , and hence also for α.

(9.5) Corollary. Assume that 2 ∈ A is a unit and let N1 , . . . , Ns be pairwise nonisomorphic A-modules with local endomorphism rings. The homomorphism,
∆ : AutA (N1 ) × · · · × AutA (Ns ) −→ AutA (N1 ⊕ · · · ⊕ Ns ) ,
given by ∆(ϕ1 , . . . , ϕs ) = d1 (ϕ1 ) · · · ds (ϕs ), induces a surjective homomorphism,
∆ab : AutA (N1 )ab ⊕ · · · ⊕ AutA (Ns )ab −→ AutA (N1 ⊕ · · · ⊕ Ns )ab .
Proof. By Proposition (9.4) every element in AutA (N1 ⊕ · · · ⊕ Ns ) is a product of
automorphisms of the form di (·) and eij (·). As 2 ∈ A is a unit, Lemma (9.2) yields
that every element of the form eij (·) is a commutator; thus in AutA (N1 ⊕· · ·⊕Ns )ab
every element is a product of elements of the form di (·), so ∆ab is surjective.

As noted above, Lemma (9.2), and consequently also Corollary (9.5), holds without the assumption that 2 ∈ A is a unit provided that s > 3.
In the following, we write [ · ]m : R ։ R/m = k for the quotient homomorphism.
(9.6) Proposition. Let (R, m, k) be any commutative local ring such that 2 ∈ R
is a unit. Assume that m is not isomorphic to R and that the endomorphism ring
EndR (m) is commutative and local. There is an isomorphism of abelian groups,
∼
=

δ : AutR (R ⊕ m)ab −→ k ∗ ⊕ AutR (m) ,
given by


α11
α21

α22
α22




7−→ [α11 (1)]m , α11 α22 − α21 α12 .

Proof. First note that the image of any homomorphism α : m → R is contained in m.
Indeed if Im α * m, then u = α(a) is a unit for some a ∈ m, and thus α(u−1 a) = 1.
It follows that α is surjective, and hence a split epimorphism as R is free. Since m
is indecomposable, α must be an isomorphism, which is a contradiction.
Therefore, given an endomorphism,




HomR (R, R) HomR (m, R)
α11 α12
∈ EndR (R ⊕ m) =
,
HomR (R, m) HomR (m, m)
α21 α22
we may by (co)restriction view the entries αij as elements in the endomorphism
ring EndR (m). As this ring is assumed to be commutative, the determinant map
EndR (R ⊕ m) −→ EndR (m)

given by

(αij ) 7−→ α11 α22 − α21 α12

preserves multiplication. If (αij ) ∈ AutR (R ⊕ m), then Proposition (9.4) implies
that α11 ∈ AutR (R) and α22 ∈ AutR (m), and thus α11 α22 ∈ AutR (m). By applying
Lemma (9.3) to ϕ = α11 α22 − α21 α12 and ψ = α21 α12 we get ϕ ∈ AutR (m), and
hence the determinant map is a group homomorphism AutR (R ⊕ m) → AutR (m).
The map AutR (R ⊕ m) → k ∗ defined by (αij ) 7→ [α11 (1)]m is also a group homomorphism. Indeed, entry (1, 1) in the product (αij )(βij ) is α11 β11 + α12 β21 . Here
α12 is a homomorphism m → R, and hence α12 β21 (1) ∈ m by the arguments in the
beginning of the proof. Consequently one has
[(α11 β11 + α12 β21 )(1)]m = [(α11 β11 )(1)]m = [α11 (1)β11 (1)]m = [α11 (1)]m [β11 (1)]m .

24

HENRIK HOLM

These arguments and the fact that the groups k ∗ and AutR (m) are abelian show
that the map δ described in the proposition is a well-defined group homomorphism.
Evidently, δ is surjective; indeed, for [r]m ∈ k ∗ and ϕ ∈ AutR (m) one has


r1R
0
δ
= ([r]m , ϕ) .
0
r−1 ϕ
To show that δ is injective, assume that α ∈ AutR (R ⊕ m)ab with δ(α) = ([1]m , 1m ).
By Corollary (9.5) we can assume that α = (αij ) is a diagonal matrix. We write
α11 = r1R for some unit r ∈ R. Since one has δ(α) = ([r]m , rα22 ) we conclude that
r ∈ 1 + m and α22 = r−1 1m , that is, α has the form


r1R
0
with
r ∈ 1+m.
α=
0
r−1 1m
Thus, proving injectivity of δ amounts to showing that every automorphism α of the
form above belongs to the commutator subgroup of AutR (R ⊕ m). As r − 1 ∈ m the
map (r − 1)1R gives a homomorphism R → m. Since r(r−1 − 1) = 1 − r ∈ m and
r∈
/ m, it follows that r−1 − 1 ∈ m. Thus (r−1 − 1)1R gives another homomorphism
R → m. If ι : m ֒→ R denotes the inclusion, then one has2




