Communications in Algebra, ( ), 1–25 ( )

Department of Mathematics
Ohio State University
Columbus, OH 43210 (USA)
schoutens@math.ohio-state.edu

Projective Dimension and the Singular Locus

Hans Schoutens

Abstract

For a Noetherian local ring, the prime ideals in

the singular locus completely determine the category of
finitely generated modules up to direct summands, ex-
tensions and syzygies. From this some simple homolog-
ical criteria are derived for testing whether an arbitrary
module has finite projective dimension.

Key Words: projective dimension, singular locus,

syzygies, Betti numbers

I. Introduction

In this paper, we address the problem of determining whether

a certain module Ω over a Noetherian ring R has finite projective
dimension. By the results of Jensen and others, the exact value of
this projective dimension, when finite, might depend on one’s model
of set theory (see for instance [5, §5]). Here, we will content ourselves

Copyright C 2000 by Marcel Dekker, Inc. www.dekker.com

2 Hans Schoutens

with proving its finiteness. In fact, we actually will give criteria for
Ω to have finite flat dimension (that is to say, admitting a finite
flat resolution). It is well-known that this is equivalent with having
finite projective dimension (see for instance [5, Proposition 5.6]).
Moreover, for Noetherian local rings, flat dimension is at most the
dimension of the ring ([1, Theorem 2.4]).
Since this is essentially a local issue, I will in the remainder of this
introduction assume that R is moreover local with maximal ideal
m and residue field k. If Ω is finitely generated, then the vanishing
of TorR n (Ω, k) suffices to conclude that Ω has flat (whence projec-
tive) dimension at most n − 1. The Local Flatness Criterion (see
for instance [7, Theorem 22.3]) can be used to extend this to certain
non-finitely generated modules, albeit chiefly in case n = 1. However,
for an arbitrary R-module Ω, the vanishing of a single Tor module
will not suffice. In fact, k-rigidity fails in this generality. That is
to say, the vanishing of TorR n (Ω, k) does not necessarily entail the
vanishing of the higher Tor modules TorR m (Ω, k), with m > n. How-
ever, I will prove in Theorem V.4 that k-rigidity holds whenever n
is at least the dimension of R, under the additional assumption that
R is Cohen-Macaulay. Although the proof uses the degeneration of
spectral sequences and does not work without the Cohen-Macaulay
assumption, it is conceivable that this latter assumption can actually
be removed from the statement.
Nonetheless, this restriction on the asymptotical behavior of the
Betti numbers does not yet solve our original problem. To this end,
we need to show that TorR n (Ω, ·) is the zero functor, for some n ≥ 1,
and in fact, it suffices to show this for the restriction of TorRn (Ω, ·) to
the category modR of finitely generated R-modules. In homological
algebra, one can distinguish three rules of inference for the vanishing
of a functor: (1) if the functor is additive, vanishing is preserved un-
der direct summands; (2) if the functor is exact in the middle, then
vanishing on the outer two modules in a short exact sequence en-
tails the vanishing on the inner module; (3) if we have a collection of
derived functors, then vanishing is transferred between two consec-
PROJECTIVE DIMENSION AND THE SINGULAR LOCUS 3

utive derivatives by taking (co-)syzygies.* In conclusion, we need to

understand the structure of the category modR up to the formation
of direct summand, extension and syzygy. This will be formalized in
this paper by the notion of syzygycal net (see Section VI for details).
If we want to study a Tor functor in a single dimension, we should
not use the third inference rule; the corresponding weaker notion
is that of a net. In other words, a net is a subclass of modR closed
under direct summands and extensions. The key observation is that
modR , as a net, can be built up from a relative small collection of
(cyclic) modules: it suffices to take all R/I where I is either a prime
ideal in the singular locus of R or otherwise a parameter ideal (an
ideal generated by as many elements as its height). If R is moreover
Cohen-Macaulay, then any parameter ideal is generated by a regular
sequence and therefore has finite projective dimension. Therefore,
the following result (Theorem VI.8 in the text) is immediate under
this additional assumption, whereas if R is not Cohen-Macaulay, a
more detailed study of Koszul homology is required.
Main Theorem. Let R be a Noetherian ring. Any finitely generated
R-module can be built up from cyclic modules of the form R/p with p
a prime ideal in the singular locus of R, by taking direct summands,
extensions and syzygies.
It follows from a careful analysis of the way in which a finitely
generated R-module is obtained from the singular locus by the three
inference rules, that, for some a > 0, the vanishing of TorR n (Ω, R/p)
for all n = a, . . . , a + d and all p in the singular locus of R, implies
that Ω has finite projective dimension. Using k-rigidity in high di-
mension (as explained above), it suffices to show this for a single
value n ≥ d (in the non-local case, we need n > d) under the ad-
ditional Cohen-Macaulay assumption. In fact, we can improve this
even further to obtain the following result.
Corollary. Let R be a d-dimensional local Cohen-Macaulay ring
with residue field k. Let Ω be an arbitrary R-module. If the Betti

*
There is in fact a fourth rule, deformation by means of regular elements, which
is implicitly used in the proofs of Theorems V.4 and VI.10.
4 Hans Schoutens

number βaR (p; Ω) vanishes, that is to say, if TorR

a (Ω, R/p)p = 0, for
some a ≥ d and for all p in the singular locus of R, then Ω has finite
projective dimension.
In particular, if R is Cohen-Macaulay with an isolated singularity,
then the vanishing of the single Tor module TorR a (Ω, k), for some
a ≥ d, implies that Ω has finite projective dimension.
Without the Cohen-Macaulay assumption, we cannot formulate
such a criterion using the vanishing of Tor in just a single dimension.
In stead, we need to require that we have vanishing for all n in some
interval of length d + 1. Presumably, this is not an optimal result
and it would be interesting to reduce the size of such a test interval;
of course, if k-rigidity in high dimensions holds, then we can reduce
this again to a single value for n.

