arXiv:1103.4726v1 [math.AC] 24 Mar 2011
CRITERIA FOR FLATNESS AND INJECTIVITY
NEIL EPSTEIN AND YONGWEI YAO
Abstract. Let R be a commutative Noetherian ring. We give criteria for
flatness of R-modules in terms of associated primes and torsion-freeness of
certain tensor products. This allows us to develop a criterion for regularity
if R has characteristic p, or more generally if it has a locally contracting endomorphism. Dualizing, we give criteria for injectivity of R-modules in terms
of coassociated primes and (h-)divisibility of certain Hom-modules. Along the
way, we develop tools to achieve such a dual result. These include a careful
analysis of the notions of divisibility and h-divisibility (including a localization
result), a theorem on coassociated primes across a Hom-module base change,
and a local criterion for injectivity.
1. Introduction
The most important classes of modules over a commutative Noetherian ring R,
from a homological point of view, are the projective, flat, and injective modules.
It is relatively easy to check whether a module is projective, via the well-known
criterion that a module is projective if and only if it is locally free. However, flatness
and injectivity are much harder to determine.
It is well-known that an R-module M is flat if and only if TorR
1 (R/P, M ) = 0
for all prime ideals P . For special classes of modules, there are some criteria for
flatness which are easier to check. For example, a finitely generated module is flat
if and only if it is projective. More generally, there is the following Local Flatness
Criterion, stated here in slightly simplified form (see [Mat86, Section 22] for a
self-contained proof):
Theorem 1.1 ([Gro61, 10.2.2]). Suppose that R → S is a homomorphism of Noetherian rings, I an ideal of R such that IS ⊆ Jac(S) (the Jacobson radical of S),
and M is a finite S-module. Then M is flat over R if and only if M/IM is flat
over R/I and TorR
1 (R/I, M ) = 0.
For injectivity, there are very few extant tests. For general modules, there is
the Baer criterion, which says that an R-module M is injective if and only if
Ext1R (R/P, M ) = 0 for all prime ideals P . It is well-known that a torsion-free
module M over an integral domain R is injective if and only if it is divisible (and
we prove a slightly more general version of this in Corollary 4.4). But as most
injective modules are not torsion-free, this criterion has limited usefulness.
In this work, we determine criteria for flatness and injectivity by checking torsionfreeness, associated primes, various sorts of divisibility, and coassociated primes,
Date: December 26, 2010.
2010 Mathematics Subject Classification. Primary 13C11; Secondary 13C05.
Key words and phrases. injective module, flat module, torsion-free module, divisible module,
h-divisible module, associated prime, coassociated prime.
The second author was partially supported by the National Science Foundation DMS-0700554.
1
2
NEIL EPSTEIN AND YONGWEI YAO
and then reducing to questions of flatness and injectivity over the total quotient
ring. In particular, for reduced rings, we get criteria for flatness in terms of torsionfreeness and associated primes, and criteria for injectivity in terms of (h-)divisibility.
We feel that our criteria for flatness are interesting even in the case where M
is a finitely generated R-module. Indeed, the first hint of Theorem 2.3 came when
the authors [EY] were investigating certain generalizations of Hilbert-Kunz multiplicity and needed a criterion to determine when certain direct summands of the
e
R-module R1/p were free, where R is a reduced F -finite Noetherian ring of prime
characteristic p > 0. Another application of our results to prime characteristic
algebra (or any other situation involving a locally contracting endomorphism) may
be found in Section 3, culminating in Theorem 3.10.
Here is a description of the contents of the sections to follow.
In Section 2, we recall a standard result about a condition on associated primes
satisfied by flat modules, and in Theorems 2.2 and 2.3, we show that in some sense
this condition mostly characterizes flatness. Several characterizations for flatness
there are given, using notions of associated primes, torsion-freeness, and certain
tensor products. We also give an example to show that the condition on associated
primes is not enough to guarantee flatness for non-reduced rings.
As discussed above, Section 3 gives an application of Theorem 2.3. Indeed,
after globalizing the notion of a locally contracting endomorphism, we exhibit in
Theorem 3.10 equivalent conditions for regularity of a reduced ring that has a
locally contracting endomorphism (e.g. any ring of characteristic p).
In Section 4, we discuss two non-equivalent notions of divisibility of a module:
divisible and h-divisible modules. We recall and demonstrate some dualities with
the notion of torsion-freeness, and we show that both of the divisibility notions
localize over a ring without embedded primes.
In Section 5, we recall the notions of attached and coassociated primes. In
Theorem 5.3, we demonstrate a strong dual (involving coassociated primes and
Hom modules) to the standard result from Section 2 on associated primes and
tensor product.
Our final section gives criteria for injectivity. Theorem 6.1 is a dual to the usual
local flatness criterion. Theorem 6.2 gives criteria for injectivity of a module in
terms of h-divisibility of Hom modules. The final result of this paper, Theorem 6.3,
gives a long list of equivalent conditions for injectivity (and a few conditions for being an injective cogenerator) when a ring is reduced, making use of all the concepts
discussed in the paper.
2. Flatness criteria
First, recall the following standard result (in simplified form below). Here we
use the usual convention that Ass 0 = ∅.
Theorem 2.1 ([Mat86, Theorem 23.2]). Let R be a Noetherian ring, and let L, M
be R-modules, where M is flat.
(1) If P ∈ Spec R such
S that M/P M 6= 0, then Ass(M/P M ) = {P }.
(2) Ass(L ⊗R M ) = P ∈Ass L Ass(M/P M ).
For a ring R, we say that a module M is torsion-free if every zero-divisor on M is
a zero-divisor on R (i.e. every R-regular element is M -regular). It is equivalent to
say that the natural map M → M ⊗R Q is injective, where Q is the total quotient
CRITERIA FOR FLATNESS AND INJECTIVITY
3
S
ring of R. For a collection X of subsets of a set S, we use the notation
X to
S
mean the union of all the elements of X. In particular, X ⊆ 2S but X ⊆ S.
Theorem 2.2. Let R be a Noetherian ring, and M an R-module (not necessarily finitely generated). Let Q be the total quotient ring of R. The following are
equivalent:
(a) M is flat.
(b) M ⊗R Q is flat over Q, and AssR (L ⊗R M ) ⊆ AssR L for every R-module
L.
(c) M ⊗R Q is flat over Q, and L ⊗R M is torsion-free for every torsion-free
R-module L.
(d) M ⊗R Q is flat over Q, and P ⊗R M is torsion-free for every P ∈ Spec R.
(e) M ⊗R Q is flat over Q, and TorR
1 (R/P, M ) is torsion-free for every P ∈
Spec R.
Suppose that R → S is a homomorphism of Noetherian rings, m ∈ Max(R), mS ⊆
Jac(S), and M is a finite S-module. Then the above conditions are equivalent to:
(d′ ) M ⊗R Q is flat over Q, and m ⊗R M is R-torsion-free.
(e′ ) M ⊗R Q is flat over Q, and TorR
1 (R/m, M ) is R-torsion-free.
In any case, the following are equivalent:
(i) M is faithfully flat.
(ii) M ⊗R Q is flat over Q, and AssR (L ⊗R M ) = AssR L for every R-module
L.
(iii) M is flat and AssR (L ⊗R M ) = AssR L whenever L is a simple R-module.
Proof. (a) =⇒ (b): This follows from the base-extension property of flat modules,
along with Theorem 2.1.
