THE GENERATING HYPOTHESIS IN THE DERIVED
CATEGORY OF A RING
MARK HOVEY, KEIR LOCKRIDGE, AND GENA PUNINSKI
Abstract.We show that a strong form (the fully faithful version) of the
generating hypothesis, introduced by Freyd in algebraic topology, holds *
*in the
derived category of a ring R if and only if R is von Neumann regular. Th*
*is
extends results of the second author [Loc05]. We also characterize rings*
* for
which the original form (the faithful version) of the generating hypothe*
*sis holds
in the derived category of R. These must be close to von Neumann regular*
* in
a precise sense, and, given any of a number of finiteness hypotheses, mu*
*st be
von Neumann regular. However, we construct an example of such a ring that
is not von Neumann regular, and therefore does not satisfy the strong fo*
*rm of
the generating hypothesis.
Introduction
The generating hypothesis was introduced by Peter Freyd [Fre66] in algebraic
topology, where it is the assertion that any map f :X -!Y of finite spectra tha*
*t is
0 on stable homotopy groups is in fact null homotopic. The generating hypothesi*
*s is
widely considered to be one of the most important and difficult problems in sta*
*ble
homotopy theory. It has many implications for the structure of the stable homot*
*opy
ring ss*S0 of the sphere, implying for example that it is totally non-coherent *
*[Fre66]
and that the p-completion ss*S0pis a self-injective ring [Hov ]. Somewhat surpr*
*isingly,
Freyd proved that the generating hypothesis in fact implies that the map
[X, Y ] -!Hom ss*S0(ss*X, ss*Y )
from maps of finite spectra to maps of their stable homotopy modules is not only
injective but also surjective. That is, the generating hypothesis implies that*
* the
stable homotopy functor is fully faithful on finite spectra.
One approach to understanding the generating hypothesis is to look at analogo*
*us
questions in other categories. Following the second author [Loc05], we say that*
* a
ring R satisfies the generating hypothesis if whenever f :X -!Y is a map of
perfect complexes in the derived category D(R) of R and H*f = 0, then f = 0.
Recall that a perfect complex is a bounded chain complex of finitely generated
projective (right) modules, and that f = 0 in D(R) exactly when f is chain ho-
motopic to 0 (for maps of perfect complexes). Perfect complexes are the algebra*
*ic
analogue of finite spectra, as they are the small objects in D(R). Thus R satis*
*fies
the generating hypothesis exactly when the homology functor is faithful on perf*
*ect
complexes. Let us also say that R satisfies the strong generating hypothesis
if the homology functor is fully faithful on perfect complexes.
The second author noticed [Loc05, Section 4] that the homology functor is fai*
*th-
ful on all of D(R) if and only if all right R-modules are projective; that is, *
*if and
____________
Date: October 5, 2006.
1
2 MARK HOVEY, KEIR LOCKRIDGE, AND GENA PUNINSKI
only if R is semisimple. Since perfect complexes are the small objects of D(R)
and finitely presented modules are the small R-modules, it is natural to conjec-
ture (as the second author did in [Loc05]) that the homology functor is faithfu*
*l on
perfect complexes (that is, R satisfies the generating hypothesis) if and only *
*if all
finitely presented right R-modules are projective; that is, if and only if R is*
* von
Neumann regular. The second author verified that all von Neumann regular rings
do satisfy the generating hypothesis, and proved that if R satisfies the genera*
*ting
hypothesis and is either commutative or right coherent, then R is von Neumann
regular [Loc05].
In this paper, we first prove that R satisfies the strong generating hypothes*
*is if
and only if R is von Neumann regular. We then consider the generating hypothesi*
*s,
in effect asking whether the generating hypothesis implies the strong generating
hypothesis. We prove that R satisfies the generating hypothesis if and only if *
*all
short exact sequences of finitely presented modules split, and all submodules o*
*f flat
modules are flat. This makes R close to von Neumann regular, and in fact if R is
local or satisfies one of several finiteness hypotheses it forces R to be von N*
*eumann
regular. However, we construct an example of a ring that satisfies the generati*
*ng
hypothesis but is not von Neumann regular. Over this ring, then, the homology
functor is faithful on perfect complexes but not full.
The authors would like to thank Grigory Garkusha for many helpful discussions.