 

1R −r−1 ι
1R
0
1R ι
1R
0
r1R
0
.
=
0
1m
(r − 1)1R 1m
0 1m
(r−1 − 1)1R 1m
0
r−1 1m
The right-hand of this equality is a product of matrices of the form eij (·), and since
2 ∈ R is a unit the desired conclusion now follows from Lemma (9.2).

10. Examples
We begin with a trivial example.
(10.1) Example. If R is regular, then there are isomorphisms,
K1 (mod R) ∼
= K1 (proj R) ∼
= KC (R) ∼
= R∗ .
1

The first isomorphism is by Quillen’s resolution theorem [24, §4 thm. 3], the second
one is mentioned in (3.6), and the third one is well-known; see e.g. [27, exa. (1.6)].
Theorem (2.12) confirms this result, indeed, as M = R is a representation generator
for MCM R = proj R one has AutR (M )ab = R∗ . As there are no Auslander–Reiten
sequences in this case, the subgroup Ξ is generated by the empty set, so Ξ = 0.
We now illustrate how Theorem (2.12) applies to compute K1 (mod R) for the ring
R = k[X]/(X 2 ). The answer is well-known to be k ∗ , indeed, for any commutative
artinian local ring R with residue field k one has K1 (mod R) ∼
= k ∗ by [24, §5 cor. 1].
(10.2) Example. Let R = k[X]/(X 2 ) be the ring of dual numbers over a field k
with char(k) 6= 2. Denote by inc : proj R → mod R the inclusion functor. The homomorphism K1 (inc) may be identified with the map,
µ : R∗ −→ k ∗

given by

a + bX 7−→ a2 .

Proof. The maximal ideal m = (X) is the only non-free indecomposable maximal
Cohen–Macaulay R-module, so M = R⊕m is a representation generator for MCM R;
see (2.1.1). There is an isomorphism k → EndR (m) of R-algebras given by a 7→ a1m ,
2 The identity comes from the standard proof of Whitehead’s lemma; see e.g. [27, lem. (1.4)].

K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE

25

in particular, EndR (m) is commutative. Via this isomorphism, k ∗ corresponds to
AutR (m). The Auslander–Reiten sequence ending in m is
ι

X

0 −→ m −→ R −→ m −→ 0 ,


2
where ι is the inclusion. The Auslander–Reiten homomorphism Υ = −1
2 : Z →Z
is injective, so Theorem (2.12) can be applied. Note that for every a1m ∈ AutR (m),
where a ∈ k ∗ , there is a commutative diagram,
0

ι

/m

/R

∼
= a1R

∼
= a1m

0


/R

ι


/m

X

X

/0

/m
∼
= a1m


/m

/0.

Applying the tilde construction (2.6) to the automorphisms a1m and a1R one gets




a1R 0
1R
0
g
g
;
and
a1R =
a1m =
0
1m
0 a1m

see Example (2.7). In view of Definition (2.10) and Remark (2.11), the subgroup Ξ
of AutR (R ⊕ m)ab is therefore generated by all elements of the form

 −1
a 1R
0
−1 g
g
g
where
a ∈ k∗ .
ξa := (a1
)(
a1
)
(
a1
)
=
m
R
m
0
a2 1 m
Denote by ω the composite of the isomorphisms,
AutR (R ⊕ m)ab

δ
∼
=

/ k ∗ ⊕ AutR (m)

∼
=

/ k∗ ⊕ k∗ ,

where δ is the isomorphism from Proposition (9.6). As ω(ξa ) = (a−1 , a) we get that
ω(Ξ) = {(a−1 , a) | a ∈ k ∗ } and thus ω induces the first group isomorphism below,
AutR (R ⊕ m)ab /Ξ

ω
∼
=

/ (k ∗ ⊕ k ∗ )/ω(Ξ)