(b) =⇒ (c): Suppose L is torsion-free. Let x be S
a zero-divisor on L⊗S
R M . Then
there
exists
p
∈
Ass
(L
⊗
M
)
with
x
∈
p.
So
x
∈
Ass
(L
⊗
M
)
⊆
AssR L ⊆
R
R
R
R
S
Ass R. That is, x is a zero-divisor on R. Thus, L ⊗R M is torsion-free.
(c) =⇒ (d): Prime ideals are obviously torsion-free.
(d) =⇒ (d′ ) and (e) =⇒ (e’): Trivial.
(d) =⇒ (e) (or (d′ ) =⇒ (e′ )): Fix P ∈ Spec R (or fix P := m). Consider the
following commutative diagram:
0 −−−−→
TorR
1 (R/P, M )
hy
d
−−−−→
d⊗1Q
P ⊗M
gy
0 −−−−→ TorR
1 (R/P, M ) ⊗ Q −−−−→ P ⊗ M ⊗ Q
The top row is exact because TorR
1 (R, M ) = 0. The fact that P ⊗ M is torsion-free
implies that g is injective. Thus g ◦d = (d⊗1Q )◦h is injective, whence h is injective.
(e) (or (e′ )) =⇒ (a): To show M is flat, it suffices to show TorR
1 (R/P, M ) = 0
for all P ∈ Spec (R) (or in the situation of (e′ ), the Local Flatness Criterion implies
it is enough to do so when P = m). Starting with the short exact sequence 0 →
P → R → R/P → 0, tensoring with M gives us the exact sequence
0 → TorR
1 (R/P, M ) → P ⊗ M → R ⊗ M
4
NEIL EPSTEIN AND YONGWEI YAO
Then tensoring with Q gives us the exact sequence
0
y
TorR
1 (R/P, M ) ⊗ Q
d⊗1Q y
P ⊗M ⊗Q
d′ ⊗1Q y
M ⊗Q
(M ⊗ Q) ⊗Q (Q ⊗ P )
ey
(M ⊗ Q) ⊗Q (Q ⊗ R)
Since M ⊗R Q is flat over Q and Q is flat over R, the map e (hence also the map
R
d′ ⊗ 1Q ) is injective. Thus, TorR
1 (R/P, M ) ⊗ Q = 0, but since Tor1 (R/P, M ) is
R
R-torsion-free, Tor1 (R/P, M ) = 0.
Now we prove the equivalence of the conditions for faithful flatness:
(i) =⇒ (ii): Since M is faithfully flat, M/pM ∼
= R/p ⊗R M 6= 0 for any
p ∈ Ass L, so by Theorem 2.1, Ass(L ⊗R M ) = Ass L.
(ii) =⇒ (iii): This follows because we know that (b) implies (a).
Finally, we show that (iii) =⇒ (i): Let N 6= 0 be any R-module. Then N
contains a non-zero cyclic R-module which maps onto a simple R-module, say L.
By assumption, Ass(L ⊗R M ) = Ass L 6= ∅, which implies L ⊗R M 6= 0. As M is
flat, L ⊗R M 6= 0 forces N ⊗R M 6= 0.
We get a particularly nice statement in the case where R is reduced.
Theorem 2.3. Let R be a reduced Noetherian ring, and let M be an R-module.
The following are equivalent:
(a) M is flat.
(b) AssR (L ⊗R M ) ⊆ AssR L for every R-module L.
(c) L ⊗R M is torsion-free for every torsion-free R-module L.
(d) P ⊗R M is torsion-free for every P ∈ Spec R.
(e) TorR
1 (R/P, M ) is torsion-free for every P ∈ Spec R.
If, in addition, R → S is a homomorphism of Noetherian rings, m ∈ Max(R),
mS ⊆ Jac(S), and M is a finite S-module, the above conditions are equivalent to:
(d′ ) m ⊗R M is torsion-free.
(e′ ) TorR
1 (R/m, M ) is torsion-free.
In any case, the following are equivalent:
(i) M is faithfully flat.
(ii) AssR (L ⊗R M ) = AssR L for every R-module L.
(iii) M is flat and AssR (L ⊗R M ) = AssR L whenever L is a simple R-module.
Proof. Since R is reduced, Q := Q(R) is a product of finitely many fields, and hence
all Q-modules are flat over Q. Thus all implications follow from Theorem 2.2.
Note that the assumptions on flatness over Q in Theorem 2.2 cannot be omitted.
Example 2.4. Let R := Z/(4). Then since Spec R = {2R} has only one element,
every nonzero R-module N satisfies Ass N = Spec R. However, M := R/(2) is not
CRITERIA FOR FLATNESS AND INJECTIVITY
5
flat over R. To see this, consider the canonical injection j : 2R ֒→ R. Note that
j ⊗R 1M is the zero map, even though 2R ⊗R M ∼
= 2R 6= 0, which means that M
is not R-flat.
3. Regularity of local rings with a locally contracting
endomorphism
We begin this section with a general fact about associated primes via restriction
of scalars through a ring homomorphism. It has been proved by Yassemi in greater
generality as [Yas98, Corollary 1.7], but we provide the following, simpler proof for
the convenience of the reader.
Proposition 3.1. Let f : R → S be a homomorphism of commutative rings, where
S is Noetherian. Let f ∗ : Spec S → Spec R be the associated map on spectra. Let
M be an S-module. Then f ∗ (AssS M ) = AssR M .
Proof. First let q ∈ AssS M . That is, q = annS z for some z ∈ M . Then f ∗ (q) =
annR z, so that f ∗ (q) ∈ AssR M . Thus, f ∗ (AssS M ) ⊆ AssR M .
For the other inclusion, let p ∈ AssR M . Then p = annR z for some z ∈ M .
Letting W := R \ p and localizing R, S and M at W , we may assume without loss
of generality that R is quasi-local with maximal ideal p. Now let N := Sz ⊆ M .
Clearly N 6= 0, so since S is Noetherian, there is some q ∈ AssS N ⊆ AssS M . Then
p ⊆ f ∗ (q), but since p is maximal it follows that p = f ∗ (q). Thus, f ∗ (AssS M ) ⊇
AssR M , which finishes the proof.
Discussion 3.2. Take any ring homomorphism g : R → S and any q ∈ Spec S.
Composing further with the localization map S → Sq , we get a ring homomorphism
R → Sq . For p ∈ Spec R, this latter homomorphism extends to a map g ′ : Rp → Sq ,
defined by g ′ (a/x) = g(a)/g(x), if and only if g −1 (q) ⊆ p. Moreover, g ′ is a local
homomorphism if and only if p = g −1 (q). That is, each prime ideal q ∈ Spec S
induces a unique local homomorphism gq : Rg−1 (q) → Sq given by gq (a/x) =
g(a)/g(x). Hence in the case where R = S, we get an induced local endomorphism
gq : Rq → Rq if and only if ϕ−1 (q) = q.
We fix a ring R and a ring endomorphism ϕ : R → R. For any R-module M , let
M denote the (R-R)-bimodule which is isomorphic to M as an abelian group, where
the image of any z ∈ M in e M is denoted by e z, and whose bimodule structure is
given by a · e z · b := e (ϕe (a)bz) for any a, b ∈ R and any z ∈ M . In other words,
the right module structure is the same as its original structure, but the left module
structure is via restriction of scalars through the endomorphism ϕe . As a final
piece of necessary notation, for any R-module L, let F e (L) := L ⊗R e R, considered
as a right R-module. That is, F e (L) is the (left) R-module whose module structure
is that of the right R-module L ⊗R e R. We call F the functor induced by ϕ.