All R-modules M will be right R-modules in this paper, so that, for example,
D(R) is the unbounded derived category of right R-modules. The differential d in
a chain complex P will lower dimension, so that dn :Pn -!Pn-1. We will denote
kerdn by ZnP and imdn by Bn-1P . If M is an R-module, then Dn(M) denotes
the complex which is M in degree n and n - 1 and 0 elsewhere, with dn being the
identity. SnM denotes the complex that is M in degree n and 0 elsewhere.
1. The strong generating hypothesis
We begin by recalling the second author's characterization of semisimple ring*
*s.
Lemma 1.1. Suppose P is a perfect complexLof R-modules with both BnP and
HnP projective for all n. Then P ~= nSn(HnP ) in D(R). In this case, the
natural map [P, Q] -!Hom R (H*P, H*Q) is an isomorphism for all complexes Q.
Proof.We have
Pn ~=ZnP Bn-1P ~=BnP HnP Bn-1P.
L
>FromLthis it follows that P ~= nDn(Bn-1P ) Sn(HnP ), which is isomorphic
to nSn(HnP ) in D(R).
A chain map from SnHnP to a complex Q is the same thing as a map f :HnP -!
ZnQ, and such a map is chain homotopic to 0 exactly when there is a map
D :HnP -! Qn+1 such that dD = f. Since HnP is projective, f is chain ho-
motopic to 0 if and only if f lands in BnQ. Using projectivity of HnP again, we
conclude that [SnHnP, Q] ~=Hom R(HnP, HnQ).
Proposition 1.2. A ring R is semisimple if and only if the homology functor is
faithful on D(R). Furthermore, in this case, the homology functor is in fact fu*
*lly
faithful on D(R).
Proof.Suppose the homology functor is faithful in D(R). Take two R-modules M
and N, and take a projective resolution P* of M. Then an element of Exts(M, N)
THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF A RING 3
is represented by a map from P* to N, thought as a complex concentrated in degr*
*ee
s. This map is necessarily 0 in homology when s > 0. Thus Exts(M, N) = 0 for all
s > 0 and all M, N, so every R-module is projective and R is semisimple.
On the other hand, if R is semisimple, then Lemma 1.1 implies that homology
is fully faithful.
The analogue for the generating hypothesis is the following theorem.
Theorem 1.3. A ring R satisfies the strong generating hypothesis if and only if*
* R
is von Neumann regular. In this case, the natural map
[P, Q] -!Hom R (H*P, H*Q)
is an isomorphism for all perfect complexes P and arbitrary complexes Q.
Recall that R is von Neumann regular if and only if, for every x 2 R, there is
a y 2 R with x = xyx. The standard reference for von Neumann regular rings
is [Goo91 ]; the book [Lam99 ] takes an approach based on module categories, so
contains some different and useful results about von Neumann regular rings. A
standard characterization is that R is von Neumann regular if and only if all R-
modules are flat, which is true if and only if all finitely presented R-modules*
* are
projective.
Proof.Suppose R satisfies the strong generating hypothesis. Then,
ann`annr(Rx) = Rx
for all x 2 R, by [Loc05, Proposition 2.7]. Now take x 2 R, and consider the pe*
*rfect
complex P with Pi = R if i = 0, 1 and Pi = 0 otherwise, with the differential
P1 -! P0 being left multiplication by x. This complex has H0(P ) = R=xR and
H1(P ) = annr(x). By the strong generating hypothesis, there exists a chain map
OE: P -! P such that H1(OE) = 0 and H0(OE) = 1, the identity of R=xR. Translati*
*ng,
this means there exist elements a, b 2 R such that xa = bx with a 2 ann`annr(x)
(so that H1(OE) = 0) and b = 1 + xc for some c 2 R (so that H0(OE) = 1). But th*
*en
a = dx for some d 2 R, so we have
xdx = xa = bx = (1 + xc)x = x + xcx.
This means that x = x(d - c)x. Since x was arbitrary, R is von Neumann regular.
Conversely, suppose R is von Neumann regular, and P is a perfect complex.
In a von Neumann regular ring, finitely generated submodules of projectives are
projective [Lam99 , p.44], so BnP is finitely generated projective for all n. *
*Then
ZnP , as the kernel of the (necessarily split) surjection Pn -!Bn-1P , is also *
*finitely
generated projective for all n. Hence HnP is finitely presented, and so is proj*
*ective
for all n. Now Lemma 1.1 implies that homology is fully faithful on maps out of
perfect complexes.
Recall from [Loc05] that if R is either commutative or right coherent and R
satisifes the generating hypothesis, then R is von Neumann regular. Hence we get
the following corollary.