χ
∼
=

/ k∗ ;

the second isomorphism is induced by the surjective homomorphism k ∗ ⊕ k ∗ → k ∗ ,
given by (b, a) 7→ ba, whose kernel is exactly ω(Ξ). In view of Theorem (2.12) and
the isomorphisms ω and χ above, it follows that K1 (mod R) ∼
= k∗ .
Theorem (2.12) asserts that K1 (inc) may be identified with the homomorphism


r1R 0
.
λ : R∗ −→ AutR (R ⊕ m)ab /Ξ
given by
r 7−→
0
1m
It remains to note that the isomorphism χω identifies λ with the homomorphism µ

described in the example, indeed, one has χωλ = µ.
Example (10.2) shows that for R = k[X]/(X 2) the canonical homomorphism,
R∗ ∼
= K1 (proj R)

K1 (inc)

/ K1 (mod R) ∼
= k∗ ,

is not an isomorphism. It turns out that if k is algebraically closed with characteristic zero, then there exists a non-canonical isomorphism between R∗ and k ∗ .
(10.3) Proposition. Let R = k[X]/(X 2 ) where k is an algebraically closed field
with characteristic p > 0. The following assertions hold.
(a) If p > 0, then the groups R∗ and k ∗ are not isomorphic.
(b) If p = 0, then there exists a (non-canonical) group isomorphism R∗ ∼
= k∗ .

26

HENRIK HOLM

Proof. There is a group isomorphism R∗ → k ∗ ⊕ k + given by a + bX 7→ (a, b/a),
where k + denotes the underlying abelian group of the field k.
“(a)”: Let ϕ = (ϕ1 , ϕ2 ) : k ∗ → k ∗ ⊕ k + be any group homomorphism. As k is
algebraically closed, every element in x ∈ k ∗ has the form x = y p for some y ∈ k ∗ .
Therefore ϕ(x) = ϕ(y p ) = ϕ(y)p = (ϕ1 (y), ϕ2 (y))p = (ϕ1 (y)p , pϕ2 (y)) = (ϕ1 (x), 0),
which shows that ϕ is not surjective.
“(b)”: Since p = 0 the abelian group k + is divisible and torsion free. Therefore
+ ∼
k = Q(I) for some index set I. There exist algebraic field extensions of Q of any
finite degree, and these are all contained in the algebraically closed field k. Thus
|I| = dimQ k must be infinite, and it follows that |I| = |k|.
The abelian group k ∗ is also divisible, but it has torsion. Write k ∗ ∼
= T ⊕ (k ∗ /T ),
∗
n
∗
where T = {x ∈ k | ∃ n ∈ N : x = 1} is the torsion subgroup of k . For the divisible
torsion free abelian group k ∗ /T one has k ∗ /T ∼
= Q(J) for some index set J. It is
not hard to see that |J| must be infinite, and hence |J| = |k ∗ /T |. As |T | = ℵ0 it
follows that |k| = |k ∗ | = ℵ0 + |J| = |J|.
Since |J| = |k| = |I| one gets k ∗ ∼
= k∗ ⊕ k+ . 
= T ⊕ Q(J) ⊕ Q(I) ∼
= T ⊕ Q(J) ∼
The artinian ring R = k[X]/(X 2) from Example (10.2) has length ℓ = 2 and
this power is also involved in the description of the homomorphism µ = K1 (inc).
The next result shows that this is no coincidence. As Proposition (10.4) might be
well-known to experts, and since we do not really need it, we do not give a proof.
(10.4) Proposition. Let (R, m, k) be a commutative artinian local ring of length ℓ.
The group homomorphism R∗ ∼
= K1 (proj R) → K1 (mod R) ∼
= k ∗ induced by the inclusion inc : proj R → mod R is the composition of the homomorphisms,
R∗

π

/ k∗

(·)ℓ

/ k∗ ,

where π : R ։ R/m = k is the canonical quotient map and (·)ℓ is the ℓ’th power.
Our next example is a non-artinian ring, namely the simple curve singularity of
type (A2 ) studied by e.g. Herzog [19, Satz 1.6] and Yoshino [32, prop. (5.11)].
(10.5) Example. Let R = k[[T 2 , T 3 ]] where k is an algebraically closed field with
char(k) 6= 2. Denote by inc : proj R → mod R the inclusion functor. The homomorphism K1 (inc) may be identified with the inclusion map,
µ : R∗ = k[[T 2 , T 3 ]]∗ ֒→ k[[T ]]∗ .
Proof. The maximal ideal m = (T 2 , T 3 ) is the only non-free indecomposable maximal Cohen–Macaulay R-module, so M = R ⊕ m is a representation generator for
MCM R; see (2.1.1). Even though T is not an element in R = k[[T 2 , T 3 ]], multiplication by T is a well-defined endomorphism of m. Thus there is a ring homomorphism,
χ : k[[T ]] −→ EndR (m)

given by

h 7−→ h1m .