This notation agrees with many authors’ notation in the case where ϕ is the
Frobenius endomorphism. Indeed, many formulas familiar to characteristic p algebraists still hold, including F e (R/I) ∼
= R/ϕe (I)R.
Recall (e.g. [AIM06]) that for a local ring (R, m), a ring endomorphism ϕ :
R → R is contracting if there exists a positive integer n such that ϕn (m) ⊆ m2 .
Globalizing this notion, we obtain:
e
Definition 3.3. Let ϕ : R → R be a ring endomorphism, and let X ⊆ Spec R.
If for all p ∈ X we have an induced local homomorphism ϕp : Rp → Rp (i.e.
6
NEIL EPSTEIN AND YONGWEI YAO
ϕ−1 (p) = p) which is contracting, then we say ϕ is locally contracting over X. If
X = {p} for some p ∈ Spec R, we say ϕ is locally contracting at p. If X = Spec R,
we say ϕ is locally contracting.
Examples 3.4. If ϕ is a contracting (resp. locally contracting over Max R) endomorphism, so is ϕn for any positive integer n. We know of two main classes of
locally contracting endomorphisms:
• Let R be a ring of prime characteristic p > 0. Then the Frobenius endomorphism ϕ : R → R defined by a 7→ ap is a locally contracting endomorphism.
To see that it induces the identity map on spectra, let q ∈ Spec R. Since
ϕ(q) ⊆ q[p] ⊆ q, we have q ⊆ ϕ−1 (q). Conversely, suppose a ∈ ϕ−1 (q).
Then ap ∈ q, but since q is a radical ideal, a ∈ q.
• Let k be a field, let X be an indeterminate over k, and let R be either
k[X](X) , k[[X]], or k[X]/(X t) for some integer t > 0. Let n ≥ 2 be an
integer and let ϕ : R → R be the unique k-algebra homomorphism that
sends X 7→ X n . Then ϕ is a locally contracting endomorphism.
Lemma 3.5. Let ϕ : Rp→ R be a locally contracting endomorphism. Then for
all p ∈ Spec R, we have ϕ(p)R = p. In particular, ϕe (m)R is m-primary for all
maximal ideals m and all positive integers e.
Proof. The second statement follows from the first, since ϕe is a locally contracting
endomorphism. So let p ∈ Spec R. By Discussion 3.2, ϕ(p)R ⊆ p. On the other
hand, take any q ∈ Spec R such that ϕ(p)R ⊆ q. Then p ⊆ ϕ−1 (q) = q (again by
Discussion 3.2),
p which means that p is the unique minimal prime ideal of ϕ(p)R.
That is, p = ϕ(p)R, as was to be shown.
E. Kunz proved the following remarkable theorem, showing once again how central the notion of regularity is in commutative algebra:
Theorem 3.6 ([Kun69, Theorem 2.1]). Let (R, m) be a Noetherian local ring of
prime characteristic p > 0. Let ϕ : R → R be the Frobenius endomorphism, and
otherwise use notation as above. If R is regular, then e R is flat as a left R-module
for all integers e > 0. Conversely, if there is some integer e > 0 such that e R is
flat as a left R-module, then R is regular.1
Avramov, Iyengar, and Miller recently proved a broad generalization, of which
we use the following special case:
Theorem 3.7 ([AIM06, part of Theorem 13.3]). Let (R, m) be a Noetherian local
ring admitting a contracting endomorphism ϕ : R → R. Then e R is flat as a left
R-module for some ( resp. all) e > 0 if and only if ϕ(m)R is m-primary and R is
regular.
As final preparation for the main theorem of this section, we need the following
lemma:
e
1Kunz’s original theorem was stated in terms of the flatness of R over its subrings Rp , so
he made the additional necessary assumption that R was reduced. However, in our context, the
reducedness of R follows from the assumption that e R is R-flat, and hence faithfully flat. For let
e
x ∈ R such that xp = 0. Then (x) ⊗R e R = 0 by flatness of the functor (−) ⊗R e R applied to
the exact sequence 0 → (x) → R → R/(x) → 0, so x = 0 by faithful flatness. Thus R is reduced.
CRITERIA FOR FLATNESS AND INJECTIVITY
7
Lemma 3.8. S
Let R be Noetherian
and M an R-module. Then M is torsion-free
S
if and only if Ass M ⊆ Ass R. If, in particular, R has no embedded primes
( e.g. if it is reduced), then M is torsion-free if and only if Ass M ⊆ Ass R, and
for any torsion-free R-module M , W −1 M is a torsion-free (W −1 R)-module for all
multiplicative subsets W of R.
Proof. The first statement is obvious from the definition.
S
If Ass M ⊆ Ass R, then clearlyS Ass M ⊆
S So suppose R has noSembedded primes.
S
Ass R. So suppose Ass M ⊆ Ass R. Let p ∈ Ass M . Then p ⊆ Ass R, so
by prime avoidance, there is some prime q with p ⊆ q ∈ Ass R. But q is a minimal
prime, so p = q.
For the final claim, we use the usual bijective correspondence between {q ∈
AssR N | q ∩ W = ∅} and AssW −1 R (W −1 N ) (given by q 7→ W −1 q) for R-modules
N.
Example 3.9. The lack of embedded primes is a necessary condition in Lemma 3.8.
To see this, let R := k[[x, y, z]]/(x2 , xy, xz), N := R/(x, y) = k[[x, y, z]]/(x, y) ∼
= k[[z]],
and p := (x, y)R. Since every non-unit of R is a zerodivisor, every R-module is
automatically torsion-free, so that in particular N is torsion-free. However, the
image of y is a zerodivisor on Np but not on Rp , so that Np is not torsion-free.
As an application of our flatness criterion and the theorems of Kunz and AvramovIyengar-Miller, we get the following regularity criterion:
Theorem 3.10. Let R be a reduced Noetherian ring admitting a locally contracting2 endomorphism ϕ : R → R, and let F be the functor induced by ϕ ( e.g. if R
has positive prime characteristic p, we can take ϕ and F to be the Frobenius endomorphism and the Frobenius functor respectively). The following are equivalent:
(a) R is regular.
(b) e R is flat as a left R-module.
(c) For every e > 0 and every R-module L, Ass F e (L) = Ass L.
(d) ∃e > 0 such that for every R-module L, Ass F e (L) ⊆ Ass L.
(e) ∃e > 0 such that for every P ∈ Spec R, F e (P ) is torsion-free.
(f ) ∃e > 0 such that for every m ∈ Max R, F e (m) is torsion-free.
Proof. First, by Proposition 3.1, we have for any R-module M that AssR e M =
(ϕe )∗ (AssR M ). But by Discussion 3.2, (ϕe )∗ is the identity map, so (ϕe )∗ (AssR M ) =
AssR M . Also, recall from Lemma 3.5 that ϕe (m)R is m-primary for all maximal
ideals m and all positive integers e.