Corollary 1.4. If R is either commutative or coherent, then R satisfies the gen-
erating hypothesis if and only if R satisfies the strong generating hypothesis.
4 MARK HOVEY, KEIR LOCKRIDGE, AND GENA PUNINSKI
The second author also investigated the generating hypothesis from the view-
point of global stable homotopy theory. Using the results of [Loc05], we get t*
*he
following corollary.
Corollary 1.5. A ring R satisfies the strong generating hypothesis if and only *
*if,
in D(R), the thick subcategory generated by R is the collection of retracts of *
*finite
coproducts of suspensions of R.
Recall that a full subcategory of a triangulated category is called thick if *
*it
is closed under shifts, retracts, and cofibers; the thick subcategory generated*
* by
R consists of the perfect complexes. This corollary follows from [Loc05, Propo-
sition 5.1], and indicates how different stable homotopy theory must be from the
derived category of a ring if the generating hypothesis in stable homotopy is t*
*o be
true, since there are many finite spectra that are not retracts of finite copro*
*ducts
of suspensions of the sphere.
2.Rings that satisfy the generating hypothesis
Having dealt with the strong generating hypothesis, we now turn our attention
to the generating hypothesis. The object of this section to prove the following
theorem.
Theorem 2.1. A ring R satisfies the generating hypothesis if and only if R has *
*weak
global dimension at most 1 and all finitely presented R-modules are FP-injectiv*
*e.
Weak global dimension at most 1 is of course equivalent to the statement that
submodules of flat modules are flat. Recall that a module M is said to be FP-
injective if Ext1(F, M) = 0 for all finitely presented modules F ; thus all fin*
*itely
presented modules are FP-injective if and only if all short exact sequences of
finitely presented modules split. FP-injective modules seem to have been intro-
duced in [Ste70]; a good guide to the literature can be found in [Fai99, Chapte*
*r 6].
An FP-injective module is sometimes called absolutely pure, because M is FP-
injective if and only if every short exact sequence
0 -!M -!N -!P -! 0
is pure (that is, remains exact upon tensoring with any left R-module). See [La*
*m99 ,
Theorem 4.89(5)] for a proof of this equivalence.
To compare the rings of Theorem 2.1 with von Neumann regular rings, the
following lemma is helpful.
Lemma 2.2. A ring R is von Neumann regular if and only if every R-module is
FP-injective.
This lemma is well-known, but does not appear in [Goo91 ] or [Lam99 ], so we
include the proof for the convenience of the reader.
Proof.Suppose R is von Neumann regular, and M is an R-module. Choose a short
exact sequence E
0 -!M -!I -!N -!0
where I is injective. Since N is necesarily flat, this sequence is pure [Lam99*
* ,
Theorem 4.85]. Hence, if F is finitely presented, Hom (F, E) is still exact [La*
*m99 ,
Theorem 4.89(5)], and so Ext1(F, M) = 0 and M is FP-injective.
THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF A RING 5
Conversely, if every module is FP-injective, another application of [Lam99 , *
*The-
orem 4.89(5)] shows that every short exact sequence is pure. Then [Lam99 , Theo-
rem 4.85] shows that every module is flat, as required.
We now begin the proof of Theorem 2.1. Our first task is to characterize the
homology groups of perfect complexes.
Proposition 2.3. Suppose R is a ring. An R-module M is a homology module of
a perfect complex of R-modules if and only if there exists a finitely presented*
* module
F such that M embeds in F and the quotient F=M embeds in a projective module.
Furthermore, in this case, there is a perfect complex P such that Pn = 0 unless
n = 0, 1, 2 and M = H1P .
Proof.Suppose M = HnP , where each Piis a finitely generated projective module.
Then we have a short exact sequence
0 -!M -!Pn=BnP -dn!Bn-1P -! 0,
Pn=BnP is finitely presented and Bn-1P embeds in the projective module Pn-1.
Conversely, suppose M embeds in the finitely presented module F and the quo-
tient F=M embeds in the projective module P0, which we can assume is finitely
generated since F is so. Choose a presentation
P2 d2-!P1 p-!F -! 0
of F , where P1 and P2 are finitely generated projectives. Define the map d1: P*
*1 -!
P0 to be the composite
P1 p-!F -! F=M -!P0.
This defines a three-term perfect chain complex P . Pulling back the presentati*
*on
of F through the inclusion M -!F shows that H1P ~=M.
