THE GHOST AND WEAK DIMENSIONS OF RINGS AND RING
SPECTRA
MARK HOVEY AND KEIR LOCKRIDGE
Abstract.The primary object of this paper is to prove the conjecture of
[HL08a] explaining how to recover the weak dimension of a ring from its *
*derived
category. In the process, we develop a theory of weak dimension, which w*
*e call
ghost dimension, for the generalized rings, known as ring spectra, that *
*arise
in algebraic topology.
Introduction
In a previous paper [HL08a ], the authors considered the problem of recovering
the weak dimension of a ring R from the derived category D(R), together with its
distinguished object R. In that paper, the authors defined the ghost dimension
of R, gh. dim.R, and proved that gh. dim.R w. dim.R, with equality holding
when R is coherent or has weak dimension 1. In the present paper, we prove that
gh. dim.R = w. dim.R for all rings R.
The point of doing this, besides its intrinsic interest, is to allow consider*
*ation of
weak dimension for more general kinds of rings. In algebraic topology, for exam*
*ple,
there is a notion of a ring spectrum, or, more precisely, an S-algebra E [EKMM9*
*7 ].
Such an S-algebra has no elements in the usual sense. There is a category of (r*
*ight)
E-modules, but it is not abelian. However, there is a derived category D(E) of
E, and it shares many of the formal properties of the derived category D(R) of *
*an
ordinary ring R; in particular, D(E) is a compactly generated triangulated cate*
*gory,
and there are derived tensor products and derived Hom objects. In fact, every r*
*ing
R has an associated Eilenberg-MacLane S-algebra HR, and D(HR) is equivalent
to D(R). To define invariants of such S-algebras E, then, one way to proceed is
to define usual ring invariants, such as the weak or global dimension, in terms*
* of
D(R), and then apply this definition to D(E) as well.
The second author did this for the (right) global dimension in his thesis, an*
*d we
now summarize this. For further details, see [HL08a ]. Define a map f :X -!Y in
D(E) to be a ghost if D(E)(E, f)* = 0. In the case that E is an ordinary ring R*
*, a
ghost is then just a map that induces the zero homomorphism on homology. If E is
an S-algebra, a ghost is a map that induces the zero homomorphism on homotopy
groups. The second author shows that the right global dimension of a ring R is *
*the
least n for which every composite of n + 1 ghosts in D(R) is null, or 1 if ther*
*e is
are arbitrarily long nonzero composites of ghosts. We can then define the global
dimension of an S-algebra E in an analogous fashion. The authors did this and
investigated the S-algebras of global dimension 0 in [HL08b ].
Weak dimension is more complicated, and there seem to be many possible defi-
nitions. A major goal of this paper is to elucidate the different possibilities*
* and to
____________
Date: March 26, 2009.
1
2 MARK HOVEY AND KEIR LOCKRIDGE
find the correct one. The ghost dimension of an S-algebra E or a ring R is the
least n such that every composite
X0 f1-!X1 f2-!. .f.n+1---!Xn+1
of n + 1 ghosts in D(E) (or D(R)), where X0 is a compact object, is null (or 1 *
*if
there are arbitrarily long nonzero composites of ghosts out of compact objects)*
*. Re-
call that X is compact in a triangulated category C if the functor C(X, -) pres*
*erves
coproducts. In particular, the compact objects of D(R) are the perfect complexes
(complexes quasi-isomorphic to bounded complexes of finitely generated projec-
tives), and the compact objects of D(E) are the retracts of finite cell E-modul*
*es.
The ghost dimension of a ring was discussed in [HL08a ], as mentioned above.
In addition, a version of weak dimension closely related to Rouquier's defini*
*tion
of the dimension of a triangulated category [Rou08 ] is defined in [HL08b ]. Ne*
*ither of
these uses the notion of a flat E-module. This obvious oversight was made becau*
*se
of a difficulty with flat modules that we now recall. If E is an S-algebra, and*
* F* is
a flat left E* = ss*E = D(E)(E, E)*-module, then we can form a homology theory
(a coproduct-preserving exact functor to abelian groups) on D(E) that takes M
to M* E* F*. One would like to say that Brown representability for homology
theories then forces there to be a left E-module F with ss*F ~=F*. Unfortunatel*
*y,
Brown representability does not hold for a general ring spectrum [Nee97, CKN01 *
*],
so flat modules may not be realizable. This worrying phenomenon led us to doubt
the utility of flat modules. However, we use them in this paper. One of the
surprising things we discover is the following. Call X 2 D(E) flat if X* is a f*
*lat
E*-module. Then we prove that X is flat if and only if every ghost f whose doma*
*in
is X is phantom, in the sense that D(E)(A, f) = 0 for all compact A 2 D(E).
