Boundary of Cohen-Macaulay cone andasymptotic behavior of system of ideals
Kazuhiko Kurano
Meiji University

1

Introduction

On a smooth projective variety, we can deﬁne the intersection number for agiven divisor and a given curve. By this pairing, we can deﬁne the numericalequivalence on divisors and curves. We get a (ﬁnitely generated) lattice if
we divide the set of Weil divisors or curves by the numerical equivalence. In
order to study the intersection pairing, we have some concepts of ”positive”
elements, e.g., ample, base point-free, nef, etc.. Consider the cone spannedby positive elements in the lattice tensored with the ﬁeld of real numbers.
This cone gives us many informations on the given algebraic variety.
In this note, we are interested in the intersection pairing around a ﬁxed
singular point of a scheme, or the vertex of the aﬃne cone of a smoothprojective variety. Let R be a Noetherian (Cohen-Macaulay) local ring corresponding to the given point. We ﬁrst deﬁne a pairing between a ﬁnitely
generated module, and a module of ﬁnite length and ﬁnite projective dimension. Consider the Grothendieck group of ﬁnitely generated R-modules, anddivide it by the numerical equivalence. Then, we get a ﬁnitely generated
lattice. It is natural to think that Cohen-Macaulay modules are positive elements under the pairing. So, we study the cone spanned by Cohen-Macaulay
modules in the numerical Grothendieck group tensored with R.
We always assume that R is a d-dimensional Noetherian Cohen-Macaulay
local domain such that one of the following conditions are satisﬁed1 :1

If either (a) or (b) is satisfied, there exists a regular alteration of Spec R by de Jong’s

(a) R is a homomorphic image of an excellent regular local ring containing
Q.
(b) R is essentially of ﬁnite type over a ﬁeld, Z or a complete DVR.
In this note, modules are always assumed to be ﬁnitely generated.

2Intersection pairing on Spec R and the CohenMacaulay cone

Let G0 (R) be the Grothendieck group of ﬁnitely generated R-modules. The
symbol [M ] means the element in G0 (R) corresponding to an R-module M .
Let CR be the category of modules of ﬁnite length and ﬁnite projective dimension. Here, note that R/(x1 , . . . , xd ) ∈ CR for a system of parameters
x1 , . . . , xd . In particular, CR is not empty.2 For L ∈ CR , we deﬁneχL : G0 (R) −→ Z by χL ([M ]) =∑Rii (−1) ℓR (Tori (L, M )).

Consider the map
CR × G0 (R) → Z deﬁned by (L, [M ]) 7→ χL ([M ]).

(1)

Here, we deﬁne numerical equivalence as follows. For α, β ∈ G0 (R),defα ≡ β ⇐⇒ χL (α) = χL (β) for any L ∈ CR .
Here, we put
G0 (R) = G0 (R)/{α ∈ G0 (R) | α ≡ 0}.
By Theorem 3.1 and Remark 3.5 in [9], we have the following result.
Theorem 1 G0 (R) is a finitely generated torsion-free abelian group.

theorem [6].
2
By the new intersection theorem due to Roberts, we know that, for a Noetherian local
ring R, CR is not empty if and only if R is Cohen-Macaulay.