II. Nets

II.1. Definition. Let R be a ring. The collection of all (isomor-

phism classes of) finitely generated R-modules will be denoted by
modR . Let N ⊂ modR be a class of finitely generated R-modules. We
say that N is a net, if N is closed under extensions and direct sum-
mands. In other words, if the following two conditions are satisfied.
(Net) If we have a short exact sequence of finitely generated R-
modules
0→K→M →N →0
for which K and N belong to N , then so does M .
(DirSum) If the above sequence is split exact (that is to say, M ∼
=
K ⊕ N ) and M belongs to N , then so do both K and N .
In particular, if R belongs to a net N , then so does any finitely
generated free module, and, more generally, any finitely generated
projective module. The intersection of an arbitrary number of nets is
again a net. Therefore, for each class of finitely generated R-modules
K, there exists a smallest net containing it. We will denote this net
by net(K) and call it the net generated by K.
PROJECTIVE DIMENSION AND THE SINGULAR LOCUS 5

Recall that a subset V of Spec R is called stable under special-

ization, if p ∈ V and p ⊂ q ∈ Spec R imply q ∈ V . In particular, a
Zariski closed subset is stable under specialization.
For an arbitrary subset V of Spec R, let us denote by supp(V )
the collection of all finitely generated R-modules M which support
in V , that is to say, M belongs to supp(V ), if Mp 6= 0 implies p ∈ V ,
for every prime ideal p of R. Recall that for a finitely generated R-
module M , the support Supp M is equal to the Zariski closed set
defined by the annihilator, AnnR (M ), of M .
II.2. Lemma. Let R be a Noetherian ring and let V be a subset of
Spec R. If V is stable under specialization, then supp(V ) is a net,
and, moreover, as such, it is generated by all cyclic modules of the
form R/p with p ∈ V .
Proof. Let N be the net generated by all cyclic modules R/p with
p ∈ V . We need to show that supp(V ) = N . Let M be a finitely gen-
erated R-module with Supp M ⊂ V . There is a filtration by finitely
generated R-modules
0 = M0 ⊂ M1 ⊂ · · · ⊂ Mn = M
such that each Mi+1 /Mi is isomorphic to some R/p, with p ∈ Supp M
(see for instance [7, Theorem 6.4]). By an inductive use of rule (Net),
it follows that M ∈ N .
Conversely, we need to show that if M ∈ N , then its support lies
inside V . This is trivial for the modules R/p with p ∈ V , since the
support of R/p consists of the Zariski closed set defined by p and
since V is stable under specialization. If the support of a module
lies in V , then so does the support of any of its direct summands.
Therefore, by an inductive argument, it suffices to prove that if
0→K→M →N →0
is exact with the support of K and N contained in V , then so is
the support of M . However, this is clear, since always Supp M =
Supp K ∪ Supp N .

II.3. Example (Nets and their generators). Let us apply the

lemma to various choices of V . The resulting nets are then generated
6 Hans Schoutens

by all cyclic modules of the form R/p with p ∈ V .

1. Let V be Spec R. This is trivially stable under specialization
and supp(V ) = modR .
2. Fix some h ≥ 0 and let V be the subset of Spec R consisting
of all prime ideals p of height at least h. This is clearly stable
under specialization and supp(V ) is the net Ih consisting of
all finitely generated R-modules whose annihilator has height
at least h.
3. Fix some h ≥ 0 and let V be the subset of Spec R consisting
of all prime ideals p of depth at least h. Again this is stable
under specialization and supp(V ) is the net Gh consisting of
all finitely generated R-modules of grade at least h.
4. Fix some h ≥ 0 and let V be the subset of Spec R consisting
of all prime ideals p for which R/p has dimension at most h.
Again this is stable under specialization and supp(V ) is the net
Dh consisting of all finitely generated R-modules of dimension
at most h.
5. Let V be the subset of Spec R consisting of all maximal ide-
als. This is trivially stable under specialization and supp(V )
consists of all finitely generated R-modules of finite length.
The usefulness of nets becomes apparent by the following result.
II.4. Proposition. Let R be a ring and let F be an additive functor
from the category of R-modules to an abelian category. Suppose F
is exact in the middle. Let K be a collection of finitely generated R-
modules such that F(K) = 0, for each K ∈ K. Then F(M ) = 0, for
each M ∈ net(K).
Proof. Recall that a (covariant) functor is called exact in the middle,
if any short exact sequence
0→K→M →N →0
transforms into an exact sequence
F(K) → F(M ) → F(N ).
For a contravariant functor the definition is the same apart from re-
versing the arrows. The statement is now immediate using induction
PROJECTIVE DIMENSION AND THE SINGULAR LOCUS 7

on the number of times the rules (Net) and (DirSum) are used.
II.5. Remark. For R an arbitrary ring and Ω an arbitrary R-module,
the following are additive functors which are exact in the middle:
TorR i i
i (Ω, ·), ExtR (Ω, ·) and ExtR (·, Ω), for any i, and, more generally,
so is any derived functor of a left or right exact functor.
II.6. Corollary. Let (R, m) be a Noetherian local ring with residue
field k. Let Ω be an arbitrary R-module. If TorR
n (Ω, k) = 0, for some
n ∈ N, then TorR n (Ω, M ) = 0, for every R-module M of finite length.
Proof. Follows immediately from (5) in Example II.3 in combination
with Proposition II.4 and Remark II.5.
Of course, the same is true for any other additive functor which
is exact in the middle. For some more applications of Corollary II.6,
see [8].

III. Singular Locus

The singular locus of a Noetherian ring R is the collection of all

prime ideals p of R for which Rp is not regular. We will denote it by
Sing R. In this paper, an ideal I is called a parameter ideal, if it is
generated by as many elements as its height (the reader should be
aware that this is not always the standard usage of the term). With
the phrase I generates locally one of its minimal primes, we will
mean that for some minimal prime p of I, we have that IRp = pRp.
Note that if I is moreover a parameter ideal, then such a minimal
prime is necessarily in the regular locus of R. The key result of this
paper is the following theorem.
III.1. Theorem. Let R be a Noetherian ring. Then modR , as a net,
is generated by all cyclic modules of the form R/I, with I either a
prime ideal in the singular locus of R or else a parameter ideal which
generates locally one of its minimal primes.
Proof. Let N be the net generated by all R/I with I either in
the singular locus of R or else a parameter ideal which generates
locally one of its minimal primes. By (1) in Example II.3 (or by a
8 Hans Schoutens

simple induction on the number of generators), modR is generated as

a net by all cyclic modules. Therefore, towards a contradiction, we
may assume that at least one R/a does not belong to N . Let a be
maximal among all such ideals. Let p be a minimal prime of a. If
a 6= p, then we have an exact sequence

0 → R/p → R/a → R/b → 0

with a b. However, by maximality both R/p an R/b belong to N .