Next, for each maximal ideal m, we consider the commutative diagram:
f
R −−−−→ Rm
(ϕe )m y
ϕe y
f
R −−−−→ Rm
2This theorem is not the most general theorem one can obtain. For instance, if one takes an
endomorphism ϕ which is only locally contracting over Max R, and one replaces all the terms
F e (L) (including the case L = m) with L ⊗R e R (i.e. the left-module structure), then one still
obtains: (b) =⇒ (c) =⇒ (d) =⇒ (e) =⇒ (a). If moreover ϕ(m)R is m-primary for all
m ∈ Max R, then all the statements are equivalent. However, we stated the theorem in its present
form since, in general, the left-module structure of L ⊗R e R tends to be more complicated than
that of F e (L).
8
NEIL EPSTEIN AND YONGWEI YAO
Suppose that R is regular. Then for all maximal ideals m, Rm is regular. Hence
by Theorem 3.7 and the fact that ϕ(m)R is m-primary, e (Rm ) is flat as a left Rm module. But the commutativity of the diagram above shows that e (Rm ) ∼
= (e R)m
e
as left R-modules. Since ( R)m is R-flat for all maximal ideals m, it follows that
e
R itself is flat over R. That is, (a) implies (b).
The equivalence of statements (b)-(e) follows from Theorem 2.3, since as left
R-modules, e (F e (M )) = M ⊗R e R for any R-module M (so that in particular, by
the first paragraph of this proof, Ass(F e (L)) = Ass(e (F e (L))) = Ass(L ⊗R e R)),
and since e R is flat over R only if it is faithfully flat. This last statement follows
because for any maximal ideal m, (R/m) ⊗R e R ∼
= e (R/ϕe (m)R) 6= 0.
As for statement (f), clearly (e) implies (f). Now suppose (f) is true. By
Lemma 3.8, FRe (m)m is a torsion-free Rm -module for all m. But tracing through the
commutative diagram above, we see that FRe m (mm ) ∼
= FRe (m)m , which, as we have
already seen, is torsion-free. Then by Theorem 2.3, e (Rm ) is flat over Rm , so that
by Theorem 3.7, Rm is regular. Since this holds for all maximal ideals m, R itself
is regular.
4. Divisibility
The concept of divisibility of a module has not been used as much as that of
torsion-freeness. Indeed, there are a number of non-equivalent notions! Most of the
work on divisibility has been done in the context of integral domains, as in [FS92],
but one may extend many of the notions to a much more general context, as was
done in [AHHT05]. Given a commutative ring R and a multiplicative subset W of
R, let S = W −1 R.3 Following the definitions given by Fuchs and Salce [FS92] in the
case of integral domains, we say in general that an R-module M is W -torsion-free
if the multiplication map w : M → M is injective for all w ∈ W (or equivalently,
the natural map M → S ⊗R M is injective). We say M is W -divisible if the map
w : M → M is surjective for all w ∈ W . We say M is hW -divisible if M is an Rquotient of a free S-module. In the case where W is the set of all non-zerodivisors
of R, we use the terms torsion-free, divisible, and h-divisible respectively.
Note that hW -divisibility implies W -divisibility, for if z ∈ M , w ∈ W , and π :
F ։ M is a surjective R-linear map with F a free (W −1 R)-module and π(u) = z,
then wg((1/w)u) = z.
We start this section with a couple of characterizations of hW -divisibility.
Lemma 4.1. Let R be a ring, M an R-module, W a multiplicative subset of R,
and S := W −1 R. The following are equivalent:
(a) M is hW -divisible
(b) The natural evaluation map e : HomR (S, M ) → M which sends f 7→ f (1)
is surjective.
(c) There is a surjective R-linear map from an S-module onto M .
Proof. First suppose M is hW -divisible. Let F be a free S-module such that there
is surjective R-linear map p : F ։ M . Pick z ∈ M , and f ∈ F such that p(f ) = z.
Since F is an S-module, there is a unique S-linear map g : S → F (which is
therefore R-linear) such that g(1) = f . Then (p ◦ g) : S → M is an R-linear map
3In this generality, we have to allow the possibility that 0 ∈ W , in which case S = W −1 R is
the zero ring. But this causes no real problems, as the only S-module is 0, which is injective, flat,
divisible, and torsion-free over S.
CRITERIA FOR FLATNESS AND INJECTIVITY
9
such that e(p ◦ g) = p(g(1)) = p(f ) = z. Thus, e is surjective, which means that
(a) implies (b). Clearly (b) implies (c).
To show that (c) implies (a), suppose there is a surjective R-linear map π : G ։
M from some S-module G. Let p : F ։ G be an S-linear map where F is a free
S-module. Then the map π ◦ p : F ։ M demonstrates that M is hW -divisible.
Next, we give a characterization of modules which are both W - (or hW -) divisible
and W -torsion-free.
Lemma 4.2. Let R be a ring, W a multiplicative set, and M an R-module. Let
S := W −1 R. The following are equivalent:
(a)
(b)
(c)
(d)
(e)
The
The
The
The
The
canonical localization map g : M → W −1 M is an isomorphism.
R-module structure on M extends to an S-module structure.
canonical evaluation map h : HomR (S, M ) → M is an isomorphism.
R-module M is W -torsion-free and hW -divisible.
R-module M is W -torsion-free and W -divisible.
Proof. (a) =⇒ (b): Use the S-module structure of W −1 M .
(b) =⇒ (a): This follows from the universality of the localization map.
(b) =⇒ (c): Since S ⊗R S ∼
= S, we have:
HomR (S, M ) ∼
= HomR (S, HomS (S, M )) ∼
= HomS (S ⊗R S, M )
∼ HomS (S, M ) ∼
=
= M.
(c) =⇒ (b): Use the S-module structure of HomR (S, M ).
[(a) and (c) together] =⇒ (d): g is injective and h is surjective.
(d) =⇒ (e): Any hW -divisible module is W -divisible.
(e) =⇒ (a): Since the injectivity of g is one definition of W -torsion-freeness, it
suffices to show g is surjective. Let z ∈ M and w ∈ W . By W -divisibility of M ,
there is some y ∈ M with wy = z. Clearly g(y) = z/w.
For an integral domain R, Matlis [Mat64] defined an R-module M to be hdivisible if it was a quotient of an injective module, so the casual reader may think
there is a conflict of terminology. However, we have the following:
Proposition 4.3. Let R be a reduced ring with only finitely many minimal primes
( e.g. a reduced Noetherian ring, or any integral domain). Let Q be its total ring
of fractions, and let M be an R-module. Then M is a quotient of an injective Rmodule ( i.e. h-divisible in the sense of Matlis) if and only if it is an R-quotient of
a free Q-module ( i.e. h-divisible in our sense).
Proof. Let N be any Q-module, free or not. Since Q is a product of finitely
many fields, N is injective over Q. From the natural isomorphism of functors
HomR (−, N ) ∼
= HomQ (Q ⊗R −, N ), the R-flatness of Q combined with the Qinjectivity of N implies that N is R-injective. Thus, any R-quotient of N is an
R-quotient of an injective R-module.
Conversely, let E be an injective R-module. There is some free R-module T =
⊕i∈I R and a surjection p : T ։ E. Let F := ⊕i∈I Q and j : T ֒→ F the obvious
injection. Since E is R-injective, the map p : T → E extends along j to a map
q : F → E, and q is surjective because p is. Thus, if M is a quotient of the injective
R-module E, it is the R-quotient of the free Q-module F as well.
10
NEIL EPSTEIN AND YONGWEI YAO
As a corollary, we get an extension to reduced rings of a well-known criterion
involving injectivity, torsion-freeness, and divisibility.