Corollary 2.4. Suppose P is a perfect complex. Then each cycle module ZnP is
a homology module of some perfect complex.
Proof.Note that ZnP is a submodule of the finitely presented module Pn and the
quotient Pn=ZnP embeds in the projective module Pn-1.
We now take a significant step towards Theorem 2.1 by showing how FP-injective
modules get involved.
Theorem 2.5. Let R be a ring, and let Q be an arbitrary object of D(R). Then
the generating hypothesis with target Q is true in D(R) if and only if HnQ is F*
*P-
injective for all n. In particular, R satisfies the generating hypothesis if an*
*d only if
all homology modules of perfect complexes are FP-injective.
The generating hypothesis with target Q is the statement that any map
f :P -! Q in D(R) where P is a perfect complex and H*f = 0 has f = 0. So R
satisfies the generating hypothesis if and only if R satisfies the generating h*
*ypothesis
with target Q for all perfect complexes Q.
Note in particular that this theorem and Lemma 2.2 imply that R satisfies the
generating hypothesis with target Q for all (not necessarily perfect) Q, if and*
* only
if R is von Neumann regular.
6 MARK HOVEY, KEIR LOCKRIDGE, AND GENA PUNINSKI
Proof.Suppose first that the generating hypothesis with target Q holds, and con-
sider a finitely presented module F and an integer n. Choose a finite presentat*
*ion
Pn dn-!Pn-1 -!F -! 0
of F , so that, by letting Pi= 0 for i 6= n, n - 1, we get a perfect complex P**
* with
Hn-1P* = F . To prove that
Ext1(F, HnQ*) = 0,
it suffices to show that any map
f :Pn=ZnP -! HnQ*
___ ___
extends to a map g :Pn-1 -! HnQ* with gdn = f, where dn is the map induced
by dn.
Since Pn is projective, there is a map OEn :Pn -!Qn such that the composite
Pn OEn--!Qn q-!Qn=BnQ
is the composite
Pn p-!Pn=ZnP -f!HnQ* i-!Qn=BnQ.
Now let OEn-1: Pn-1 -!Qn-1 be the zero map. Then OE: P* -!Q* is a chain map.
Indeed, write dn :Qn -!Qn-1 as dn = rq. Then
dnOEn = rqOEn = rifp = 0
since ri = 0. Furthermore, OE induces the zero map on homology, because if x 2
ZnP , then qOEnx = 0, so OEnx is a boundary.
If the generating hypothesis is true, then OE must be chain homotopic to 0. T*
*his
gives us maps Dn-1: Pn-1 -! Qn and Dn :Pn -! Qn+1 such that dnDn-1 = 0
and Dn-1dn_+ dn+1Dn = OEn. Since_dn :Qn -!Qn-1 factors through Qn=BnQ as
dn = dnq, we conclude that dnqDn-1 = 0, so there exists a map g :Pn-1 -!HnQ*
such that ig = qDn-1. Of course, we claim that g is the desired extension. To s*
*ee
this, apply q to the relation
Dn-1dn + dn+1Dn = OEn
to get
qDn-1dn = ifp origdn = ifp.
___
Writing dn = dnp and using_the_fact that i is a monomorphism and p is an epimor-
phism, we conclude that gdn = f, as required.
Now suppose that every homology group of Q is FP-injective, and OE: P* -!Q* is
a map of perfect complexes that induces 0 on homology. We will construct a chain
homotopy Dn :Pn -!Qn+1 such that dn+1Dn + Dn-1dn = OEn by induction on n.
Our induction hypothesis will be that we have constructed Di for i n - 1 and
that OEn - Dn-1dn, which is a map from Pn to Qn, in fact lands in the boundaries
BnQ. Getting started is easy since P is bounded below. For the induction step,
our hypothesis gives us the commutative square below,
____OE
Pn+1=Zn+1P ----! n+1Qn+1=Bn+1Q
? ?
dn+1?y ?ydn+1
Pn --------!OE BnQ
n-Dn-1dn
THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF A RING 7
_____
where OEn+1exists because OE_is_zero on homology, so must take cycles to bounda*
*ries.
We will construct a lifting Dn :Pn -! Qn+1=Bn+1Q in this square. First of all,
there is obviously a map
En :Pn -!Qn+1=Bn+1Q
such that dn+1En = OEn - Dn-1dn, simply because Pn is projective. Then
_____ _____
dn+1(OEn+1- Endn+1) = dn+1OEn+1- OEndn+1 + Dn-1dndn+1 = 0.