By rule (Net), then so does R/a, contradiction. Therefore a = p. By
our assumption, we must have that p ∈ / Sing R, that is to say, that
Rp is regular. I claim that there exists a parameter ideal I inside
p of height ht p, such that IRp = pRp. This is an easy exercise in
prime avoidance, but for sake of completeness, I will give a proof by
induction on the height h of p. The case h = 0 is trivial, since Rp is
then a field. Suppose h > 0. In particular, p is not contained in any
minimal prime of R. By prime avoidance, we can find x1 ∈ p, not in
p2 nor in any minimal prime p0j of R. In particular, x1 R has height
one. If h = 1 we are done, since regularity implies that x1 Rp = pRp.
Otherwise, if h > 1, we have that p is neither a minimal prime p1j
of x1 R nor can it be equal to x1 R + p2 , by Nakayama’s Lemma.
By prime avoidance, we can find x2 ∈ R, not in x1 R + p2 nor in any
p0j or p1j . It follows that (x1 , x2 )R has height 2. Continuing this
way, we find x1 , . . . , xh ∈ p, such that their images in pRp/p2 Rp are
linearly independent and such that (x1 , . . . , xh )R has height h. Since
Rp is regular, the first condition implies that (x1 , . . . , xh )Rp = pRp
(see for instance [7, Theorem 14.2]).
Since p is a minimal prime of I, we can find s ∈ R, such that
p = (I :R s). Since IRp = pRp, one checks that s ∈ / p. We have an
equality

I = p ∩ (I + sR).

Indeed, one inclusion is immediate, so take z ∈ p ∩ (I + sR). Hence

we can write z = i + sa, with i ∈ I and a ∈ R. Since sa = z − i ∈ p
and since s ∈
/ p, we get a ∈ p. Since p = (I : s), we get that sa belongs
to I, whence so does z, as claimed.
In general, if a and b are ideals in a ring R, then we have an exact
PROJECTIVE DIMENSION AND THE SINGULAR LOCUS 9

sequence
s t
0 → R/(a ∩ b) −→ (R/a) ⊕ (R/b) −→ R/(a + b) → 0,
where s sends an element x + (a ∩ b) to the pair (x + a, x + b) and t
sends a pair (x + a, y + b) to the element (x − y) + (a + b). Applied
to the present situation with a = p and b = I + sR, we get an exact
sequence
0 → R/I → R/p ⊕ R/(I + sR) → R/(p + sR) → 0. (1)
By maximality, R/(p + sR) belongs to N and by construction so
does R/I. Using the rules (Net) and (DirSum), it then follows from
(1) that also R/p belongs to N , contradiction.
III.2. Remark. If R is moreover Cohen-Macaulay, then any parame-
ter ideal is generated by an R-regular sequence. Under this additional
assumption, fix some h ∈ N. Consider the net Gh consisting of all
finitely generated R-modules of grade at least h, as discussed in (3)
in Example II.3. By Lemma II.2, the net Gh is generated by all R/p
with p a prime ideal of depth at least h. Analyzing the above proof,
we see that the collection of all R/I already generate Gh , where I is
either an ideal generated by an R-regular sequence of length at least
h or a prime ideal in the singular locus of R of depth at least h.
However, in the presence of embedded associated primes, funny
things can happen: it is very well possible that p belongs to the
regular locus of R but its depth is strictly less than its height. If R is
not Cohen-Macaulay, then the nets Gh are of lesser use. Instead, one
can use the nets Ih , introduced in (2) in Example II.3, consisting of
all finitely generated R-modules whose annihilator has height at least
h. The argument in the proof of the Theorem shows that this net is
generated by all R/I, with I an ideal of height at least h which is
either a parameter ideal locally generating one of its minimal primes
or else a prime ideal in the singular locus of R.
The following example shows some of the complications that arise
in the absence of the Cohen-Macaulay property.
III.3. Example. Let k be a field and let
R := (k[X, Y, Z]/I)m
10 Hans Schoutens

where I is the ideal generated by X 2 , XY and XZ and where m

is the maximal ideal generated by X, Y and Z. One checks that
R has dimension 2 and depth 0, so that it is not Cohen-Macaulay.
Moreover, Sing R = {m}. In this case, we cannot expect that we can
strengthen the assertion in the Theorem by taking only parameter
ideals generated by a regular sequence. Indeed, in this case, there
are no such, except for the zero ideal. However, the net generated
by k = R/m and R is not modR . Indeed, suppose the contrary. Let
g = XR and p = (X, Y )R. Consider the exact sequence
0 → R p → Rg → Ω → 0
Since Rg is flat and Rp ⊗ k = 0, we get after tensoring this sequence
with k that TorR 1 (Ω, k) = 0. Therefore, if modR would be equal to
net(k, R), then by Corollary II.6, we would have that TorR 1 (Ω, ·) is
identically zero on modR . Consequently, Ω would be flat. However,
tensoring the above sequence with R/Y R yields TorR ∼
1 (Ω, R/Y R) =
Rp/Y Rp and the latter is isomorphic to k(p), showing that Ω is not
flat. Note that Rp ∼ = k(Z)[Y ](Y ) is a DVR, although p has depth 0.
III.4. Corollary. Let R be a d-dimensional Cohen-Macaulay ring
and let Ω be an R-module. If TorR a (Ω, R/p) = 0, for some a ≥ d and
all p ∈ Sing R, then Ω has finite projective dimension.
Proof. Since this is a local problem, we may assume without loss of
generality that R is local with maximal ideal m. Moreover, as the con-
clusion holds trivially for regular local rings, we may assume that R
is not regular. Let I be a parameter ideal which, moreover, generates
locally one of its minimal primes. In particular, this last condition
forces that minimal prime to be in the regular locus. Therefore, any
such parameter ideal has height h at most d − 1. Moreover, since
R is Cohen-Macaulay, I is generated by an R-regular sequence of
length h by [7, Theorem 17.4]. It is well-known (see for instance [2,
Corollary 1.6.14]) that R/I has projective dimension h. Therefore,
we get that
TorR
a (Ω, R/I) = 0.