Corollary 4.4. Let R be a reduced ring with only finitely many minimal primes.
Let M be a torsion-free R-module. The following are equivalent:
(a) M is h-divisible.
(b) M is divisible.
(c) M is injective.
Proof. (a) =⇒ (b): Any h-divisible module is divisible.
(b) =⇒ (c): By Lemma 4.2, M is a Q-module, where Q is the total quotient
ring of R. But then by the first paragraph of the proof of Proposition 4.3, M is
R-injective.
(c) =⇒ (a): This follows from Proposition 4.3.
Although every h-divisible module is divisible, the converse is often not true.
Matlis [Mat60] showed that for an integral domain R, the two notions are equivalent
if and only if projdimR Q ≤ 1. On the other hand, Kaplansky [Kap66] showed
that if R is a polynomial ring in n ≥ 2 variables over an uncountable field, then
projdimR Q ≥ 2.
However, for certain classes of modules, the two notions are always equivalent.
For a Noetherian ring R, recall that an injective cogenerator is an injective Rmodule E with the property that for any nonzero R-module
M , HomR (M, E) 6= 0.
L
(These always exist. A popular choice is to let E := m∈Max(R) ER (R/m).)
For an R-module M , let M ∨ := HomR (M, E), and similarly for morphisms to
create a (contravariant) duality functor.
Proposition 4.5 ([Ric06, Proposition 1.4 and main text]). Let R be a Noetherian
ring, E an injective cogenerator for R, and (−)∨ := HomR (−, E) the corresponding
duality functor. Let M , N be R-modules.
(1) For any R-linear map g : M → N , g is injective ( resp. surjective) if and
only if g ∨ is surjective ( resp. injective).
(2) M is injective ( resp. flat) if and only if M ∨ is flat ( resp. injective).
i
∨
∨ ∼
(3) TorR
i (M, N ) = ExtR (M, N ) for all i ≥ 0.
∨
(4) If M is finitely generated, then ExtiR (M, N )∨ ∼
= TorR
i (M, N ) for all i ≥ 0.
This duality functor helps in examining properties of torsion-freeness and divisibility:
Lemma 4.6. Let R be a Noetherian ring. Let W be a multiplicative subset of R.
Let E be an injective cogenerator for R, and (−)∨ the corresponding duality functor.
Let L be an R-module. We have:
(1) L is W -divisible if and only if L∨ is W -torsion-free.
(2) The following are equivalent:
(a) L∨ is W -divisible.
(b) L is W -torsion-free.
(c) L∨ is hW -divisible.
(3) In particular, if R is complete local and L is Noetherian or Artinian, then
L is W -divisible if and only if it is hW -divisible.
w
w
Proof. (1): L is W -divisible ⇐⇒ L → L is surjective for all w ∈ W ⇐⇒ L∨ → L∨
is injective for all w ∈ W ⇐⇒ L∨ is W -torsion-free.
CRITERIA FOR FLATNESS AND INJECTIVITY
11
w
(2): (a) ⇐⇒ (b): L∨ is W -divisible if and only if ∀w ∈ W , L∨ → L∨ is
w
surjective, if and only if ∀w ∈ W , L → L is injective, if and only if L is W -torsionfree.
(b) =⇒ (c): Let S := W −1 R. If L is W -torsion-free then the natural map
j : L → S ⊗R L is injective, whence j ∨ : (S ⊗R L)∨ → L∨ is surjective. But since
(S ⊗R L)∨ is an S-module, Lemma 4.1 shows that L∨ is hW -divisible.
(c) =⇒ (a): Any hW -divisible module is W -divisible.
Statement (3) follows from Matlis duality.
Finally, it is worth giving an analogue of the localization portion of Lemma 3.8
for divisible and h-divisible modules:
Proposition 4.7. Let R be a Noetherian ring with no embedded primes, and M
an R-module. Let W be a multiplicative subset of R. If M is divisible ( resp.
h-divisible) over R, then W −1 M is divisible ( resp. h-divisible) over W −1 R.
Proof. If 0 ∈ W , then W −1 R is the zero-ring, so all modules over it are h-divisible.
Thus, we may assume that 0 ∈
/ W.
We begin by setting up some notation: Let X1 := {p ∈ Min R |Sp ∩ W 6= ∅},
and X2 :=SMin R \ X1 = {p ∈ Min R | p ∩ W = ∅}.
R \ ( Min R) and
S Let S := T
T := R \ ( X2 ). We will use the convention that ∅ = 0 and ∅ = R.
Next, we prove two claims:
Claim 1:4 Take any c ∈ T (i.e., c/1 is a non-zerodivisor of W −1 R). We can find
w ∈ W and r ∈ R such that wr = 0 and wc + r is a non-zerodivisor of R.
Proof of Claim 1. By assumption Min R = Ass R.
Since the primes in Min R are mutually incomparable,Tand since every
S p ∈ X1
intersects
W
,
prime
avoidance
allows
us
to
choose
w
∈
(
X
)
∩
W
\
(
X2 ) and
1
T
S
S
T
S
r ∈ ( X2 ) \ ( X1 ). As c ∈
/ X
,
we
have
wc
∈
(
X
)
\
(
X
).
2
1
2
√
T
Note that wr ∈ Min R = 0, so there is some n with wn rn = 0. Replace r
and w with rn and wn respectively and all the above conditions hold.
Now, for any p ∈ X1 , note that wc ∈ p but r ∈
/ p, so that wc + r ∈
/ p. Similarly,
for
any
q
∈
X
,
we
have
r
∈
q
but
wc
∈
/
q,
so
that
wc
+
r
∈
/
q.
That
is, wc + r ∈
/
2
S
S
S
S
( X1 )∪( X2 ) = Min R = Ass R, which means that wc+r is a non-zerodivisor
of R.
Claim 2: W −1 Q(R) ∼
= Q(W −1 R), where Q(−) is the function that sends each
ring to its total quotient ring.
Proof of Claim 2. Note that S, T , and W are all unital semigroups
Let SW (resp. T W ) be the unital semigroup generated by S and W (resp. T
and W ). Clearly, we have W −1 Q(R) ∼
= (SW )−1 R and Q(W −1 R) ∼
= (T W )−1 R.
Since SW ⊆ T W , the latter ring is a localization of the former. On the other
hand, let c ∈ T = T W (where equality holds because W ⊆ T ), and pick w, r as
in Claim 1. Then w2 c = w(wc + r) ∈ W S, so that c/1 is already invertible in
(SW )−1 R = W −1 Q(R).
Now suppose M is divisible. Let c/1 be a non-zerodivisor of W −1 R (so that
c ∈ T ), let w, r be as in Claim 1, and let x ∈ M . Then there is some y ∈ M such
4The idea for this claim came from the proof of [AHH93, Lemma 3.3(a)].
12
NEIL EPSTEIN AND YONGWEI YAO
that x = (wc + r)y. But then
wc wr
wc
wc + r
=
+
=
1
1
w
1
in W −1 R, since w ∈ W and wr = 0. So
wc + r y
wc y
c wy
x
=
· =
· = ·
1
1
1
1 1
1 1
in W −1 M . That is, x/1 is divisible by c/1. Thus, W −1 M is divisible as a (W −1 R)module.