_____
Hence OEn+1-Endn+1 is a map from Pn+1=Zn+1P to Hn+1Q*. Since Hn+1Q*_is FP-
injective, there is a map Fn :Pn -!Hn+1Q* such that Fndn+1 = OEn+1- Endn+1.
Hence ___
Dn = En + Fn :Pn -!Qn+1=Bn+1Q
defines a lift in our commutative square._
We now choose Dn :Pn -! Qn+1 lifting Dn , which we can do because Pn is
projective._ Then_one_can easily check that dn+1Dn = OEn - Dn-1dn, and also,
because Dndn+1 = OEn+1, that OEn+1 - Dndn+1 lands in Bn+1Q. This completes the
induction step and the proof.
We can now prove Theorem 2.1.
Proof of Theorem 2.1.Suppose the generating hypothesis holds in D(R). In view
of Theorem 2.5, we need only show that R has weak dimension at most 1. Since
Tor*(-, M) commutes with direct limits, it suffices to show that the weak dimen*
*sion
of any finitely presented module is at most 1. Since any finitely presented mod*
*ule
is a homology group of a perfect complex, it is enough to show that the cycles
ZnP and the boundaries BnP are flat for all perfect complexes P and integers n.
But ZnP is itself a homology group of a perfect complex by Corollary 2.4, and so
Theorem 2.5 implies that ZnP is FP-injective. This means that the short exact
sequence
0 -!ZnP -! Pn -!Bn-1P -! 0
is pure. Now choose a left R-module M and apply - R M to this short exact
sequence. By purity, it remains exact, and so the Tor long exact sequence shows
that TorR1(Bn-1P, M) = 0. Since M was arbitrary, Bn-1P is flat. But then ZnP ,
as a kernel of a surjection of flat modules, is also flat.
Conversely, assume R has global weak dimension at most 1 and all finitely pre-
sented R-modules are FP-injective. We need to show that an arbitrary homology
group M of a perfect complex is FP-injective, by Theorem 2.5. By Proposition 2.*
*3,
there is a finitely presented module F and an exact sequence
0 -!M -!F -! F=M -!0,
where F=M embeds in a projective module. Since R has global weak dimension
at most 1, F=M is flat. But then the above exact sequence is pure [Lam99 , Theo-
rem 4.85]. Applying Hom R(N, -) to this sequence we get a long exact sequence
0 -!Hom R (N, M) -!Hom R (N, F ) -!Hom R (N, F=M)
-! Ext1R(N, M) -!Ext1R(N, F ) -!. . .
If N is finitely presented, though, the map Hom R(N, F ) -! Hom R(N, F=M) is
surjective, since our original sequence is pure [Lam99 , Theorem 4.89(5)]. By h*
*y-
pothesis, Ext1R(N, F ) = 0, so we conclude that Ext1R(N, M) = 0. Thus M is
FP-injective.
8 MARK HOVEY, KEIR LOCKRIDGE, AND GENA PUNINSKI
3.Examples and counterexamples
In this section, we give conditions under which rings that satisfy the genera*
*ting
hypothesis must be von Neumann regular, and also give an example of a ring that
satisfies the generating hypothesis yet is not von Neumann regular, and thus do*
*es
not satisfy the strong generating hypothesis.
Theorem 3.1. A ring R is von Neumann regular if and only if the generating
hypothesis holds in D(R) and finitely generated flat submodules of projective r*
*ight
R-modules are projective.
Proof.Assume that the generating hypothesis holds in D(R) and finitely generated
flat submodules of projectives are projective. We will show that all finitely p*
*resented
modules, and hence all modules, are flat. Given a finitely presented module M,
choose a perfect complex P with M ~= HnP for some n. We then have a short
exact sequence
0 -!BnP -! ZnP -! M -!0.
Now BnP is finitely generated and flat (since it is a submodule of Pn) by Theo-
rem 2.1. By hypothesis, then, BnP is finitely generated projective. Hence BnP is
FP-injective by Theorem 2.1 again, and so the above exact sequence splits. Thus
M is a summand of ZnP , which is flat as well, since it is also a submodule of *
*Pn.
So M is flat.
Conversely, if R is von Neumann regular, then any finitely generated submodule
of a projective module is projective [Lam99 , Example 2.32(d)].
This immediately gives the following corollary, implicit in [Loc05].