By Theorem III.1 and Proposition II.4, we conclude that TorR

a (Ω, ·)
PROJECTIVE DIMENSION AND THE SINGULAR LOCUS 11

vanishes identically on modR , showing that Ω has flat dimension (at

most a − 1). Since any flat module has finite projective dimension,
the claim follows.

There is a similar criterion for finite injective dimension.

If ExtaR (R/p, Ω) = 0, for some a ≥ d and all p ∈ Sing R,
then Ω has finite injective dimension.
Indeed, the question is again local and by [7, §18 Lemma 1], it
suffices to show that ExtaR (·, Ω) vanishes on modR to conclude that
Ω has finite injective dimension at most a − 1.

IV. Big Cohen-Macaulay Modules

We apply the results from the previous section to obtain a flat-

ness criterion for balanced big Cohen-Macaulay modules over a lo-
cal Cohen-Macaulay ring. Recall that an arbitrary module Ω over a
Noetherian local ring (R, m) is called a big Cohen-Macaulay module,
if there exists a system of parameters (x1 , . . . , xd ) in R, such that
(x1 , . . . , xd ) is a Ω-regular sequence. We call Ω moreover balanced,
if this is true for every system of parameters. Note that if R itself
is Cohen-Macaulay, then Ω is a (balanced) big Cohen-Macaulay if
(every) some maximal R-regular sequence is Ω-regular. In particular,
any flat R-module is a balanced big Cohen-Macaulay module. For a
regular local ring, the converse also holds; see [3, p. 77], [4, Proof of
Theorem 9.1] or the argument in the proof below. The following is a
generalization to local Cohen-Macaulay rings.
IV.1. Theorem. Let (R, m) be a local Cohen-Macaulay ring with
an isolated singularity. Let k denote the residue field of R and d its
dimension. For an R-module Ω, the following are equivalent.
1. Ω is flat;
2. Ω is a balanced big Cohen-Macaulay module of finite projective
dimension;
3. Ω is a balanced big Cohen-Macaulay module with TorR 1 (Ω, k) =
0.
12 Hans Schoutens

In fact, for the equivalence of (1) and (2), we do not need to assume
that R has an isolated singularity.
Proof. Assume first that R is only Cohen-Macaulay. It follows that
any system of parameters is an R-regular sequence. As being a reg-
ular sequence is preserved by flatness, (1) implies (2). To prove that
(2) implies (1), one can use essentially the same argument as in the
parenthetical remark in [3, p. 77] or in the proof of [4, Theorem
9.1]. For sake of convenience, I repeat this argument here. Let n
be the maximum of all i ≥ 1 for which TorR i (Ω, ·) is not identically
zero. Note that n is finite, since Ω has finite projective dimension.
By (1) in Example II.3, there is some prime ideal p of R for which
TorRn (Ω, R/p) 6= 0. Let (x1 , . . . , xh ) be a maximal regular sequence
in p. Since R is Cohen-Macaulay, h is the height of p, so that p is a
minimal prime of R/(x1 , . . . , xh )R. Therefore, we have a short exact
sequence
0 → R/p → R/(x1 , . . . , xh )R → C → 0
for some finitely generated R-module C. The Tor long exact sequence
gives an exact sequence
TorR R R
n+1 (Ω, C) → Torn (Ω, R/p) → Torn (Ω, R/(x1 , . . . , xh )R)

The left most module in this sequence is zero by maximality of n

whereas the right most is zero since (x1 , . . . , xh ) is also Ω-regular.
Therefore, TorRn (Ω, R/p) = 0, contradiction.
So remains to show the equivalence of (3) with the first two con-
ditions under the additional assumption that R has an isolated sin-
gularity. Clearly (1) implies (3). Therefore, assume that (3) holds
and we seek to show that then so does (2). We only need to show
that Ω has finite projective dimension. Let (x1 , . . . , xd ) be a maximal
R-regular sequence. By assumption, (x1 , . . . , xd ) is also Ω-regular. In
particular, we have, for every j > 0, that
TorR
j (Ω, R/(x1 , . . . , xd )R) = 0. (2)
Since (x1 , . . . , xd )R is m-primary, we can find an ascending chain of
ideals ai with a0 = (x1 , . . . , xd )R and am = m, such that ai+1 /ai ∼
= k,
for all i. Let us show by lexicographical induction on the pair (j, i)
PROJECTIVE DIMENSION AND THE SINGULAR LOCUS 13

that all TorRj (Ω, R/ai ) = 0, for j ≥ 1 and i = 0, . . . , m. When j = 1,

this follows from our assumption that TorR 1 (Ω, k) = 0 and Corol-
lary II.6, since each ai is m-primary. Therefore, let j > 1. Equal-
ity (2) proves the case i = 0, hence we may also assume that i > 0.
By construction, we have an exact sequence
0 → k → R/ai−1 → R/ai → 0.
From the Tor long exact sequence we get that
TorR R R
j (Ω, R/ai−1 ) → Torj (Ω, R/ai ) → Torj−1 (Ω, k).

By induction on i, the first of these modules is zero and by induction

on j, so is the last. This proves the claim. In particular, we showed
that TorRj (Ω, k) vanishes, for all j ≥ 1. By Corollary III.4 we get that
Ω has finite projective dimension, as required.

Using Kunz’s Theorem, we get the following criterion for regu-

larity. Recall that for a ring of prime characteristic p, the Frobenius
endomorphism is defined by Fp : x 7→ xp . Let us write RFp for R
viewed as an R-module via Fp .
IV.2. Corollary. Let R be a reduced Cohen-Macaulay ring of prime
characteristic p. Then R is regular if, and only if, RFp has finite
projective dimension.
Proof. One direction is of course just Serre’s homological char-
acterization of regularity. Therefore assume R is a reduced Cohen-
Macaulay ring such that RFp has finite projective dimension over
R. Since everything is preserved under localization, we may assume
that R is local. Clearly, if (x1 , . . . , xn ) is R-regular, then the same
is true for (xp1 , . . . , xpn ), showing that RFp is a balanced big Cohen-
Macaulay module. Theorem IV.1 then yields that RFp is flat over R.
By Kunz’s Theorem (see for instance [6, Theorem107]), it follows
that R is regular.