Finally, suppose M is h-divisible. Then it is a homomorphic image of a free
Q(R)-module F . But then W −1 F is a free module over W −1 Q(R) ∼
= Q(W −1 R)
(by Claim 2), and the localization at W preserves the surjection onto M , so W −1 M
is a (W −1 R)-homomorphic image of a free Q(W −1 R)-module. Thus W −1 M is hdivisible over W −1 R.
5. Coassociated primes and Hom modules along a base change
Let R be a commutative Noetherian ring.
MacDonald [Mac73] and Kirby [Kir73] independently proved a dual for Artinian
modules to the Lasker-Noether primary decomposition theorem for Noetherian
modules. (In the sequel, we use Macdonald’s terminology.) In this theory, a nonzero
R-module M is called secondary if for all r ∈ R, the map r : M → M (left multiplication by r) is either surjective or nilpotent.
The annihilator of such a module
√
always has a prime radical, and if p = ann M for a secondary R-module M , M is
called p-secondary. One says that an R-module M has a secondary representation
if one may express it as a finite sum of the form:
n
X
Mi ,
M=
i=1
where each Mi is pi -secondary, for a set {p1 , . . . , pn } of pairwise-distinct prime
ideals. Macdonald shows that if such a representation exists, then it has uniqueness
properties dual to those enjoyed by primary decompositions. In particular, if one
takes an irredundant such representation, then the set of prime ideals involved is
unique. If such a representation exists, then the pi are called the attached primes
of M , and we write Att M = {p1 , . . . , pn }. Macdonald and Kirby show that any
Artinian module has a secondary representation.
However, not every R-module has a secondary representation. There is, however,
a notion of coassociated primes (Coass) of an arbitrary R-module, by Chambless,
dual to that of the associated primes:
Definition 5.1. [Cha81, p. 1134] Let R be a Noetherian ring, M an R-module,
and P ∈ Spec R. We say that P is coassociated to M if there is some nonzero
factor module N of M which is hollow (i.e. N cannot be written as the sum of two
proper submodules) such that P = {a ∈ R | aN 6= N }. The set of all such primes
is written Coass M .
We collect below some established facts about Coass and Att:
Proposition 5.2. Let R be a commutative Noetherian ring:
(1) If M ։ N is a surjection of R-modules, then Coass N ⊆ Coass M . [Cha81,
Lemma 3 on p. 1136]
CRITERIA FOR FLATNESS AND INJECTIVITY
13
(2) If 0 → L → M → N → 0 is a short exact sequence of R-modules, then
Coass
M ⊆ Coass L ∪ Coass N . [Zös83, Lemma 2.1(a)]
S
(3)
Coass M = {x
√ ∈ R | xM 6= M }. [Zös83, Folgerung 1 on p. 129]
T
(4)
Coass M ⊇ ann M . [Zös83, Folgerung 1 on p. 129]
(5) If M has a secondary representation, then Att M = Coass M . [Yas95,
Theorem 1.14]
(6) Let M be a nonzero R-module. Then Coass M 6= ∅. [Yas95, Theorem 1.9]
(7) Let p ∈ Spec R, q a p-primary ideal, and E an injective R-module. Then
(0 :E q) is either 0 or p-secondary. [Sha76, Lemma 2.1]
(8) Let g : R → S be a ring homomorphism, with S Noetherian, and let M
be an S-module. Consider M to be an R-module via restriction of scalars
along g. Then g ∗ (CoassS M ) ⊆ CoassR M . [DAT99, Theorem 1.7(i)]
Theorem 5.3. Let g : R → S be a map of Noetherian rings. Let L be a finite
R-module, and let M be an S-module which is injective over R (via the R-module
structure given by restriction of scalars).
(1) For any p ∈ Spec R, we S
have g ∗ (CoassS (0 :M p)) = CoassR (0 :M p) ⊆ {p}.
(2) CoassS HomR (L, M ) = p∈AssR L CoassS (0 :M p).
Proof. To prove (1) we use parts (5)–(8) of Proposition 5.2. Namely, let L :=
(0 :M p). If L = 0, then both CoassR L and CoassS L are empty. Otherwise,
by 5.2(6), we have CoassS L 6= ∅ and CoassR L 6= ∅. But then by 5.2(7), L is
p-secondary as an R-module, so that by 5.2(5), CoassR L = {p}. Then by 5.2(8),
∅ 6= g ∗ (CoassS L) ⊆ CoassR L = {p}, so that g ∗ (CoassS L) = {p}, completing the
proof.
It remains to prove (2). For one inclusion, let p ∈ AssR L. Then there is an
injection 0 → R/p → L, which when we apply HomR (−, M ), yields an exact
sequence
HomR (L, M ) → (0 :M p) → 0.
Now it follows from Proposition 5.2(1) that CoassS (0 :M p) ⊆ CoassS HomR (L, M ).
For the reverse inclusion, let P ∈ CoassS HomR (L, M ). Since L is finitely generated and R is Noetherian, setting AssR L = {p1 , . . . , pn } the R-submodule 0 ⊆ L
has a minimal primary decomposition
0 = K1 ∩ · · · ∩ Kn ,
such that L/Ki is pi -coprimary. We get an injection
0 → L ֒→
n
M
L
,
Ki
i=1
which leads to a surjection
n
M
i=1
HomR (L/Ki , M ) ։ HomR (L, M ) → 0.
14
NEIL EPSTEIN AND YONGWEI YAO
Thus, by Proposition 5.2(1 and 2)
n
M
HomR (L/Ki , M ))
CoassS HomR (L, M ) ⊆ CoassS (
i=1
=
n
[
CoassS HomR (L/Ki , M )
i=1
Since P ∈ CoassS HomR (L, M ), we have P ∈ CoassS HomR (L/Ki , M ) for some i.
Let L′ := L/Ki for this choice of i, and let p := pi .
Claim 1: g −1 (P ) = p. (That is, g ∗ (P ) = p.)
Proof of Claim 1. Since L′ is p-coprimary, there is some positive
p integer k such that
pk L′ = 0. Hence g(p)k HomR (L′ , M ) = 0. That is, g(p) ⊆ annS HomR (L′ , M ).
But then by Proposition 5.2(4), g(p) ⊆ Q for every Q ∈ CoassS HomR (L′ , M ), so
that in particular, p ⊆ g −1 (P ).
Conversely, let x ∈ R \ p. Since L′ is p-coprimary, we get an exact sequence pf
R-modules
x
0 → L′ → L′ ,
which induces an exact sequence of S-modules:
g(x)
HomR (L′ , M ) → HomR (L′ , M ) → 0
But then by Proposition 5.2(3), g(x) is not in any coassociated prime of HomR (L′ , M ).
In particular, g(x) ∈
/ P , whence x ∈
/ g −1 (P ). This shows that g −1 (P ) ⊆ p, which
completes the proof of the Claim.
Since L′ is finitely generated over the Noetherian ring R, there is a prime filtration
0 = N0 ⊂ N1 ⊂ · · · ⊂ Nt = L ′ ,
∼ R/qj , where the qj are prime ideals of R.
such that for each 1 ≤ j ≤ t, Nj /Nj−1 =
Claim 2: For each 1 ≤ j ≤ t,
CoassS HomR (Nj , M ) ⊆
j
[
CoassS HomR (R/ql , M )
l=1
Proof of Claim 2. We proceed by induction on j. When j = 1, we have N1 ∼
= R/q1 ,
so there is nothing to prove.