Corollary 3.2. A ring R is von Neumann regular if and only if R satisfies the
generating hypothesis and is right coherent.
Proof.If R is right coherent, then a finitely generated submodule of a projecti*
*ve
module is finitely presented. If it is also flat, then it is projective.
There are a great many rings where finitely generated flat modules are known
to be projective [PR04 ]. The following theorem contains some cases of this, wh*
*ich
are somewhat less satisfactory since not all von Neumann regular rings satisfy *
*the
hypotheses.
Theorem 3.3. Suppose the generating hypothesis holds in D(R) and one of the
following hypotheses holds.
(1) R is local (unique maximal right ideal).
(2) R is semiperfect (every finitely generated module has a projective cover*
*).
(3) R is reduced (no nonzero nilpotents)and has finite uniform dimension (R
contains no infinite direct sum of nonzero right ideals).
(4) R has zero Jacobson radical and finite uniform dimension.
(5) R is right nonsingular (the only element whose right annihilator is esse*
*ntial
in R is 0)and has finite uniform dimension.
(6) R is simple (no nontrivial two-sided ideals)and has finite uniform dimen-
sion.
Then R is von Neumann regular.
THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF A RING 9
Note that these conditions may not all be independent of each other. For ex-
ample, the authors suspect that if R is both right FP-injective (as it must be *
*if it
satisfies the generating hypothesis) and has finite uniform dimension, then R m*
*ay
have to be semiperfect.
Proof.For a local, semiperfect, or right nonsingular ring with finite uniform d*
*imen-
sion, every finitely generated flat module is projective; the local case is due*
* to Endo
and can be found in [Lam99 , Theorem 4.38]. The semiperfect case is due to Bass
and is [Lam99 , Exercise 4.21]. The right nonsingular case is due to Sandomier-
ski [San68, Corollary 1,p. 228]. Every reduced ring is right nonsingular by [La*
*m99 ,
Lemma 7.8]; since the singular elements form a two-sided ideal, every simple ri*
*ng is
also right nonsingular [Lam99 , Section 7A]. If R is FP-injective, or in fact o*
*nly has
Ext1(R=aR, R) = 0 for all a 2 R, then having zero Jacobson radical is equivalent
to being right nonsingular, by [NY95 , Theorem 2.1].
Not every von Neumann regular ring has finite uniform dimension. They all,
however, are right nonsingular [Lam99 , Corollary 7.7]. This leads to the follo*
*wing
theorem.
Theorem 3.4. A ring R is von Neumann regular if and only if it satisfies the
generating hypothesis, is right nonsingular, and its maximal right ring of quot*
*ients
Q is a flat left R-module.
The maximal right ring of quotients of R is the endomorphism ring of the inje*
*ctive
hull of R as a right R-module, and is much studied in ring theory. See [Lam99 ,
Section 13] for an introduction. When R is right nonsingular, Q is just equal t*
*o the
injective hull of R.
Proof.Sandomierski [San68, Theorem 2.9] proves that if R is right nonsingular a*
*nd
the maximal right ring of quotients Q is flat as a left R-module, then finitely*
* gen-
erated flat submodules of free R-modules (and hence also of projective R-module*
*s)
are projective. Theorem 3.1 completes the proof.
Theorem 3.5. There exists a ring S that satisfies the generating hypothesis but*
* is
not von Neumann regular.
Of course, such a ring will not satisfy the strong generating hypothesis. Bef*
*ore
proving this theorem, we need the following lemmas.
Lemma 3.6. Every principal right ideal of a ring R is flat if and only if whene*
*ver
ab = 0 in R there is an x 2 R such that ax = 0 and xb = b.
Proof.Consider the short exact sequence
0 -!annr a -!R ax--!aR -!0.
By [Lam99 , Theorem 4.23], aR is flat if and only if for every b 2 annra, there*
* is a
map ` :R -!annr(a) with `(b) = b. Translating, this means that aR is flat if and
only if whenever ab = 0, there is an x such that ax = 0 and xb = b.
Lemma 3.7. A ring R has global weak dimension 1 if and only if for every
integer m and every pair of m x m matrices A, B over R with AB = 0, there is an
m x m matrix X over R such that AX = 0 and XB = B.
10 MARK HOVEY, KEIR LOCKRIDGE, AND GENA PUNINSKI
Proof.In view of Lemma 3.6, the matrix condition of this lemma is equivalent to
every principal right ideal of Mm (R) being flat, for all m 1. We will use t*
*he
Morita equivalence between R and Mm (R) to prove that this is equivalent to R
having global weak dimension 1. Indeed, if R has global weak dimension 1, so
does Mm (R) [Lam99 , p. 481], and so every ideal of Mm (R) is flat.