This gives us two different notions of flat dimension. The one most similar to *
*the
algebraic situation we call the constructible flat dimension, con. flat dim.X.
It is a measure of how many steps one needs to construct X from flat objects of
D(E). We reserve the term flat dimension, flat dim.X, for the smallest n such
that every composite of n + 1 ghosts with domain X is phantom. This seems
algebraically strange, but has better properties. This gives several more noti*
*ons
of weak dimension: the maximal constructible flat (resp. flat) dimension of a
compact E-module, and the maximal constructible flat (resp. flat) dimension of *
*an
arbitrary E-module. We show that all of these are equal to the ghost dimension,
except possibly the maximal constructible flat dimension of an arbitrary E-modu*
*le.
The conjecture of [HL08a ] that gh. dim.R = w. dim.R then follows.
In the end, we are left with three definitions of weak dimension for an S-alg*
*ebra
E. There is gh. dim.E, which coincides with the maximal flat dimension of any
object. There is the maximal constructible flat dimension of any object, which
agrees with gh. dim.E for E = HR, but possibly not in general. And there is
the Rouquier dimension Rouq. dim.E, which agrees with gh. dim.E when E* is
coherent. We prove that the ghost dimension is right-left symmetric, which we
have been unable to do with any of the other definitions. Hence we argue that t*
*he
ghost dimension is the proper version of weak dimension for S-algebras E.
This subject sorely needs examples, in order to be sure that all these defini*
*tions
are in fact distinct. It should be possible to find an ordinary ring R such th*
*at
the Rouqier dimension of D(R) is distinct from the other dimensions. Such an
example would involve serious analysis of the derived category of a non-coherent
THE GHOST AND WEAK DIMENSIONS OF RINGS AND RING SPECTRA 3
ring. To determine whether the constructible flat dimension is different from t*
*he
flat dimension would seem to require a new idea.
Note that all modules we use in this paper are right modules unless explicitly
stated otherwise. The reader who is interested only in ordinary rings can read
R everywhere the symbol E or E* appears, read "chain complex of R-modules"
whenever the term "E-module" appears, and read H*X everywhere X* appears,
for X an E-module.
1. Ghost dimension and Rouquier dimension
For an S-algebra E or a ring R, the authors have previously considered two
different possible definitions related to weak dimension, which we now discuss.
First of all, we can define the Rouquier dimension to be the maximum number of
steps needed to build a compact object of D(E) from finitely many copies of E
(along the lines of Rouquier [Rou08 ]). In more detail, given a class A of obje*
*cts
of D(E), define n inductively as follows. Define 0 to be the collection o*
*f all
retracts of coproducts of suspensions of elements of A, and define an object X *
*to
be in n if and only if it is a retract of an object eXfor which there is an *
*exact
triangle
A -!Y -! eX-!A
where A 2 0, and Y 2 n-1. If A is a class of compact objects, we define
nfsimilarly, with 0fbeing the collection of all retracts of finite coprod*
*ucts
of suspensions of elements of A, and then using the same induction procedure to
define nf. Then nfconsists of compact objects in n, but there may be
compact objects in n that are nevertheless not in nf.
We define the Rouquier dimension of E (or R), Rouq. dim.E, to be the small-
est n such that nfis all of the compact objects, or 1 if no such n exists. T*
*his
was called the weak dimension in [HL08b ], but that seems inappropriate, since *
*we
do not know that it agrees with the weak dimension when E is an ordinary ring R.
We define the ghost dimension of E (or R), gh. dim.E, to be the smallest n such
that n contains all the compact objects. We also define the projective dimen-
sion, proj. dim.X, of a given object X to be the smallest n such that X 2 n.
This was called the ghost length in [HL08a ]. Then gh. dim.E is the supremum of
proj. dim.X for X compact.
The following proposition explains the connection to the definition given in *
*the
introduction.