So let j > 1, and assume the claim for j − 1. We have a short exact sequence of
R-modules
0 → Nj−1 → Nj → R/qj → 0,
which induces a short exact sequence of S-modules
0 → HomR (R/qj , M ) → HomR (Nj , M ) → HomR (Nj−1 , M ) → 0.
Thus, by Proposition 5.2(2),
CoassS HomR (Nj , M ) ⊆ (CoassS HomR (Nj−1 , M )) ∪ CoassS HomR (R/qj , M )
⊆
as was to be shown.
j
[
CoassS HomR (R/ql , M ),
l=1
CRITERIA FOR FLATNESS AND INJECTIVITY
15
In particular, since L′ = Nt ,
CoassS HomR (L′ , M ) ⊆
t
[
CoassS HomR (R/qj , M ),
j=1
so that since P ∈ CoassS HomR (L′ , M ), there is some j such that
P ∈ CoassS HomR (R/qj , M ) = CoassS (0 :M qj ).
But then by part (1), g −1 (P ) ∈ CoassR (0 :M qj ) ⊆ {qj }, so that p = g −1 (P ) = qj ,
and P ∈ CoassS (0 :M p), completing the proof since p ∈ AssR L.
6. Injectivity criteria
We begin this section with a partial dual of Theorem 1.1. This result may be
known to experts; however, we could not find it in the literature, and so we give a
proof of it here for completeness.
Theorem 6.1 (A local injectivity criterion). Let (R, m, k) → (S, n, ℓ) be a local
homomorphism of Noetherian local rings, and let M be an Artinian S-module.
Then M is injective over R if and only if Ext1R (k, M ) = 0.
Proof. We only need to prove the “if” direction. So suppose Ext1R (k, M ) = 0,
and assume for the moment that S = Ŝ – that is, S is n-adically complete. Let
E ′ := ES (ℓ).
First note that for any ideal I of R, we have
HomS (Ext1 (R/I, M ), E ′ ) ∼
= TorR (R/I, HomS (M, E ′ )).
R
1
To see this, consider the (natural) right-exact sequence
M → HomR (I, M ) → Ext1R (R/I, M ) → 0,
then apply the exact functor HomS (−, E ′ ), and note that HomS (HomR (I, M ), E ′ ) ∼
=
I ⊗R HomS (M, E ′ ) in a natural way.
1
′
′
′
Thus, TorR
1 (k, HomS (M, E )) = HomS (ExtR (k, M ), E ) = HomS (0, E ) = 0, but
′
by Matlis duality we have that HomS (M, E ) is a finite S-module, so by the Local
Flatness Criterion, we have that HomS (M, E ′ ) is flat as an R-module. In particular,
for any ideal I of R, we have
0 = TorR (R/I, HomS (M, E ′ )) ∼
= HomS (Ext1 (R/I, M ), E ′ ).
1
R
′
But since E is an injective cogenerator in the category of S-modules, it follows
that Ext1R (R/I, M ) = 0. Since I was arbitrary, Baer’s criterion then shows that M
is injective over R.
Finally, remove the assumption that S is complete. Since M is Artinian over S,
it has a natural structure as an Artinian module over Ŝ, in such a way that the
restriction of scalars to S gives the original S-module structure of M . Thus, we
may replace the map R → S with the composite map R → S → Ŝ, and then by
the above argument, it follows that M is injective over R.
Theorem 6.2. Let R be a Noetherian ring and M an R-module. Let Q be the total
quotient ring of R. The following are equivalent:
(a) M is injective.
(b) HomR (Q, M ) is injective over Q, and HomR (L, M ) is h-divisible for every
torsion-free R-module L.
16
NEIL EPSTEIN AND YONGWEI YAO
(c) HomR (Q, M ) is injective over Q, and HomR (P, M ) is h-divisible for every
P ∈ Spec R.
Moreover, if (R, m, k) → (S, n, ℓ) is a local homomorphism of Noetherian local rings
and M is an Artinian S-module, then the following condition is also equivalent to
the above three:
(c′ ) HomR (Q, M ) is injective over Q, and HomR (m, M ) is an h-divisible Rmodule.
Proof. (a) =⇒ (b): For the first statement, recall that if A → B is any ring
homomorphism and I is an injective A-module, then HomA (B, I) is an injective
B-module. For the second statement, let L be a torsion-free R-module. Since L
is torsion-free, the natural map L → L ⊗R Q is injective. Since M is an injective
module, this implies that the map C := HomR (L ⊗R Q, M ) → HomR (L, M ) is
surjective. But as C is a Q-module, Lemma 4.1 now implies that HomR (L, M ) is
h-divisible.
(b) =⇒ (c): Every ideal of R is torsion-free.
(c) =⇒ (c′ ): Obvious.
(c) (or (c′ )) =⇒ (a): By a version of Baer’s criterion, we need only show that
the natural map HomR (R, M ) → HomR (P, M ) is surjective for every P ∈ Spec R.
(In the situation of (c′ ), it suffices (by Theorem 6.1) to show this map is surjective
for P = m. So in that case, we fix the notation P = m for the remainder of the
proof.) Consider the commutative square:
h
HomR (Q, M ) −−−−→ HomR (Q ⊗R P, M )
gy
g′ y
f
HomR (R, M ) −−−−→
HomR (P, M )
Since the natural map P → R is injective and Q is flat over R, the corresponding
map Q ⊗R P → Q of Q-modules is also injective. Then since HomR (Q, M ) is
injective over Q, we have that
HomQ (Q, HomR (Q, M )) → HomQ (Q ⊗R P, HomR (Q, M ))
is surjective. But up to canonical isomorphism, the displayed map may be identified
with the map h. Hence h is surjective.
On the other hand, the source module of g is canonically isomorphic to the
module HomR (Q, HomR (P, M )), in such a way that for any α : Q → HomR (P, M ),
g(α) = α(1). By Lemma 4.1, the fact that HomR (P, M ) is h-divisible means that
g is surjective.
We have shown that g ◦ h = f ◦ g ′ is surjective, which implies that f is surjective,
as required.
In the reduced case, we get a longer list of equivalent conditions for injectivity,
as well as some equivalent conditions for being an injective cogenerator.
Theorem 6.3. Let R be a reduced Noetherian ring and let M be an R-module.
The following are equivalent:
(a) M is injective
(b) CoassR HomR (L, M ) ⊆ AssR L for every finitely generated R-module L.
(c) HomR (L, M ) is h-divisible for every torsion-free module L.
CRITERIA FOR FLATNESS AND INJECTIVITY
17
(d) HomR (L, M ) is divisible for every finite torsion-free module L.
(e) HomR (P, M ) is h-divisible for every P ∈ Spec R.
(f ) HomR (P, M ) is divisible for every P ∈ Spec R.
(g) Ext1R (R/P, M ) is divisible for every P ∈ Spec R.
Moreover, if (R, m, k) → (S, n, ℓ) is a local homomorphism of Noetherian local rings
and M is an Artinian S-module, then the following conditions are also equivalent
to M being injective:
(f′ ) HomR (m, M ) is a divisible R-module.
(g′ ) Ext1R (k, M ) is a divisible R-module.
In any case, the following are equivalent:
(i) M is an injective cogenerator.
(ii) CoassR HomR (L, M ) = AssR L for every finitely generated R-module L.
(iii) M is injective and CoassR HomR (L, M ) = AssR L whenever L is a simple
R-module.
Proof. Note first that conditions (a), (c) and (e) are equivalent by Theorem 6.2.