Note that if R is moreover local, then R is regular if, and only

if, TorR Fp
1 (R , k) = 0, where k is the residue field of R. Indeed, by
the Local Flatness Criterion ([7, Theorem 22.3]) the vanishing of
TorR Fp
1 (R , k) implies that R → R
Fp (that is to say, the homomor-
14 Hans Schoutens

phism Fp ) is flat and Kunz’s Theorem then shows that R is regular.

For some other, related consequences of Theorem IV.1, see [9].

V. Asymptotic behavior of Betti numbers

V.1. Definition. Let R be a Noetherian ring and let Ω be an arbi-

trary R-module. The n-th Betti number of Ω at the prime p, is the
(possibly infinite) dimension of the k(p)-vector space
R
Torn p (Ωp, k(p)) = TorR
n (Ω, R/p)p,

where k(p) denotes the residue field of p, that is to say, k(p) =

Rp/pRp. We denote n-th Betti number by βnR (p; Ω), or simply, by
βn (p; Ω) if the ring is understood.

The following result gives a restriction on the asymptotic behavior

of the Betti numbers.

V.2. Theorem (k-rigidity in high dimensions). Let (R, m) be a

d-dimension local Cohen-Macaulay ring with residue field k and let
Ω be an arbitrary R-module. If TorR
a (Ω, k) = 0, for some a ≥ d, then
TorR
m (Ω, k) = 0, for all m ≥ a.

Proof. We will induct on the dimension d of R. Suppose first that

d = 0. If a = 0, so that 0 = TorR 0 (Ω, k) = Ω ⊗R k, whence Ω = mΩ,
we get that Ω = 0, as m is nilpotent. Suppose next that a = 1. By
Corollary II.6, it follows that TorR
1 (Ω, M ) = 0, for every R-module M
of finite length. However, since R is Artinian, every finitely generated
R-module has finite length. Therefore, Ω is in fact flat, and the claim
holds trivially. For a ≥ 2, we can take syzygies to reduce to the case
a = 1.
Next, suppose that d = 1. Again, by taking syzygies, we may re-
duce to the case that a = 1. Since R is Cohen-Macaulay, we can find
an R-regular element x ∈ m, so that the standard spectral sequence

TorR/xR
p (TorR R
q (Ω, R/xR), k) =⇒ Torp+q (Ω, k)
PROJECTIVE DIMENSION AND THE SINGULAR LOCUS 15

degenerates into an exact sequence

R/xR
Torm−1 (AnnΩ (x), k) → TorR R/xR
m (Ω, k) → Torm (Ω/xΩ, k) → . . .
R/xR
→ AnnΩ (x) ⊗ k → TorR
1 (Ω, k) → Tor1 (Ω/xΩ, k) → 0, (3)
where AnnΩ (x) denotes the submodule of all elements in Ω an-
nihilated by x. Since by assumption TorR 1 (Ω, k) vanishes, so does
R/xR
Tor1 (Ω/xΩ, k) by (3). Therefore, we obtain by the above zero
R/xR
dimensional case that all Torm (Ω/xΩ, k) vanish, for m ≥ 1. Us-
ing (3) once we more, we get that AnnΩ (x) ⊗ k vanishes. Again by
R/xR
the zero dimensional case we obtain that all Torm (AnnΩ (x), k)
vanish. Therefore, so do all TorR
m (Ω, k), by (3).
Finally, assume d ≥ 2 and let x ∈ m be R-regular. Let
0→Π→Φ→Ω→0
be exact, with Φ free. In particular, we obtain, for all p ≥ 1, that
TorR ∼ R
p+1 (Ω, k) = Torp (Π, k). (4)
Moreover, since Π is a submodule of a free module, we have that x
is also Π-regular, so that
TorR ∼ R/xR
m (Π, k) = Torm (Π/xΠ, k), (5)

for all m ≥ 1. By assumption TorR a (Ω, k) = 0, where a ≥ d ≥ 2, so

R/xR
that by (4) and (5), we have that Tora−1 (Π/xΠ, k) = 0. Therefore,
R/xR
Torm (Π/xΠ, k) = 0, for all m ≥ a − 1, by our induction hypoth-
esis. By (4) and (5) again, it follows that TorR
p (Ω, k) = 0, for all
p ≥ a.

V.3. Example. In low dimensions, however, k-rigidity fails, even

for regular local rings, as the following example shows. Let (R, m) be
a regular local ring of dimension d ≥ 2. Let E be the injective hull of
the residue field of R. It is well-known that all βnR (m; E) = 0, except
when n = d, in which case the Betti number is one (see for instance
[2, Exercise 3.3.26]).
I do not know of any counterexample to the Theorem without the
Cohen-Macaulay condition.
16 Hans Schoutens

Our next goal is to replace in Corollary III.4 the requirement that

TorR R
n (Ω, R/p) vanishes, by the weaker condition that (Torn (Ω, R/p))p
R
vanishes, that is to say, that βn (p; Ω) = 0.

V.4. Theorem. Let R be a d-dimensional Cohen-Macaulay ring

and let Ω be an R-module. If for some a ≥ d, we have that βaR (p; Ω) =
0, for all p in the singular locus of R, then Ω has finite projective
dimension.