It is not hard to see that χ is injective. To prove that it is surjective, i.e. that one
has EndR (m)/k[[T ]] = 0, note that there is a short exact sequence of R-modules,
0 −→ k[[T ]]/R −→ EndR (m)/R −→ EndR (m)/k[[T ]] −→ 0 .
To see that EndR (m)/k[[T ]] = 0, it suffices to argue that the R-module EndR (m)/R
is simple. As noted in the beginning of the proof of Proposition (9.6), the inclusion m ֒→ R induces an isomorphism EndR (m) ∼
= HomR (m, R), so by applying

K-GROUPS FOR RINGS OF FINITE COHEN–MACAULAY TYPE

27

HomR (−, R) to the short exact sequence 0 → m → R → k → 0, it follows that
EndR (m)/R ∼
= Ext1R (k, R) .
The latter module is isomorphic to k since R is a 1-dimensional Gorenstein ring.
Note that via the isomorphism χ, the group k[[T ]]∗ corresponds to AutR (m).
The Auslander–Reiten sequence ending in m is
(1 −T )t

0

/m

(1 −T )t

/m

/ R⊕m

(T 2 T )

/m
/0.

2
Since the Auslander–Reiten homomorphism Υ = −1
1 : Z → Z is injective, Theorem (2.12) can be applied. We regard elements in R ⊕ m as column vectors. Let
α = h1m ∈ AutR (m), where h ∈ k[[T ]]∗ , be given. Write h = f +gT for some f ∈ R∗
and g ∈ R. It is straightforward to verify that there is a commutative diagram,
0

0

(1 −T )

t

/m


f g
∼
= β = gT 2 f

∼
= γ = (f −gT )1m


/m

(T 2 T )

/ R⊕m

/ R⊕m

(T



2

/0

∼
= α = (f +gT )1m


/m
T)

/0.

Note that β really is an automorphism; indeed, its inverse is given by


f −g
.
β −1 = (f 2 − g 2 T 2 )−1
−gT 2 f
We now apply the tilde construction (2.6) to α, β, and γ; by Example (2.7) we get:




1
0
1
0
α̃ =
, β̃ = β, and γ̃ =
.
0 f + gT
0 f − gT
In view of Definition (2.10) and Remark (2.11), the subgroup Ξ of AutR (R ⊕ m)ab
is therefore generated by all the elements


f
−g(f − gT )
−1
2
2 2 −1
.
ξh := α̃β̃ γ̃ = (f − g T )
−gT 2(f + gT ) f (f 2 − g 2 T 2 )
Denote by ω the composite of the isomorphisms,
AutR (R ⊕ m)ab

δ
∼
=

/ k ∗ ⊕ AutR (m)

1 ⊕ χ−1
∼
=

/ k ∗ ⊕ k[[T ]]∗ ,

where δ is the isomorphism from Proposition (9.6). Note that δ(ξh ) = ([f ]m , 1m ) =
(h(0), 1m ) and hence ω(ξh ) = (h(0), 1). It follows that ω(Ξ) = k ∗ ⊕ {1} and thus ω
induces a group isomorphism,
∼
=

ω : AutR (R ⊕ m)ab /Ξ −→ (k ∗ ⊕ k[[T ]]∗ )/ω(Ξ) = k[[T ]]∗ .
In view of this isomorphism, Theorem (2.12) shows that K1 (mod R) ∼
= k[[T ]]∗ . Theorem (2.12) also asserts that K1 (inc) may be identified with the homomorphism


f 1R 0
∗
.
λ : R −→ AutR (R ⊕ m)ab /Ξ
given by
f 7−→
0
1m
It remains to note that the isomorphism ω identifies λ with the inclusion map µ
described in the example, indeed, one has ωλ = µ.


28

HENRIK HOLM

Acknowledgements
It is a pleasure to thank Peter Jørgensen without whom this paper could not
have been written. However, Peter did not wish to be a coauthor of this manuscript.
We are also grateful to Marcel Bökstedt and Charles A. Weibel for valuable input
on the Gersten–Sherman transformation, and to Christian U. Jensen for pointing
out Proposition (10.3). Finally, we thank Viraj Navkal and the anonymous referee
for their thoughtful comments and for making us aware of the paper [23].

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Department of Mathematical Sciences, Faculty of Science, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark
E-mail address: holm@math.ku.dk
URL: http://www.math.ku.dk/~holm/