Remark 2 MCM (Maximal Cohen-Macaulay) modules behave as ”positiveelements” under the pairing (1) by the following reason.
Let L be an object in CR . Then, by Auslander-Buchsbaum formula, we
have
depth L + pdR L = depth R = d.
Then, we have pdR L = d. Let F. be the minimal free resolution of L. Then,
it is very easy to check that the complex F. has a depth sensitive property,
i.e., for any module N , we have
depth N = d − max{i | Hi (F. ⊗R N ) ̸= 0}.
We say that M is a MCM module if depth M = d. By the depth sensitivity, if M is MCM, then TorR
i (L, M ) = 0 for any i > 0. Therefore, wehaveχL ([M ]) = ℓR (L ⊗R M ) > 0.
By Auslander-Buchsbaum formula, any MCM module over a regular local
ring is free. We say that a ring R is of ﬁnite (Cohen-Macaulay) representation
type if there are only ﬁnitely many isomorphism classes of indecomposable
MCM’s. If R is of ﬁnite representation type, then R has only isolated singularity. It was proved that a Gorenstein local ring of ﬁnite representation
type has a simple singularity. Simple singularities are of ﬁnite representation
type. We refere the reader to Yoshino [17] for the representation theory of
MCM’s.
Bad Cohen-Macaulay rings have many MCM’s in general. But, if we do
not assume that R is Cohen-Macaulay, it is not known whether there exists
an MCM module. This open problem is called the small Mac conjecture [5].
Example 3
1. If L = R/(x1 , . . . , xd ) for a system of parameters x1 , . . . , xd ,then χL ([R]) ̸= 0. Hence, G0 (R) ̸= 0.
2. If d ≤ 2, then rank G0 (R) = 1. See Proposition 3.7 in [9].
3. Let X be a smooth projective variety with embedding X ,→ Pn . Let
R (resp. D) be the aﬃne cone (resp. the very ample divisor) of thisembedding. Then, we have the following commutative diagram:
∼

∼

∼

ϕ

G0 (R)Q −→ A∗ (R)Q ←−↓↓CH · (X)Q /D · CH · (X)Q↓··
G0 (R)Q −→ A∗ (R)Q ←− CHnum(X)Q /D · CHnum(X)Q(a) By the commutativity of this diagram, ϕ is a surjection. Therefore,
we have··rank G0 (R) ≤ dimQ CHnum(X)Q /D · CHnum(X)Q .

(2)·(b) If CH · (X)Q ≃ CHnum(X)Q , then ϕ is an isomorphism ([9], [15]).
In this case, the equality holds in (2).

(c) There exists an example such that ϕ is not an isomorphism [15].
Further, Roberts and Srinivas [15] proved the following: Assume
that the standard conjecture and Bloch-Beilinson conjecture aretrue. Then ϕ is an isomorphism if the deﬁning ideal of R is generated by polynomials with coeﬃcients in the algebraic closure of
the prime ﬁeld.
4. It is conjectured that G0 (R)Q ≃ Q if R is complete intersection isolated
singularity with d even.
It is true if R is the aﬃne cone of a smooth projective variety X overC ([2]). In fact, since we have an injectioniCHhom(X)Q −→ H 2i (X, Q) = Q

and the natural surjectionii(X)Q ̸= 0,(X)Q −→ CHnum
CHhomi(X)Q = Q for each i = 0, 1, . . . , dim X. Here, remark
we know CHnum
that H 2i (X, Q) = Q since the dimension of X is odd. Then, we have··(X)Q = Q.
(X)Q /D · CHnum
CHnum

Therefore, the rank of G0 (R) is one by 3 (a) as above.

Definition 4 We define the Cohen-Macaulay cone as follows:∑
CCM (R) =R≥0 [M ] ⊂ G0 (R)R .
M :M CM

Here G0 (R)R = G0 (R) ⊗Z R.
We refer the reader to [1] for basic properties on Cohen-Macaulay cones.
It is easy to see that the dimension of the cone is equal to the rank of G0 (R).
Further, we have
G0 (R)R ⊃ CCM (R)− ⊃ CCM (R) ⊃ Int(CCM (R)− ) = Int(CCM (R)) ∋ [R],where CCM (R)− is the closure of CCM (R) with respect to the classical topology on G0 (R)R , and Int(−) is the interior.
If R is of ﬁnite representation type, then CCM (R) is a strongly convex
polyhedral cone, in particular CCM (R)− = CCM (R).
We have no example that CCM (R)− is not equal to CCM (R), or CCM (R)is not a polyhedral cone.
Remark that, for any L ∈ CR , χL induces χL which makes the following
diagram commutative:χL
G0 (R) −→Z↓↗ χL
G0 (R)The map χL induces(χL )R : G0 (R)R −→ R.
Let x1 , . . . , xd be a system of parameters. Consider the map
χR/(x) : G0 (R) −→ Z.
Let K. be the Koszul complex with respect to x. This map satisﬁes
χR/(x) ([M ]) = rank M · χR/(x) ([R]),since K. is the minimal free resolution of R/(x) and K. admits this property.
Therefore, we have a map
rk : G0 (R) −→ ZandrkR : G0 (R)R −→ R
deﬁned by rk([M ]) = rank M . (Here, rk = χR/(x)1 ([R]) χR/(x) .)Let F be the kernel of the map rk. Then, F is generated by cycles [M ]with dim M < d. Thus, we have
G0 (R) = Z[R] ⊕ F

and G0 (R)R = R[R] ⊕ FR .