Proposition 1.1. Suppose E is an S-algebra or an ordinary ring, and X 2 D(E).
Then proj. dim.X n if and only if every composite of n + 1 ghosts with domain
X is the zero map. Furthermore, proj. dim.X proj. dim.E*X*, with equality
when proj. dim.X = 0 and also when E is an ordinary ring and X is the projective
resolution of a module M.
This proposition is the content of Proposition 1.1, the proof of Proposition *
*1.3,
and Lemma 1.4 of [HL08a ], although Proposition 1.1 of [HL08a ] is really due to
Christensen [Chr98, Theorem 3.5].
We commonly call the objects P with proj. dim.P = 0 projective, as this
proposition implies P is projective if and only if P* is a projective E*-module*
*. We
note that the universal coefficient spectral sequence of [EKMM97 , Theorem IV*
*.4.1]
4 MARK HOVEY AND KEIR LOCKRIDGE
implies that if P is projective then the natural map
D(E)(P, X) -!Hom E*(P*, X*)
is an isomorphism for all X 2 D(E). The converse is also true, for if this natu*
*ral
map is an isomorphism, then there are no nonzero ghosts with domain P .
The following lemma gives the most obvious relationship between ghost dimen-
sion and Rouquier dimension.
Lemma 1.2. Suppose E is an S-algebra or an ordinary ring. Then
gh. dim.E Rouq. dim.E,
with equality holding when gh. dim.E = 0.
Proof.The inequality is clear. If gh. dim.E = 0, then every compact object is a
retract of a coproduct of suspensions of E, so is also a retract of a finite co*
*product
of suspensions of E.
Note that S-algebras with gh. dim.E = 0 are called von Neumann regular,
because if R is an ordinary ring, gh. dim.R = 0 if and only if R is von Neumann
regular (see [HL08b ]).
The ghost dimension and the Rouquier dimension agree when E* is coherent, as
we see in the following proposition.
Proposition 1.3. Suppose E is an S-algebra or an ordinary ring for which E*
is coherent. Then X 2 nfif and only if X 2 n and X is compact. Thus
gh. dim.E = Rouq. dim.E.
There is no reason to think that gh. dim.E = Rouq. dim.E if E* is not coheren*
*t,
even if E is an ordinary ring R, but we do not know a counterexample.
Proof.We first prove the well-known fact that, since E* is coherent, for every
compact object X of D(E), X* is a finitely presented E*-module. Consider the
class C of X for which X* is a finitely presented E*-module. We claim that C is
a thick subcategory. Given this, since C contains E, it contains all the compa*
*ct
objects as required (this is well-known; see [HPS97 , Theorem 2.1.3] for a gene*
*ral
approach to this fact). To prove that C is thick, we must show that it is clos*
*ed
under suspensions and retracts, which is obvious in this case, and also if we h*
*ave
an exact triangle
X f-!Y -g!Z h-! X
in which X, Y are in C, then Z 2 C. Given such an exact triangle, Z* is an exte*
*n-
sion of cokerf* by ker( f*). Finitely presented modules are always closed under
cokernels of morphisms and extensions [Lam99 , Lemma 4.54]. If E* is coherent,
finitely presented modules are closed under kernels of morphisms as well, and so
Z* is finitely presented. Indeed, if g :M -! N is a morphism of finitely presen*
*ted
modules over a coherent ring, then M= kerf ~=imf is a finitely generated submod-
ule of the finitely presented module N, so is finitely presented. Hence kerf mu*
*st
be a finitely generated submodule of the finitely presented module M, so is fin*
*itely
presented.
Now suppose X 2 n and X is compact. By induction, because X* is finitely
presented, we can choose finite coproducts Piof suspensions of E and exact tria*
*ngles
Xi+1-! Pi-fi!Xi-ffii! Xi+1
THE GHOST AND WEAK DIMENSIONS OF RINGS AND RING SPECTRA 5
where X0 = X and fi is onto on homotopy, so ffii is a ghost. Of course each Xi *
*is
compact. Then consider the exact triangle
Xi+1-! Yi-! X -! i+1Xi+1.
By using the 3 x 3 lemma (well-known, but stated in [HPS97 , Lemma A.1.2]) on
the square
X - ---! iXi
flfl ?
fl ?y
X - ---! i+1Xi+1
we see that there is an exact triangle
i-1Pi-! Yi-1-! Yi-! iPi.