Conversely, suppose every principal right ideal of Mm (R) is flat for all m *
* 1.
Suppose I is an m-generated right ideal of R. Then I corresponds under the Mori*
*ta
equivalence to a principal right ideal of Mm (R) [Lam99 , Remark 17.23(C)]. This
principal ideal is flat, and so I is flat as well, since Morita equivalences pr*
*eserve
flatness [Lam99 , p. 481].
Hence all finitely generated ideals of R are flat. Since Torcommutes with dir*
*ect
limits, all ideals of R are flat. But then R has weak dimension 1 [Lam99 ,
Lemma 4.66].
Proof of Theorem 3.5.We will use the method of [PRZ95 ], who introduce and
study indiscrete rings. For us, the salient property of indiscrete rings is th*
*at all
finitely presented modules over an indiscrete ring are FP-injective [PRZ95 , Th*
*eo-
rem 2.4]. Thus, we must find an indiscrete ring that also has weak dimension on*
*e.
The construction given in [PRZ95 , p. 359] begins with a finite-dimensional alg*
*ebra
R of finite representation type over an infinite field F . Because we want to *
*end
up with something of weak dimension one, we will take R to have right (and left)
global dimension 1. For example, we can take R to be the ring of 2 x 2 upper
triangular matrices over F , which is a classical example of a ring of right (a*
*nd left)
global dimension 1 that is not von Neumann regular [Lam99 , Example 2.36].
The method of [PRZ95 ] is then to construct a map o :R -!MnR and then let
S = Ro be the direct limit
Mn2o
S = Ro = colim(R o-!MnR Mno---!Mn2R ----! . .).
Then Prest, Rothmaler, and Ziegler show that S is always indiscrete. Now, in our
case, our ring R has global dimension 1, and therefore all of the MkR also have
global dimension 1 since they are Morita equivalent to R. Now, if we take a pai*
*r of
m x m matrices A, B over S with AB = 0, then we can choose k large enough such
that A, B are actually matrices over MnkR, and AB = 0 as such matrices. Then
Lemma 3.7 shows that there is a matrix X over MnkR, and hence over S, with
AX = A and XB = 0. Thus Lemma 3.7 implies that S has weak dimension 1,
and S cannot be von Neumann regular because R is not (see [PRZ95 , p. 359]).
The indiscrete rings of [PRZ95 ], of which our counterexample S is one, have
been generalized by Garkusha and Generalov [GG99 ] to the class of almost regul*
*ar
rings, in which all (left or right) finitely presented modules are FP-injective*
*. The
indiscrete rings are the simple almost regular rings.
We also note that the ring S of Theorem 3.5 is in fact weakly semihereditary
in the sense of Cohn [Coh85 , p. 13]. This means that if A and B are (not neces*
*sarily
square) matrices such that AB = 0, then there is an idempotent matrix E such th*
*at
AE = A and EB = 0. Since hereditary implies weakly semihereditary, each Mn(R)
in the above proof is weakly semihereditary, and so the same argument shows that
S is as well. One can then use (the left module version of) Lemma 3.7 to see th*
*at
weakly semihereditary implies global weak dimension 1.
THE GENERATING HYPOTHESIS IN THE DERIVED CATEGORY OF A RING 11
We now turn to some questions we have not been able to answer. First of all,
the stable homotopy category in topology is more like D(R) for a graded ring R
(or, better yet, a differential graded algebra R), though, it must be stressed,*
* these
categories are still much simpler than the stable homotopy category. We have not
considered the generating hypothesis for these R.
We could ask whether there is a ring R that satisfies the generating hypothes*
*is
for right R-modules but not left R-modules. Such a ring could not be von Neumann
regular, of course.
Also, recall that there is a strongly convergent spectral sequence whose E2 t*
*erm
is Ext**R(H*P, H*Q) converging to D(R)(P, Q)*. It seems intuitively evident that
for the strong generating hypothesis to hold, this spectral sequence must colla*
*pse
to the 0-line for perfect complexes P and Q. This is in fact true, since in thi*
*s case R
is von Neumann regular, hence coherent, so the homology groups H*P are finitely
presented modules and therefore projective.