Example 5
1. Put R = k[x, y, z, w](x,y,z,w) /(xy − zw), where k is a ﬁeld.
Then, F = Z[R/(x, z)] ≃ Z. This ring has only three indecomposable
MCM modules, R, (x, z) and (x, w).
Then, the Cohen-Macaulay cone is spanned by[(x, z)] = ([R], −[R/(x, z)]) and [(x, w)] = ([R], [R/(x, z)])in G0 (R)R = R[R] ⊕ FR .
2. Put R = k[x1 , x2 , . . . , x6 ](x1 ,x2 ,...,x6 ) /(x1 x2 + x3 x4 + x5 x6 ), where k isa ﬁeld. Then, F = Z[R/(x1 , x3 , x5 )] ≃ Z. This ring has only threeindecomposable MCM modules, R, M1 and M2 , where M1 and M2 are
MCM modules of rank 2.
Then, the Cohen-Macaulay cone is spanned by[M1 ] = (2[R], [R/(x1 , x3 , x5 )]) and [M2 ] = (2[R], −[R/(x1 , x3 , x5 )])in G0 (R)R = R[R] ⊕ FR .
The Cohen-Macaulay cone of this ring is not spanned by classes of
MCM modules of rank one.
3. Put R = k[x, y, z, w](x,y,z,w) /(xy−f1 f2 · · · ft ), where k is an algebraicallyclosed ﬁeld of characteristic zero. Here, we assume that f1 , f2 , . . . , ft
are pairwise coprime linear forms in k[z, w]. In this case, we have
F = (⊕i Z[R/(x, fi )]) /Z([R/(x, f1 )] + · · · + [R/(x, ft )]) ≃ Zt−1 .
We can prove that the Cohen-Macaulay cone is minimally spanned by
the following 2t − 2 MCM’s of rank one.
{{(x, fi1 fi2 · · · fis ) | 1 ≤ s < t, 1 ≤ i1 < i2 < · · · < is ≤ t}
This ring is of ﬁnite representation type if and only if t ≤ 3.

3

Maximal Cohen-Macaulay modules of rankone

Let R be a Noetherian standard graded normal domain such that R0 is aﬁeld of characteristic zero. Assume that R has at most isolated singularity,and [HR2 + (R)]0 = 0. Then, it is known that R has at most ﬁnitely many
MCM modules of rank one. (Prof. Flenner kindly taught me this result.)
If R is a Noetherian local ring with at most isolated singularity. Assume
that R is a complete intersection and dim R ≥ 4. Then, R is factorial. In
particular, R is only one MCM modules of rank one.
Assume that R is a Noetherian local ring of dimension 2. Even if R is a
hypersurface with at most isolated singularity, R may have inﬁnitely many
MCM modules of rank one. (For example, the aﬃne cone of an elliptic curve
actually has inﬁnitely many MCM modules of rank one.)
So, it is important to consider the case of dim R = 3. In this section(Theorem 9), we show that R has only ﬁnitely many MCM’s of rank one if R
is a 3-dimensional isolated hypersurface singularity with desingularization.
By the following result, we know that CCM (R)− is a strongly convex cone,that is, CCM (R)− does not contain a line through the origin.
Lemma 6 Let R be a d-dimensional Cohen-Macaulay local domain which
satisfies (a) or (b) in the introduction. Then, (rkR )−1 ∩CCM (R)− is a compact
set.
Corollary 7 Assume that R is a Cohen-Macaulay local domain. Then, forany positive integer r,{[M ] ∈ G0 (R) | M is a MCM module of rank r }is a finite subset of G0 (R).
Further, assume that R is a normal domain. Then, we have the determinant map (or the first Chern class map) c1 : G0 (R) → Ad−1 (R).
We can also deﬁne numerical equivalence on Ad−1 (R). Then, we deﬁne
the class group modulo numerical equivalence to be
Ad−1 (R) = Ad−1 (R)/ ≡ .