Hence Yiis in if. On the other hand, the map X -! n+1Xn+1 is the composite
of n + 1 ghosts, so it is null since X 2 n, using Proposition 1.1. Hence X i*
*s a
retract of Yn, so X 2 nf.
2.Flat dimension
We now offer a different approach to the weak dimension of an S-algebra or an
ordinary ring using flat modules. As discussed in the introduction, we did not *
*use
these in [HL08a ] because of the fundamental issue that homology functors are n*
*ot
always representable in D(E) for an S-algebra E, or even a ring R.
However, we can still define F to be the class of objects F in D(E) such that*
* F* is
flat over E*. In this case, we say that F is flat (as an object of D(E)). We ca*
*n then
define an E-module X 2 D(E) to have constructible flat dimension n, written
con. flat dim.X = n, if X 2 n. Note that con. flat dim.X proj. dim.X,
since every projective is flat. We can then consider the maximal constructible *
*flat
dimension of any object in D(R), or of just a compact object in D(R). Both of
these are possible candidates for something like weak dimension. In principle, *
*we
could also consider a definition similar to the Rouquier dimension, using compa*
*ct
flat objects to resolve arbitrary compact objects, but we will see that a compa*
*ct
flat object is projective, so this would just recover Rouquier dimension.
Proposition 2.1. We have con. flat dim.X flat dim.X* for all X 2 D(E). In
particular, the maximal constructible flat dimension of an object in D(E) is bo*
*unded
above by w. dim.E*.
Proof.There is nothing to prove if flat dim.X* is infinite, so suppose flat dim*
*.X* =
n. Then by beginning a projective resolution of X*, we get an exact sequence of
E*-modules
0 -!F -! Pn-1 dn-1---!.d.1.-!P0 d0-!X* -!0
where each Piis projective over E* and F is flat over E*. This gives us short e*
*xact
sequences
0 -!Ki+1-! Pi-! Ki-! 0
for i n - 1, where Ki = kerdi-1, K0 = X*, and Kn = F . Because the Pi are
projective, these short exact sequences are uniquely realizable by exact triang*
*les in
D(E)
Xi+1-! Qi-! Xi-! Xi+1
6 MARK HOVEY AND KEIR LOCKRIDGE
where X0 = X, (Xi)* = Ki and (Qi)* = Pi. In more detail, Pi is a retract of a
direct sum of copies of E*. Thus we can let Qi be the corresponding retract of a
coproduct of copies of E. Then one checks that a map out of Qi to any object Y
is equivalent to a map Pi-! Y*. This gives us exact triangles of the form
i-1Xi-! Yi-! X -! iXi
for all i, where the last map is the composite of the maps jXj -! j+1Xj+1.
Using the 3 x 3 lemma on the commutative square
X - ---! iXi
flfl ?
fl ?y
X - ---! i+1Xi+1
gives us exact triangles
i-1Qi-! Yi-! Yi+1-! iQi
for all i. In particular, con. flat dim.Yi i - 1. Now the exact triangle
n-1Xn -!Yn -!X -! nXn
shows that con. flat dim.X n, since (Xn)* is flat.
We now give an alternative characterization of the flat objects in D(E). Reca*
*ll
that a phantom map in D(E) is a map f :X -! Y such that fg = 0 for all
g :A -!X where A is compact. We need the following lemma.
Lemma 2.2. Suppose E is an S-algebra. A map f :X -!Y is phantom in D(E)
if and only if ss*(f ^E Z) = 0 for all left E-modules Z if and only if ss*(f ^E*
* Z) = 0
for all compact left E-modules Z.
Proof.Spanier-Whitehead duality implies that ss*(f ^E Z) = 0 for all compact le*
*ft
E-modules Z if and only if D(E)(W, f) = 0 for all compact right E-modules W ,
which of course is the definition of f being phantom. It remains to show that, *
*under
this condition, ss*(f ^E Z) = 0 for all left E-modules Z. But ss*(- ^E Z) is a *
*ho-
mology theory on D(E), and phantom maps vanish on all homology theories [CS98 ,
Proposition 1.1]. The reader should note that Christensen and Strickland are wo*
*rk-
ing in the ordinary category of spectra, where Spanier-Whitehead duality is int*
*ernal,
but the same proof will work for a general S-algebra E as long as we remember t*
*hat
Spanier-Whitehead duality shifts from left to right E-modules and vice versa.