Proof. By localizing, we may assume from the start that R is a

d-dimensional local Cohen-Macaulay ring with residue field k and
maximal ideal m. Moreover, the statement is trivial if R is regular,
so that we may assume that m lies in the singular locus. By Theo-
R (p; Ω) = 0, for m ≥ a and p ∈ Sing R. Note that each
rem V.2, all βm
Rp is Cohen-Macaulay of dimension at most d.
As in the proof of Corollary III.4, for I a parameter ideal of height
at most d − 1, we have that

TorR
m (Ω, R/I) = 0 (6)

for all m ≥ d, since any such ideal is generated by an R-regular se-

quence.
I claim that TorRm (Ω, R/p) = 0, for all prime ideals p which are in
the singular locus Sing R of R and all m ≥ a. Assuming the claim,
the assertion then follows from Corollary III.4. To prove the claim,
we will perform a downward induction on the height h of p. By
assumption, the case p = m, that is to say, h = d, holds, so that we
may assume h < d. By Remark III.2, the net Ih+1 , introduced in (3)
of Example II.3, is generated by all R/I, with I a parameter ideal of
height e ≥ h + 1 which generates locally one of its minimal primes,
together with all R/q, with q a prime ideal in the singular locus of
R of height at least h + 1. Since R is not regular, e is at most d − 1.
By induction on h, we have that TorR m (Ω, R/q) = 0, for all m ≥ a
and all prime ideals q ∈ Sing R of height at least h + 1. In view of
(6) this implies by Remark III.2 and Proposition II.4 that TorR m (Ω, ·)
vanishes on the whole net Ih+1 , for all m ≥ a. Take a height h prime
p in the singular locus of R, if any. For an arbitrary x ∈ / p, we have
PROJECTIVE DIMENSION AND THE SINGULAR LOCUS 17

an exact sequence
x
0 → R/p −→ R/p → R/a → 0. (7)
where a := xR + p. In particular, since a has height h + 1, we get
that
TorR
m (Ω, R/a) = 0, (8)
for all m ≥ a. Fix some m ≥ a. From the long exact sequence ob-
tained from (7) by tensoring with Ω, we get from (8), that
x
0 → TorR R
m (Ω, R/p)−→ Torm (Ω, R/p) → 0

is exact (note that at this point, it is crucial that we do not just have
vanishing in dimension a). In particular, x is not a zero-divisor on
TorRm (Ω, R/p). By assumption
R
(TorR
m (Ω, R/p))p = Torm (Ωp, k(p)) = 0.
p

It follows from these two observations that TorR

m (Ω, R/p) = 0 by
Lemma V.5 below.
V.5. Lemma. Let Λ be an R-module and let p be a prime ideal of
R. If Λp = 0 and x is Λ-regular, for every x ∈
/ p, then Λ = 0.
Proof. Pick any τ ∈ Λ. Since Λp = 0, there is some x ∈
/ p, such that
xτ = 0. By assumption, x is Λ-regular, showing that τ = 0.

VI. Syzygycal Nets

The goal of this section is to establish a similar result as Corol-

lary III.4 without the Cohen-Macaulay assumption. To this end, it
is useful to view the results from Section III in an alternative way.
VI.1. Definition. Let R be a ring and N a net. We call N syzygy-
cal, if it is closed under syzygies and co-syzygies. More precisely, if
the following condition is satisfied
(Syz) Given an exact sequence of finitely generated R-modules
0 → K → F → M → 0, (9)
18 Hans Schoutens

with F free, then M ∈ N if, and only if, K ∈ N .

Since any net contains the zero module (by rule (DirSum)) and
since R is a syzygy of the zero module, any syzygycal net contains
R. Therefore, any syzygycal net contains all finitely generated pro-
jective modules. Therefore, by rule (Syz), any syzygycal net contains
all finitely generated modules of finite projective dimension. In par-
ticular, if R is regular, then the only syzygycal net is modR itself.
VI.2. Definition (Meanders). We call the smallest syzygycal net
N generated by some class K of finitely generated R-modules, the
syzygycal net generated by K. To measure how far we have ’mean-
dered’ from the generators K by invoking rule (Syz), we introduce
the notion of a K-meander of a member M of N . A K-meander (or
simply, meander, if K is clear from the context) will be an interval
in Z of the form [−a, b] with a, b ∈ N. In the following recursive def-
inition of a meander of M (with respect to K), it is understood that
M is derived from K using the three rules for syzygycal nets and for
each of the intermediate modules, at least one meander has already
been defined.
1. If M ∈ K or M is projective, then [0, 0] is a meander of M .
2. If M is the direct summand of some N ∈ N , then any meander
of N is also a meander of M .
3. Suppose we have an exact sequence
0→K→M →N →0
with K, N ∈ N . If JK is a meander of K and JN a meander
of N , then JK ∪ JN is a meander of M .
4. Suppose we have an exact sequence
0→N →F →M →0
with F free and N ∈ M. If [−a, b] is a meander of N , then
[−a, b + 1] is a meander of M .
5. Suppose we have an exact sequence
0→M →F →N →0
with F free and N ∈ M. If [−a, b] is a meander of N , then
PROJECTIVE DIMENSION AND THE SINGULAR LOCUS 19

[−a − 1, b] is a meander of M .
This concludes the recursive definition of a meander of M . Of
course, a module might have infinitely many meanders, but it will
only have finitely many meanders which are minimal with respect
to inclusion. Indeed, if [−a, b] is a meander of M , then a moment’s
reflection shows that M can have at most a + b distinct minimal
meanders. Note that if M lies already in the net generated by K,
that is to say, is derived form K using only rules (Net) or (DirSum),
then [0, 0] is the unique minimal meander of M .
We will tacitly use the following fact, whose easy proof is left to
the reader. Let K ⊂ N be subclasses of finitely generated R-modules.
Suppose N lies in the syzygical net generated by K and every N ∈ N
has a K-meander contained in [−a, b]. Then any M ∈ net(N ) has
also a K-meander contained in [−a, b] (note that in any case, net(N )
is contained in the syzygycal net generated by K).
VI.3. Definition (Kernels). Let F be a right exact (covariant)
functor on the category of R-modules. We denote its left derived
functors by Ln F. Let M be an arbitrary finitely generated R-module.
We define the F-kernel, KerF (M ), of M as the collection of all n ∈ N
for which Ln F(M ) = 0.
For instance, let (R, m) be a Noetherian local with residue field
k ring and let F be the functor · ⊗R k. If M is a finitely gener-
ated R-module of projective dimension d, then KerF (M ) = [d, ∞) (if
d = ∞, this means that KerF (M ) = ∅). If K is a class of finitely gen-
erated R-modules, then KerF (K) is by definition the intersection of
all KerF (K) with K ∈ K. A similar definition can be made for left
exact functors and for contravariant functors.
VI.4. Theorem. Let R be a ring and F a right exact covariant
functor for which KerF (R) = N \ {0}. Let K be a class of finitely
generated R-modules and let N be the syzygycal net generated by K.
Let M ∈ N and let [−a, b] be a meander of M . If for some non-zero
u, v ∈ N, the interval [u, v] lies in KerF (K), then [u + b, v − a] lies in
KerF (M ).
Proof. By a recursive argument, it suffices to show this for M and
20 Hans Schoutens

[−a, b] given by one of the five formation rules in Definition VI.2.