By Proposition 3.7 and Example 4.1 in [9], we know that it is also a ﬁnitely
generated torsion-free abelian group.
Here we can prove that there exists the map c1 which makes the following
diagram commutative:c1
G0 (R) −→Ad−1 (R)↓↓c

1
G0 (R) −→Ad−1 (R)

By the commutativity of the above diagram, we have the following:
Corollary 8 Let R be a d-dimensional Cohen-Macaulay local normal domain. Assume that(*) the kernel of the natural map Ad−1 (R) −→ Ad−1 (R) is a finite group.
Then, for any positive integer r,{c1 ([M ]) ∈ Ad−1 (R) | M is a MCM module of rank r }is a finite subset of Ad−1 (R).
In particular, R has only finitely many MCM modules of rank one up toisomorphism.
Theorem 9 (Dao-Kurano, [2]) Let R be a 3-dimensional isolated hypersurface singularity with desingularization. Then, the natural map
A2 (R) −→ A2 (R)is an isomorphism. In particular (*) in Corollary 8 is satisfied. Therefore R
has only finitely many MCM’s of rank one.
Remark 10 Put B = ⊕n≥0 Bn = C[B1 ] = C[y0 , y1 , . . . , yn ]/I, R = BB+ ,and X = Proj(B). Assume that X is smooth over C. (Since dim R = d,dim X = d − 1.)
CH1 (X) −→CH1 (X)/c1 (OX (1))CH0 (X)= Ad−1 (R)
↓
↓f
g
1
1
CHnum (X) −→ CHnum (X)/c1 (OX (1))CH0num (X) −→ Ad−1 (R)1. Assume that R is a Cohen-Macaulay local normal ring with d ≥ 3.
Then, CH1 (X) is finitely generated and f ⊗ Q is an isomorphism.
2. Assume that the ideal I is generated by some elements in Q[y0 , y1 , . . . , yn ].
If some famous conjectures (the standard conjecture and Bloch-Beilinson
conjecture) are true, then g⊗Q is an isomorphism. (Roberts-Srinivas [15])Therefore, if R is a Cohen-Macaulay local normal ring with d ≥ 3 such that
X is defined over Q, and if some conjectures are true, then (*) is satisfied.
It is also proved in the case of positive characteristic.
If we remove the assumption that X is defined over Q, then there exists
an example that g ⊗ Q is not an isomorphism (Roberts-Srinivas [15]).
However, remark that if R is a standard graded Cohen-Macaulay normal
graded domain over C with dim R ≥ 3. Then, there exist only finitely many
MCM modules of rank one.

4

The fundamental class of a Noetherian local ring

We deﬁne the strictly nef cone SN (R), and the fundamental class µR for a
Noetherian local domain R. They satisfy the following:
G0 (R)R ⊃ SN (R) ⊃ CCM (R) − {0}∪G0 (R)Q ∋µR
The fundamental class is deeply related to the homological conjectures as inFact 15.
We are mainly interested in the problem whether µR is in such cones ornot. Theorem 18 is the main result in this section, which states that if R
is FFRT or F-rational, then µR is in CCM (R). We shall give a corollary(Corollary 21).
Definition 11 We deﬁne the strictly nef cone by
SN (R) = {α | χL (α) > 0 for any L ∈ CR }.