Proposition 2.3. Suppose E is an S-algebra or a ring, and X 2 D(E). Then the
following are equivalent:
(1) X* is a flat E*-module.
(2) Every ghost with domain X is phantom.
(3) There is an exact triangle
P -! X g-!Y -! P
where P is projective and g is phantom.
(4) Every map A -!X, where A is compact, factors through a compact projec-
tive object.
THE GHOST AND WEAK DIMENSIONS OF RINGS AND RING SPECTRA 7
(5) The natural map
X* E* Z* -!ss*(X ^E Z)
is an isomorphism for all left E-modules Z.
If E is an ordinary ring R, then X ^E Z would be the total left derived tensor
product of the chain complex X of right R-modules and the chain complex Z of
left R-modules.
Proof.For any X, there is an exact triangle
P -! X g-!Y -! P
in which P is projective and g is a ghost. Indeed, we simply take an epimorphism
from a free E*-module P* to X*. We then let P be the corresponding coproduct
of suspensions of E, which is projective, and realize the map P* -!X* as a map
P -! X. The cofiber g is then automatically a ghost, and every other ghost with
domain X factors through g.
It follows from this that part (2) and (3) are equivalent. This also means th*
*at
part (3) implies part (4), since part (3) means that a map A -! X, where A is
compact, factors through a projective, and therefore a free E-module. Since A is
compact, it must factor through a finite coproduct of suspensions of E.
We now show that part (4) implies part (1). Recall the filtered category (X)
from [HPS97 , Section 2.3] of maps from a compact object into X, and consider t*
*he
full subcategory 0(X) of maps from a compact projective into X. Given part (4),
0(X) is cofinal in (X) and itself filtered. Thus, for any homology theory H,
H(X) = colim 0(X)H(Pff) by [HPS97 , Corollary 2.3.11]. In particular, X* is a
colimit of finitely generated projective modules, so is flat.
To see that part (1) implies part (5), use the universal coefficient spectral*
* se-
quence
TorE*s,t(X*, Z*) ) sst-s(X ^E Z)
of [EKMM97 , Theorem IV.4.1].
To see that part (5) implies part (2), suppose that g is a ghost with domain *
*X.
Part (5) then implies that ss*(g ^E Z) = 0 for all left E-modules Z, which impl*
*ies
that g is phantom by Lemma 2.2.
Recall that, for a general ring R, there are finitely generated flat modules *
*which
are not projective, though of course every finitely presented flat module is pr*
*ojec-
tive. The following corollary is an analog of this fact for S-algebras.
Corollary 2.4. Suppose E is an S-algebra or an ordinary ring. If X is a compact
flat object of D(E), then X is projective.
Proof.The universal ghost out of X is phantom, and hence null. Thus X is pro-
jective, necessarily finitely generated since X is compact.
We have been unable to fully generalize Proposition 2.3 to objects X with
con. flat dim.X = n. We therefore make the following definition.
Definition 2.5. Suppose E is an S-algebra or an ordinary ring, and X 2 D(E).
We say that X has flat dimension at most n, written flat dim.X n, if every
composite of n + 1 ghosts with domain X is phantom.
We then have the following theorem.
8 MARK HOVEY AND KEIR LOCKRIDGE
Theorem 2.6. Suppose E is an S-algebra or an ordinary ring, and X 2 D(E).
Then flat dim.X con. flat dim.X, and the following are equivalent.
(1) flat dim.X n.
(2) There is an exact triangle
B -!X g-!Y -! B
where proj. dim.B n and g is phantom.
(3) Every map A -!X, where A is compact, factors through a compact object
B with proj. dim.B n.
(4) For any left E-module Z, in the universal coefficient spectral sequence
E2s,t= TorE*s,t(X*, Z*) ) sst-s(X ^E Z),
we have E1s,*= 0 for all s > n.
It would be good to find an example where flat dim.X < con. flat dim.X, or to
prove they are always equal.
Proof.We show that con. flat dim.X n implies every composition of n+1 ghosts
out of X is phantom by induction on n. The base case of n = 0 is Proposition 2.*
*3.
For the induction step, suppose con. flat dim.X n,
X g1-!Z1 g2-!. .g.n+1---!Zn+1
is the composition g of n+1 ghosts, and f :A -!X is a map from a compact object.