Next, we will show first that (a), (b), (d), (f), (g), (f′ ) and (g′ ) are equivalent:
The fact that (a) =⇒ (b) follows from Theorem 5.3.
To see that (b) =⇒ (d), let L be a torsion-free R-module. Then we have
Coass HomR (L, M ) ⊆ AssR L ⊆ Ass R,
S
S
from which it follows that Coass HomR (L, M ) ⊆ Ass R = the set of zerodivisors
of R. But then by Proposition 5.2(3), it follows that every non-zerodivisor of R
acts surjectively on
L HomR (L, M ). That is, HomR (L, M ) is divisible.
Now let E := m∈Max(R) ER (R/m), and let (−)∨ := HomR (−, E) be the associated duality functor. By Proposition 4.5(2), M is injective if and only if M ∨ is
flat. For any finite R-module L, Proposition 4.5(4) implies that HomR (L, M )∨ ∼
=
∨
(L,
M
).
Thus,
the
equivalence
of
conditions
L ⊗R M ∨ and Ext1R (L, M )∨ ∼
= TorR
1
(a), (d), (f), and (g) follows from Theorem 2.3 and Proposition 4.5. The implication
(f′ ) =⇒ (g′ ) follows from the fact that Ext1R (k, M ) is a homomorphic image of
HomR (m, M ). Moreover, it is clear that (f) =⇒ (f′ ) and (g) =⇒ (g′ ).
To see that (g′ ) =⇒ (a) under these conditions, suppose (g′ ) holds. If R is a
field, then M is clearly injective. Otherwise, let x ∈ m be a non-zerodivisor. Then
x acts surjectively on E := Ext1R (k, M ) (since E is R-divisible), but also x kills E
(since E is a k-module), so E = 0. The conclusion then follows from Theorem 6.1.
Finally, we show the equivalence of conditions (i), (ii), and (iii). To see that (i)
=⇒ (ii), let p ∈ Ass L. Then there is an injection 0 → R/p → L, which when we
apply HomR (−, M ), yields an exact sequence
HomR (L, M ) → (0 :M p) → 0.
Now, Coass(0 :M p) ⊆ Coass HomR (L, M ) by Proposition 5.2(1), so it suffices to
show that p ∈ Coass(0 :M p). We have (0 :M p) = HomR (R/p, M ) 6= 0 since M
is an injective cogenerator. But then by Proposition 5.2(6), Coass(0 :M p) 6= ∅, so
that by Theorem 5.3(1), it follows that p ∈ Coass(0 :M p).
The implication (ii) =⇒ (iii) holds because (ii) =⇒ (b) ⇐⇒ (a).
Finally, we show that (iii) =⇒ (i), we need to show that for any nonzero Rmodule N , HomR (N, M ) 6= 0. The module N contains a cyclic module C, which
maps onto a simple module L = R/m for some maximal ideal m. By assumption,
Coass HomR (L, M ) = Ass L = {m}, which certainly implies HomR (L, M ) 6= 0. But
18
NEIL EPSTEIN AND YONGWEI YAO
HomR (L, M ) is a submodule of HomR (C, M ), and since M injective, HomR (C, M )
is a quotient module of HomR (N, M ). Hence, HomR (N, M ) 6= 0.
Remark. There is, of course, the temptation to adapt the proof of Theorem 6.3 in
order to add more equivalent conditions to Theorem 6.2. This is indeed possible.
For example, when R, Q, and M are as in Theorem 6.2, then M is injective if and
only if “Q ⊗R M ∨ is flat over Q and HomR (L, M ) is divisible for every finitely
generated torsion free R-module L.” However, such sets of conditions seem too
unwieldy to merit theorem status, especially since, as far as the authors know, all
such condition sets require explicit use of the duality functor. One might hope,
for instance, that the flatness of Q ⊗ M ∨ would be equivalent to the injectivity of
HomR (Q, M ). However, one almost never has HomR (Q, M )∨ ∼
= Q ⊗R M ∨ , so such
equivalences seem unlikely.
References
[AHH93]
Ian M. Aberbach, Melvin Hochster, and Craig Huneke, Localization of tight closure and
modules of finite phantom projective dimension, J. Reine Agnew Math. 434 (1993),
67–114.
[AHHT05] Lidia Angeleri Hügel, Dolors Herbera, and Jan Trlifaj, Divisible modules and localization, J. Algebra 294 (2005), no. 2, 519–551.
[AIM06]
Luchezar L. Avramov, Srikanth Iyengar, and Claudia Miller, Homology over local
homomorphisms, Amer. J. Math. 128 (2006), no. 1, 23–90.
[Cha81]
Lloyd Chambless, Coprimary decompositions, N -dimension and divisibility: application to artinian modules, Comm. Algebra 9 (1981), no. 11, 1131–1146.
[DAT99] Kamran Divaani-Aazar and Massoud Tousi, Some remarks on coassociated primes, J.
Korean Math. Soc. 36 (1999), no. 5, 847–853.
[EY]
Neil Epstein and Yongwei Yao, Some extensions of Hilbert-Kunz multiplicity, preprint.
[FS92]
László Fuchs and Luigi Salce, S-divisible modules over domains, Forum Math. 4
(1992), no. 4, 383–394.
[Gro61]
Alexandre Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique
des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. (1961), no. 11, 5–167.
[Kap66]
Irving Kaplansky, The homological dimension of a quotient field, Nagoya Math. J. 27
(1966), 139–142.
[Kir73]
David Kirby, Coprimary decomposition of artinian modules, J. London Math. Soc. (2)
6 (1973), 571–576.
[Kun69]
Ernst Kunz, Characterizations of regular local rings for characteristic p, Amer. J.
Math. 91 (1969), 772–784.
[Mac73]
Ian G. Macdonald, Secondary representation of modules over a commutative ring,
Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome,
1971), Academic Press, London, 1973, pp. 23–43.
[Mat60]
Eben Matlis, Divisible modules, Proc. Amer. Math. Soc. 11 (1960), 385–391.
, Cotorsion modules, Mem. Amer. Math. Soc. 49 (1964), 1–66.
[Mat64]
[Mat86]
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced
Mathematics, no. 8, Cambridge Univ. Press, Cambridge, 1986, Translated from the
Japanese by M. Reid.
[Ric06]
Andrew S. Richardson, Co-localization, co-support and local homology, Rocky Mountain J. Math. 36 (2006), no. 5, 1679–1703.
[Sha76]
Rodney Y. Sharp, Secondary representations for injective modules over commutative
Noetherian rings, Proc. Edinburgh Math. Soc. (2) 20 (1976), no. 2, 143–151.
[Yas95]
Siamak Yassemi, Coassociated primes, Comm. Algebra 23 (1995), no. 4, 1473–1498.
[Yas98]
, Weakly associated primes under change of rings, Comm. Algebra 26 (1998),
no. 6, 2007–2018.
[Zös83]
Helmut Zöschinger, Linear-kompakte Moduln über noetherschen Ringen, Arch. Math.
(Basel) 41 (1983), no. 2, 121–130.
CRITERIA FOR FLATNESS AND INJECTIVITY
19
Universität Osnabrück, Institut für Mathematik, 49069 Osnabrück, Germany
E-mail address: nepstein@uni-osnabrueck.de
Department of Math and Statistics, Georgia State University, 30 Pryor St., Atlanta, GA 30303
E-mail address: yyao@gsu.edu