In case (1), the assertion is just our assumption that all positive
integers belong to KerF (R). In case (2), let M be a direct summand
of some N in N . Any right exact functor, whence also its derived
functors, are additive, so that KerF (N ) ⊂ KerF (M ), and the asser-
tion holds, since each meander of N is also a meander of M . In
case (3), let
0→K→M →N →0
be an exact sequence, where K, N ∈ N have respective meanders
[−aK , bK ] and [−aN , bN ] and where [−a, b] is obtained as the union
of these two meanders, that is to say,
a = max{aK , aN } b = max{bK , bN }
By induction, [u + bK , v − aK ] lies in KerF (K) and [u + bN , v − aN ]
lies in KerF (N ). From the long exact sequence
Ln F(K) → Ln F(M ) → Ln F(N ).
it is clear that Ln F(M ) = 0, for all n in
[u + bK , v − aK ] ∩ [u + bN , v − aN ] = [u + b, v − a].
In case (4), consider an exact sequence
0→N →F →M →0
with F free and N ∈ M with meander [−a, b − 1]. By induction,
we have that [u + b − 1, v − a] lies in KerF (N ). From the long exact
sequence of derived functors, we get isomorphisms
Ln+1 F(M ) ∼
= Ln F(N ) (10)
for all n ≥ 1. Therefore, since everything gets shifted up by one, we
have that [u + b, v − a + 1] lies in KerF (M ) and the assertion holds.
In case (5), the same reasoning holds, where this time we have to
shift everything down by one.

We turn now to the generalization of Theorem III.1 and Corol-

lary III.4. Let us first reinterpret the result of Theorem III.1 in our
new terminology.
PROJECTIVE DIMENSION AND THE SINGULAR LOCUS 21

VI.5. Corollary. Let R be a Cohen-Macaulay ring. The syzygycal

net generated by the singular locus Sing R of R, is modR .
More precisely, if S is the class of all R/p with p ∈ Sing R, then
any finitely generated R-module M has an S-meander contained in
[0, d], where d is the dimension of R. If R is moreover local but not
regular, then M has an S-meander contained in [0, d − 1].

Proof. Let N be the syzygycal generated by S. If I is a height e pa-

rameter ideal which generates locally one of its minimal primes, then
I is generated by a regular sequence of length e, since R is Cohen-
Macaulay. The Koszul complex of this sequence is a free resolution
of R/I by [2, Corollary 1.6.14]. Therefore, R/I belongs to N with
(S-)meander [0, e]. In general e ≤ d, and if R is local but not regular,
then I cannot have height d, so that e ≤ d − 1. The statement now
follows from Theorem III.1.

To extend this to the non-Cohen-Macaulay case, we need a lemma

about homology of complexes.

VI.6. Lemma. Let R be a Noetherian ring and N a syzygycal net.

Let F• be a finite complex of finitely generated free R-modules. If for
all i > 0, the homology modules Hi (F• ) belong to N , then so does
H0 (F• ).

Proof. Let F• be a finite free complex of length e, that is to say, a

complex of the form
fe fe−1 f2 f1
Fe −−→Fe−1 −−−→ . . . −−→F1 −−→F0

with each Fi a finitely generated free R-module. Let Hi := Hi (F• ),

for i = 0, . . . , e. Let Ki and Zi−1 denote respectively the kernel and
the image of fi . Put K0 := F0 and Ze := 0. By assumption, all Hi =
Ki /Zi for i = 1, . . . , e, belong to N . We prove by downward induction
on i that Zi and Ki belong to N , where the case i = e holds by
assumption (note that Ke = He ). When we have reached i = 0, we
then conclude by rule (Syz) that also H0 = F0 /Z0 belongs to N and
we are done.
Therefore, suppose the claim proven for i + 1, with 0 ≤ i < e.
22 Hans Schoutens

From the exact sequence

0 → Ki+1 → Fi+1 → Zi → 0

and induction, the cosyzygy Zi belongs to N by rule (Syz). If i = 0,

we are done, since K0 = F0 by definition. If i > 0, the exact sequence

0 → Zi → Ki → Hi → 0

our assumption and rule (Net) then show that Ki belongs to N .

VI.7. Remark. In fact, the above proof shows that H0 (F• ) belongs
to the syzygycal net generated by all Hi (F• ) with i > 0 and as such,
it has a meander [0, e + 1], where e is the length of F• , since we took
e + 1 times a cosyzygy.

VI.8. Theorem. Let R be a d-dimensional Noetherian ring. The

syzygycal net generated by the singular locus Sing R of R, is modR .
More precisely, if S is the class of all R/p with p ∈ Sing R, then
any finitely generated R-module M has an S-meander contained in
[0, d + 1], where d is the dimension of R. If R is moreover local but
not regular, then M has an S-meander contained in [0, d].

Proof. By Theorem III.1, the net modR is generated by the cyclic

modules of the form R/I, with I ∈ Sing R or I a parameter ideal
which generates locally one of its minimal primes. Therefore, it suf-
fices to show that any cyclic module of the form R/I with I =
(x1 , . . . , xh )R a parameter ideal as above, belongs to the syzygycal
net generated by S, with the indicated meander. Let K• (I) be the
Koszul complex of (x1 , . . . , xh ). If p is a prime containing I but not in
the singular locus of R, then the image of (x1 , . . . , xh ) in Rp is an Rp-
regular sequence, since Rp is in particular Cohen-Macaulay. There-
fore, K• (I) becomes acyclic after localization at p. It follows that
the homology modules Hi (K• (I)) for i > 0 all have support inside
Sing R. Therefore, by Lemma II.2 and the fact that Sing R is stable
under specialization, each Hi (K• (I)), for i > 0, belongs to the net
generated by S. By Lemma VI.6, we then get that H0 (K• (I)) = R/I
belongs to the syzygycal net generated by S. By Remark VI.7, it has
an S-meander [0, h + 1].
PROJECTIVE DIMENSION AND THE SINGULAR LOCUS 23