We must show gf = 0. Since con. flat dim.X n, there is a cofiber sequence
F -! Y -s!eXh-! F
where F is flat, con. flat dim.Y n - 1, and there are maps
X i-!eXr-!X
with ri = 1X . Since gf = (gr)(if), and gr is again a composition of n + 1 ghos*
*ts,
we can assume X = eX.
The composition hf factors through a finitely generated projective P , by Pro*
*po-
sition 2.3. This gives us a commutative diagram
-1P ----! B --t--!A --u--! P
?? ? ? ?
y f0?y f?y ?y
F ----! Y ----!sX ----!h F
whose rows are exact triangles. Note that B is necessarily a compact object, and
so the composition gn O . .O.g1sf0 is null, since flat dim.Y n - 1. Hence we *
*have
gn O . .O.g1ft = 0 sogn O . .O.g1f = vu
for some map v :P -! Zn. But then gn+1v = 0, since P is projective, and so
gn+1 O . .O.g1f = 0
as required. This completes the proof that flat dim.X con. flat dim.X.
The work of Christensen [Chr98, Theorem 3.5] implies that, for any X, there is
an exact triangle
B -!X g-!Y -! B
THE GHOST AND WEAK DIMENSIONS OF RINGS AND RING SPECTRA 9
with proj. dim.B n and g is a composite of n+1 ghosts. But then every composi*
*te
of n + 1 ghosts with domain X factors through g. Thus every composite of n + 1
ghosts is phantom if and only if g is phantom, so part (1) and part (2) are equ*
*ivalent.
Now, the universal coefficient spectral sequence for ss*(X ^E Z) is construct*
*ed
as follows. Beginning with X0 = X, we construct exact triangles
Xi+1-! Qi-hi!Xi-ki! Xi+1
as in the proof of Proposition 2.1, in which Qiis projective, hiis onto on homo*
*topy,
and ki is a ghost. We then smash them with Z and take homotopy to get our
spectral sequence of homological type. An element in ss*(X ^E Z) is detected in
E1s,*if and only if it is in the kernel of
ss*(X ^E Z) -!ss*( jXj^E Z)
for j = s but not for j = s - 1. So we must determine when the map
ss*(X ^E Z) -!ss*( nXn ^E Z)
is zero for all Z. However, comparison of this construction of [EKMM97 , Sec-
tion IV.5] with [Chr98, Theorem 3.5] shows that in fact the map X -! nXn is
the same as the universal composite of n + 1 ghosts out of X, g :X -! Y of the
previous paragraph. Lemma 2.2 then shows part (2) and part (4) are equivalent.
It is clear that if every map from a compact to X factors through a Y , compa*
*ct
or not, with proj. dim.Y n, then every composite of n + 1 ghosts out of X is
phantom, so part (3) implies part (1). For the converse, in view of part (2), i*
*t suffices
to prove that every map from a compact object to an A with proj. dim.A n fact*
*ors
through a compact B with proj. dim.B n. we proceed by induction on n. The
base case n = 0 is implied by Proposition 2.3. For the induction step, suppose *
*we
have a map f :F -! A, where F is compact and proj. dim.A n + 1. Then we
have an exact triangle
C r-!P -s!eAt-! C
with P projective, proj. dim.C n, and A is a retract of eA. It suffices to s*
*how
that the composite
F -! A -!Ae
factors through a compact B with proj. dim.B n + 1. We can therefore assume
A = eA. By the induction hypothesis, we can write tf = OEh for some h: F -! D,
where D is compact with proj. dim.D n. This gives us a map of exact triangles
-1D ----! Z ----! F --h--! D
? ? ? ?
-1OE?y _?y f?y ?yOE
C ----!rP ----!s A ----!t C
But then Z is necessarily compact, so we can write _ = jo, where the codomain Q
of o is compact projective. By taking the weak pushout (which amounts to applyi*
*ng
10 MARK HOVEY AND KEIR LOCKRIDGE
the 3 x 3 lemma), we get the following diagram, whose rows are exact triangles.
-1D ----! Z - ---! F ---h-! D
flfl ? ? fl
fl o?y oe?y flfl
-1D ----! Q - ---! B0 ----! D
? ? ? ?