By the depth sensitivity, χL ([M ]) = ℓR (H0 (L ⊗ M )) > 0 for any MCM
module M (̸= 0) and L ∈ CR . Therefore,
SN (R) ⊃ CCM (R) − {0}.
We can also deﬁne SN (R) for non-Cohen-Macaulay local ring R using some
perfect complexes instead of CR .
Definition 12 We putµR = τR −1 ([Spec R]) ∈ G0 (R)Q ,∼where τR : G0 (R)Q → A∗ (R)Q is the singular Riemann-Roch map.
G0 (R)Q −→ G0 (R)QµR
7→µR
We call the image µR in G0 (R)Q the fundamental class of R.
Remark that µR ̸= 0 since rankR µR = 1.
Put R = T /I, where T is a regular local ring. The map τR is deﬁned
using not only R but also T . Therefore, µR ∈ G0 (R)Q may depend on the
choice of T .3 However, we can prove that µR ∈ G0 (R)Q is independent of T
(Theorem 5.1 in [9]).
We shall explain why we call µR the fundamental class of R.
Remark 13
1. If X = Spec R is a d-dimensional aﬃne variety over C,
we have the cycle map cl
τcl

R
G0 (R)Q −→A∗ (R)Q −→ H∗ (X, Q)µR
7→ [Spec R] 7→µX

such that cl([Spec R]) is the fundamental class µX in H2d (X, Q) in theusual sense, where H∗ (X, Q) is the Borel-Moore homology. Here µX isthe generator of H2d (X, Q) ≃ Z.
Hence, we call µR the fundamental class of R.
3

There is no example that the map τR actually depend on the choice of T . For some
excellent rings, it had been proved that τR is independent of the choice of T (Proposition 1.2
in [8]).

2. Let R have a subring S such that S is a regular local ring and R isa localization of a ﬁnite extension of S. Let L be a ﬁnite-dimensional
normal extension of Q(S) containing Q(R). Let B be the integral
closure of R in L. Then, we have
µR = rank1R B [B] in G0 (R)Q .
[B]
In particular, µR = rank
in G0 (R)Q .
RB

3. Assume that R is of characteristic p > 0 and F-ﬁnite. Assume thatthe residue class ﬁeld is algebraically closed. By the singular RiemannRoch theorem, we have[e R]
in G0 (R)R ,e→∞ pdeµR = lim

where e R is the eth Frobenius direct image.
Example 14
1. If R is a complete intersection, then µR is equal to [R]in G0 (R)Q , therefore µR = [R] in G0 (R)Q . There exists a Gorensteinring such that µR ̸= [R]. However there exist many examples of rings
satisfying µR = [R]. Roberts ([12], [13]) proved the vanishing property
of intersection multiplicity for rings satisfying µR = [R].
2. Let R be a normal domain. Then, we haveτR
G0 (R)Q −→A∗ (R)Q = Ad (R)Q ⊕ Ad−1 (R)Q ⊕ · · ·[R]
7→ [Spec R] − K2R + · · ·[ωR ]7→ [Spec R] + K2R + · · ·If τR−1 (KR ) ̸= 0 in G0 (R)Q , then [R] ̸= µR .
Sometimes µR = 12 ([R] + [ωR ]) is satisﬁed. But it is not true in general.
3. Let R = k[xij ]/I2 (xij ), where (xij ) is the generic (m + 1) × (n + 1)matrix, and k is a ﬁeld. Suppose 0 < m ≤ n.

Then, we have
G0 (R)Q ≃ G0 (R)Q ≃ Q[a]/(am+1 )()m ( −a )n[R] 7→ 1−ea−a1−ea11= 1 + 2 (m − n)a + 24(· · · )a2 + · · ·( −a )m ( a )n[ωR ] 7→ 1−ea
1−e−a
µR 7→ 1−1
τR (KR ) 7→ (n − m)aHere, we shall explain the relationship between the fundamental class µR
and homological conjectures.
Fact 15
1. The small Mac conjecture is true if and only if µR ∈ CCM (R)for any R.
Even if R is an equi-characteristic Gorenstein ring, it is not known
whether µR is in CCM (R) or not. If R is a complete intersection, thenµR = [R] ∈ CCM (R) as in 1) in Example 14.
2. If µR = [R] in G0 (R)Q , then the vanishing property of intersection
multiplicity holds (Roberts [12], [13]).
3. Roberts [14] proved µR ∈ SN (R) if ch(R) = p > 0. Using it, he proved
the new intersection theorem in the mixed characteristic case.
4. µR ∈ SN (R) if R contains a ﬁeld (Kurano-Roberts [11]). Even if R isa Gorenstein ring (of mixed characteristic), we do not know whetherµR ∈ SN (R) or not.
5. If µR ∈ SN (R) for any R, then Serre’s positivity conjecture is true inthe case where one of two modules is (not necessary maximal) CohenMacaulay.
If µR ∈ CCM (R) for any R, then small Mac conjecture is true, and so
Serre conjecture is true in the case.
Remark 16
1. If R is Cohen-Macaulay of characteristic p > 0, then e R
is a MCM module. Since µR is the limit of [e R]/pde in G0 (R)R , µR is
contained in CCM (R)− . In the case where R is not of characteristic