Since always h ≤ d, the first assertion of the final statement is

clear. Suppose finally that R is moreover local but not regular. Since
the parameter ideal I is assumed to locally generate one of its min-
imal primes, h < d (lest the maximal ideal of R is generated by d
elements) and the last assertion follows.
VI.9. Theorem. Let R be a d-dimensional Noetherian ring and let
Ω be an R-module. If there is some a ≥ 1, such that TorR a+j (Ω, R/p) =
a+j
0 (respectively, ExtR (R/p, Ω) = 0), for all j = 0, . . . , d + 1 and all
p ∈ Sing R, then Ω has finite flat (respectively, finite injective) di-
mension (at most a + d).
Moreover, if R is local but not regular, then we only have to check
the vanishing of the Tor modules in the range j = 0, . . . , d.
Proof. Let F = Ω ⊗R · in the first case and HomR (·, Ω) in the second
case. In either case, KerF (R) = [1, ∞), so that Theorem VI.4 applies.
Let S consist of all R/p with p ∈ Sing R. By assumption, [a, a + d +
1] ⊂ KerF (S). Let M be an arbitrary finitely generated R-module.
By Theorem VI.8, the interval [0, d + 1] contains an S-meander of
M . Therefore, by Theorem VI.4, the interval [a + d + 1, a + d + 1]
lies in KerF (M ). This shows that in the first case TorR a+d+1 (Ω, ·)
vanishes on each finitely generated R-module, showing that Ω has
flat dimension at most a + d. In the second case, the vanishing of
a+d+1
ExtR (·, Ω) on modR implies that Ω has injective dimension at
most a + d, by [7, §18 Lemma 1]. The final assertion now follows
from the last statement in Theorem VI.8.
The final result is a local criterion similar to Theorem V.4, involv-
ing only Betti numbers. Since we do not know whether k-rigidity in
high dimensions holds for non-Cohen-Macaulay rings, we can only
state the following weaker version of Theorem V.4.
VI.10. Theorem. Let R be a Noetherian ring and let Ω be an R-
module. If for some a ≥ 1, we have that βa+j R (p; Ω) = 0, for all p

in the singular locus of R and all j = 0, . . . , ht p, then Ω has finite

projective dimension (and, in fact, flat dimension at most a − 1).
Proof. By localizing, we may assume from the start that R is a
Noetherian local ring with residue field k and maximal ideal m. We
24 Hans Schoutens

will prove the result by induction on the dimension d of R. If d = 0,

then either R is a field and there is nothing to prove, or we have that
TorR R
a (Ω, k) = 0. By Corollary II.6, we get that Tora (Ω, M ) = 0, for
every R-module M of finite length. Since R is Artinian, every finitely
generated R-module has finite length and hence we showed that Ω
has finite flat dimension at most a − 1. So let d > 0 and assume that
the result is proven for all lower dimensional Noetherian local rings.
If R is regular, there is nothing to prove, so we may assume that m
lies in the singular locus of R. Let p be a prime ideal different from m.
Since the singular locus of Rp is contained in the singular locus of R,
induction on the dimension yields that Ωp has finite flat dimension
at most a − 1. In particular, βj (p; Ω) = 0, for all j ≥ a and all prime
ideals p different from m.
I claim that TorR j (Ω, R/p) = 0, for all prime ideals p of R and
for all j ∈ [a, a + d − h], where h is the height of p. Assuming the
claim, we get that TorR a (Ω, R/p) vanishes for all prime ideals p,
proving that Ω has flat dimension at most a − 1 by (1) in Exam-
ple II.3. To prove the claim, we perform a downward induction on
the height h of the prime ideal p. Since the case h = d is covered
by the hypothesis (recall that R is singular), we may assume that
h < d. By (2) in Example II.3 and our induction hypothesis on h,
we get that TorR j (Ω, R/a) = 0, for all a of height at least h + 1 and
all j ∈ [a, a + d − h + 1].
Let x be an arbitrary element of R not in p, and put a := xR + p.
Tensoring the short exact sequence
x
0 → R/p −→ R/p → R/a → 0
with Ω, yields a long exact sequence
x
TorR R R
j+1 (Ω, R/a) → Torj (Ω, R/p)−→ Torj (Ω, R/p)

Since a has height h + 1, the left most module vanishes for all j
in the range [a, a + d − h]. Therefore, x is not a zero-divisor on
TorR
j (Ω, R/p), for all j ∈ [a, a + d − h]. Since βj (p; Ω) = 0, for all
j ≥ a, we get by Lemma V.5 that TorR j (Ω, R/p) = 0, for j ∈ [a, a +
d − h], as claimed.
Note that in this proof, we did not use the theory of syzygycal
PROJECTIVE DIMENSION AND THE SINGULAR LOCUS 25

nets. Moreover, the range in which the Betti numbers are required
to vanish is smaller than the one given in Theorem VI.9. Together
with k-rigidity in high dimensions (Theorem V.2), this also provides
in the Cohen-Macaulay case an alternative proof for Theorem V.4.

References

1. M. Auslander and D. Buchsbaum, Homological dimension in

Noetherian rings II, Trans. Amer. Math. Soc. 88 (1958), 194–
206.
2. W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Uni-
versity Press, Cambridge, 1993.
3. M. Hochster and C. Huneke, Infinite integral extensions and big
Cohen-Macaulay algebras, Ann. of Math. 135 (1992), 53–89.
4. C. Huneke, Tight closure and its applications, CBMS Regional
Conf. Ser. in Math, vol. 88, Amer. Math. Soc., 1996.
5. C.U. Jensen, Les foncteurs dérivés de lim
←− et leurs application en
théorie des modules, Lect. Notes in Math., vol. 254, Springer-
Verlag, 1972.
6. H. Matsumura, Commutative algebra, W.A. Benjamin, 1970.
7. , Commutative ring theory, Cambridge University Press,
Cambridge, 1986.
8. H. Schoutens, A local flatness criterion for complete modules,
preprint on http://www.math.ohio-state.edu/~schoutens,
2001.
9. , On the vanishing of Tor of the absolute integral closure,
preprint on http://www.math.ohio-state.edu/~schoutens,
2002.