-1OE?y j?y ae?y ?yOE
C ----!r P - ---!s A ----!t C
Now B0 is a compact object with proj. dim.B0 n + 1, but unfortunately aeoe need
not be equal to f. Nevertheless, we do have
taeoe = OEh = tf,
so f -aeoe = sq for some map q :F -! P . But then q = iq0for some map q0: F -! *
*Q0,
where Q0is compact projective. Altogether then, we have
f = aeoe + siq0.
This means that f factors through the compact object B0q Q0. Indeed, f is the
composite
0) ae+si
F -(oe,q--!B0q Q0---! A.
Since proj. dim.(B0q Q0) n + 1, the proof is complete.
Corollary 2.7. Suppose E is an S-algebra or an ordinary ring. If X is a compact
object of D(E), then
flat dim.X = con. flat dim.X = proj. dim.X.
In particular, gh. dim.E is the maximal (constructible or not) flat dimension o*
*f a
compact object of D(E), or 1 if there is no such maximal dimension.
Proof.We always have flat dim.X con. flat dim.X proj. dim.X. Suppose
flat dim.X = n and X is compact. Then every composition of n+1 ghosts out of X
is phantom, hence null. Thus proj. dim.X n, so proj. dim.X = flat dim.X.
Corollary 2.8. Suppose E is an S-algebra or an ordinary ring. Then gh. dim.E =
supflat dim.X as X runs through arbitrary objects of D(E).
We do not know whether this corollary remains true for the constructible flat
dimension.
Proof.Corollary 2.7 implies that
gh. dim.E supflat dim.X.
X
Now suppose gh. dim.E = n, so that flat dim.F n for all compact objects F
of D(E). Choose an arbitrary X 2 D(E), and consider the universal coefficient
spectral sequence
E2s,t= TorE*s,t(X*, Z*) ) sst-s(X ^E Z)
for an arbitrary left E-module Z. We must show that E1s,*= 0 for s > n. Since
this spectral sequence is of homological type, this means we must show that eve*
*ry
THE GHOST AND WEAK DIMENSIONS OF RINGS AND RING SPECTRA 11
element in ss*(X ^E Z) is in filtration n (and possibly lower filtration as wel*
*l). But
the functor ss*(- ^E Z) is a homology theory on right E-modules, so
ss*(X ^E Z) = colimss*(F ^E Z),
where the colimit is taken over all maps F -! X from a compact object to X. Then
any element of ss*(X ^E Z) comes from some ss*(F ^E Z), where it lies in filtra*
*tion
n. Naturality of the spectral sequence implies that it also lies in filtration*
* n in
ss*(X ^E Z).
We now point out another advantage of the ghost dimension; it is left-right
symmetric, like the usual weak dimension of rings.
Theorem 2.9. Suppose E is an S-algebra or an ordinary ring. Then
gh. dim.E = gh. dim.Eop.
Proof.The ghost dimension of E is the largest n such that there exists a right
E-module X and a left E-module Y for which E1n,*is nonzero in the universal
coefficient spectral sequence for ss*(X ^E Y ). This is obviously left-right sy*
*mmetric.
Summing up, then, we are left with three possible definitions for the weak di-
mension of an S-algebra E. We list the basic inequalities between these definit*
*ions
in the following theorem, which also proves the main conjecture of [HL08a ] that
gh. dim.R = w. dim.R for ordinary rings R.
Theorem 2.10. Suppose E is an S-algebra or an ordinary ring. Then we have
gh. dim.E = sup con. flat dim.X = sup proj. dim.X
Xcompact Xcompact
= sup flat dim.X = sup flat dim.X.
Xcompact Xarbitrary
Furthermore,
gh. dim.E sup con. flat dim.X w. dim.E*
Xarbitrary
with equality if E is an ordinary ring. Finally,
gh. dim.E Rouq. dim.E
with equality if E* is coherent.
Proof.The only thing we have not already proved is that equality holds in the f*
*irst
chain of inequalities when R is an ordinary ring. But the main result of [HL08a*
* ] is
that w. dim.R gh. dim.R, giving us the desired equalities.
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Department of Mathematics, Wesleyan University, Middletown, CT 06459
E-mail address: hovey@member.ams.org
Department of Mathematics, Wake Forest University, Winston-Salem, NC 27109
E-mail address: lockrikh@wfu.edu