However, the situation for R satisfying the generating hypothesis but not the
strong generating hypothesis is less clear. To satisfy the generating hypothesi*
*s, it
must be that every element of Exts,*(H*P, H*Q) with s > 0 does not survive the
spectral sequence. But in order not to satisfy the strong generating hypothesi*
*s,
there must be an element of Hom *(H*P, H*Q) for some perfect P and Q that
supports a differential. It would be intriguing to understand how this happens.
Finally, one could define R to satisfy the n-fold generating hypothesis if
whenever f1, . .,.fn are composable maps of perfect complexes such that H*(fi) *
*= 0
for all i, then fn O . .O.f1 = 0 in D(R). If we ask for this condition to hold*
* for
all n-tuples of composable maps with H*fi = 0, not just maps between perfect
complexes, then the second author has shown in his thesis, using work of Chris-
tensen [Chr98], that R has projective dimension n. One could then ask for
an analogous characterization of rings R, probably in terms of weak dimension,
that satisfy the n-fold generating hypothesis, or some strong version of the n-*
*fold
generating hypothesis.
References
[Chr98]J. Daniel Christensen, Ideals in triangulated categories: phantoms, ghos*
*ts and skeleta,
Adv. Math. 136 (1998), no. 2, 284-339. MR MR1626856 (99g:18007)
[Coh85]P. M. Cohn, Free rings and their relations, second ed., London Mathemati*
*cal Society
Monographs, vol. 19, Academic Press Inc. [Harcourt Brace Jovanovich Publ*
*ishers], Lon-
don, 1985. MR MR800091 (87e:16006)
[Fai99]Carl Faith, Rings and things and a fine array of twentieth century assoc*
*iative alge-
bra, Mathematical Surveys and Monographs, vol. 65, American Mathematical*
* Society,
Providence, RI, 1999. MR MR1657671 (99j:01015)
[Fre66]Peter Freyd, Stable homotopy, Proc. Conf. Categorical Algebra (La Jolla,*
* Calif., 1965),
Springer, New York, 1966, pp. 121-172. MR MR0211399 (35 #2280)
[GG99] G. A. Garkusha and A. I. Generalov, Duality for categories of finitely p*
*resented modules,
Algebra i Analiz 11 (1999), no. 6, 139-152, translation in St. Petersbur*
*g Math. J. 11
(2000), no. 6, 1051-1061. MR MR1746072 (2001a:16005)
[Goo91]K. R. Goodearl, von Neumann regular rings, second ed., Robert E. Krieger*
* Publishing
Co. Inc., Malabar, FL, 1991. MR MR1150975 (93m:16006)
[Hov] Mark Hovey, On Freyd's generating hypothesis, to appear in Q. J. Math.
[Lam99]T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics,*
* vol. 189,
Springer-Verlag, New York, 1999. MR MR1653294 (99i:16001)
[Loc05]Keir Lockridge, The generating hypothesis in the derived category of R-m*
*odules,
preprint, 2005.
12 MARK HOVEY, KEIR LOCKRIDGE, AND GENA PUNINSKI
[NY95] W. K. Nicholson and M. F. Yousif, Principally injective rings, J. Algebr*
*a 174 (1995),
no. 1, 77-93. MR MR1332860 (96i:16005)
[PR04] Gena Puninski and Philipp Rothmaler, When every finitely generated flat *
*module is
projective, J. Algebra 277 (2004), no. 2, 542-558. MR MR2067618 (2005c:1*
*6004)
[PRZ95]Mike Prest, Philipp Rothmaler, and Martin Ziegler, Absolutely pure and f*
*lat modules and
"indiscrete" rings, J. Algebra 174 (1995), no. 2, 349-372. MR MR1334216 *
*(96d:16002)
[San68]Francis L. Sandomierski, Nonsingular rings, Proc. Amer. Math. Soc. 19 (1*
*968), 225-230.
MR MR0219568 (36 #2648)
[Ste70]Bo Stenstr"om, Coherent rings and F P-injective modules, J. London Math.*
* Soc. (2) 2
(1970), 323-329. MR MR0258888 (41 #3533)
Department of Mathematics, Wesleyan University, Middletown, CT 06459
E-mail address: hovey@member.ams.org
Department of Mathematics, University of Washington, Box 354350, Seattle, WA
98195
E-mail address: lockridg@math.washington.edu
Department of Mathematics, University of Manchester, Booth Street East, Manch-
ester M13 PL, United Kingdom
E-mail address: gpuninski@maths.man.ac.uk