p > 0, we do not know whether µR is contained in CCM (R)− even if R
is Gorenstein.
2. As we have already seen, if R is Cohen-Macaulay, then [R] ∈ Int(CCM (R)) ⊂CCM (R).
There is an example of non-Cohen-Macaulay ring R such that [R] ̸∈
SN (R).4 On the other hand, it is expected that µR ∈ SN (R) for any
R. Therefore, for the non-Cohen-Macaulay local ring R, µR behavesbetter than [R] in a sence.
The fundamental class µR is deeply related to homological conjectures.
Therefore, we propose the following question.
Question 17 Assume that R is a ”good” Cohen-Macaulay local domain (forexample, equi-characteristic, Gorenstein, etc). Is µR in CCM (R)?
We can prove the following:
Theorem 18 (Kurano-Ohta [10]) Assume that R is an F-finite CohenMacaulay local domain of characteristic p > 0 with residue class field algebraically closed.
1. If R is FFRT, then µR is contained in CCM (R).
2. If R is F-rational, then µR is contained in Int(CCM (R)).
The most important point in this proof is to use the dual F-signaturedeﬁned by Sannai [16].
Remark 19 If the rank of G0 (R) is one for a Cohen-Macaulay local domain
R, then µR ∈ CCM (R).
If R is a toric ring (a normal semi-group ring over a ﬁeld k), then we canprove µR ∈ CCM (R) as in the case of FFRT without assuming that ch(k) is
positive.

4

It was conjectured that [R] ∈ SN (R).

Problem 20
1. As in the above proof, if there exists a MCM module
in Int(CCM (R)) such that its generalized F-signature or its dual Fsignature is positive, then µR is in Int(CCM (R)− ).
Without assuming that R is F-rational, do there exist such a MCM
module?
2. How do we make mod p reduction?

(the case of rational singularity)

3. If R is Cohen-Macaulay, is µR in CCM (R)− ? If R is a Cohen-Macaulay
ring containing a ﬁeld of positive characteristic, then µR in CCM (R)−
as in 1) in Remark 16.
4. If R is of ﬁnite representation type, is µR in CCM (R)?
5. Find more examples of CCM (R) and SN (R).
In order to prove the following corollary, we use the fact µR ∈ Int(CCM (R))
for some F-rational ring R.
Corollary 21 (Chan-Kurano [1]) Let d be a positive integer and p a prime
number. Let ϵ0 , ϵ1 , . . . , ϵd be integers such that
i = d,
 1
ϵi =−1, 0 or 1 d/2 < i < d,0i ≤ d/2.
Then, there exists a d-dimensional Cohen-Macaulay local ring R of characteristic p, a maximal primary ideal I of R of finite projective dimension,and positive rational numbers α, βd−1 , βd−2 ,. . . , β0 such thatℓR (R/I[pn ]) = ϵd αp

dn

+

d−1
∑
i=0

for any n > 0.

ϵi βi pin

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Kazuhiko Kurano
Department of Mathematics, School of Science and Technology,
Meiji University,
Higashimita 1-1-1, Tama-ku, Kawasaki 214-8571, Japan,
kurano@isc.meiji.ac.jp
http://www.isc.meiji.ac.jp/~kurano