THE GHOST DIMENSION OF A RING
MARK HOVEY AND KEIR LOCKRIDGE
Abstract.We introduce the concept of the ghost dimension gh. dim.R of a
ring R. This is the longest nontrivial chain of maps in the derived cate*
*gory
emanating from a perfect complex such that each map is zero on homology.*
* We
show that gh. dim.R w. dim.R, with equality if R is coherent or w. dim*
*.R =
1.
Introduction
Given a ring R, one can study R by looking at properties of elements of R,
properties of the category R-mod of right R-modules, or properties of its unbou*
*nded
derived category D(R). If we allow ourselves to remember the object R (thus
ruling out Morita and derived equivalences), these approaches all contain the s*
*ame
information. However, there are generalized rings where not all of these approa*
*ches
are available. In algebraic topology, for example, there are structured ring sp*
*ectra
E (often called S-algebras or symmetric ring spectra). For such an E, there are*
* no
elements, and there is no abelian category of E-modules. But there is a triangu*
*lated
category D(E) analogous to D(R). If one wants to define various homological
dimensions of E, then, one needs to look at how to define homological dimensions
of R solely in terms of the derived category D(R) (with its distinguished object
R). In this paper, we describe how to recover the right global dimension of R f*
*rom
D(R) and its distinguished object R, and we introduce another invariant, the gh*
*ost
dimension of R, that we originally thought would be the weak dimension or the
finitistic right global dimension of R. However, we now believe this to be a n*
*ew
invariant of R.
In more detail, given a ring R, let D(R) denote the unbounded derived category
of right R-modules. A map f :X -! Y in D(R) is called a ghost if H*f = 0.
(Note that to define a ghost, we need to remember the object R, since H*X =
D(R)(s*R, X), so ghosts are not derived invariant). A complex X is said to have
ghost dimension n, written gh. dim.X = n, if every composite
X f1-!Y1 f2-!. .f.n+1---!Yn+1
of n + 1 ghosts is 0 in D(R), and there exists a composite of n ghosts from X
that is not 0 in D(R). Since the complex R itself has ghost dimension 0, one
might think that the ghost dimension of a complex is analogous to the projective
dimension of a module. Hence, the maximal ghost dimension should be the right
global dimension of R. This is correct, and we prove it in Theorem 1.5, followi*
*ng
the second author's thesis [Loc06, Theorem 4.4.4]. Now a finitely presented mod*
*ule
is analogous to a perfect complex in D(R), so the maximal ghost dimension of a
perfect complex should be related to the maximal projective dimension of a fini*
*tely
____________
Date: January 18, 2008.
1
2 MARK HOVEY AND KEIR LOCKRIDGE
presented module, or perhaps to the weak dimension of R (since these two invari*
*ants
are equal when R is coherent). However, we have not been able to prove this, an*
*d so
we define the ghost dimension of R, written gh. dim.R, to be the maximum ghost
dimension of any perfect complex in D(R) (or 1 if there is no such maximum).
The authors and G. Puninski proved that ghost dimension 0 is equivalent to we*
*ak
dimension 0 (i.e., von Neumann regular) in [HLP07 ], although the authors refer*
*red
to R having ghost dimension 0 as R satisfying the strong generating hypothesis,
based on analogy with the stable homotopy category in algebraic topology. In the
present paper, we prove that w. dim.R gh. dim.R, with equality holding if R is
right coherent or w. dim.R 1. We have not been able to resolve the general ca*
*se,
though we suspect there is a counterexample.
Ghost dimension in the stable module category of a finite group has been stud*
*ied
in [CCM07 ].
1.Global dimension
The object of this section is to prove that r. gl. dim.R is the maximum ghost
dimension of a complex in D(R). Along the way, we prove some basic facts about
ghost dimension that we will need later.
We first recall that the ghost maps I are part of a projective class (P, I) in
D(R) [Chr98, Section 8]. Here P is the collection of all complexes isomorphic in
D(R) to a complex of projectives with zero differential. To say that (P, I) is*
* a
projective class means three things:
(1) f :X -!Y 2 I if and only if the composite fg is 0 for all g :P -! X with
P 2 P;
(2) P 2 P if and only if fg = 0 for all f :X -!Y 2 I and all g :P -! X; and
(3) For all X 2 D(R) there is a cofiber sequence
P -! X f-!Y -! P
with P 2 P and f 2 I.
These conditions are all easy to check for the ghost projective class. For exam*
*ple,
givenLX, choose a free module Pn mapping onto HnX for all n, and let P =
nSnPn, where SnPn is the complex consisting of Pn concentrated in degree n.
There is an obvious map SnPn -!X that is onto in homology, from which it follows
that the cofiber X -!Y is a ghost.
As Christensen describes in [Chr98, Theorem 3.5], there is an induced project*
*ive
class (Pn+1, In+1), where In+1 consists of the n + 1-fold composites of ghost m*
*aps,
and so, by definition, Pn+1 is the collection of complexes of ghost dimension *
* n.
In particular, from Christensen's work we deduce the following proposition.
Proposition 1.1. Let R be a ring.
(1) A complex X 2 D(R) has gh. dim.X n if and only if there is an exact
triangle
Z -!X -!P -! sZ
in D(R), where gh. dim.Z n - 1 and gh. dim.P = 0, and s is the shift
functor.
(2) X has finite ghost dimension if and only if it is in the thick subcatego*
*ry
generated by the complexes of projective modules with zero differential.
THE GHOST DIMENSION OF A RING 3
Recall that a full subcategory C is thick if it is closed under shifts and su*
*mmands,
and if two out of three objects in an exact triangle are in C, so is the third *
*(where we
view an object and its shift as the same). Note also that this means that all p*
*erfect
complexes have finite ghost dimension, since they are in the thick subcategory
generated by the complex R.
Proof.The first statement follows from [Chr98, Theorem 3.5] and the discussion
immediately preceding it. For the second statement, note first that complexes
of projective modules with zero differential have ghost dimension 0, so [Chr98,
Note 3.6] implies that everything in the thick subcategory generated by them has
finite ghost dimension. For the converse, we proceed by induction on the ghost
dimension of X. The induction step follows from part (a). The base case is clea*
*r,
since every complex of ghost dimension 0 is a complex of projectives with zero
differential.
Corollary 1.2. Suppose X is a complex of projectives such that Xi= 0 for all but
n + 1 values of i. Then gh. dim.X n.
Proof.Use induction and part (1) of Proposition 1.1.
Christensen's work also gives us the following proposition, which is a restat*
*ement
of [Chr98, Theorem 8.3].
Proposition 1.3. Suppose r. gl. dim.R n. Then every complex X in D(R) has
gh. dim.X n.
One way to think about this proposition is in terms of the spectral sequence
Es,t2= ExtsR(H*X, H*Y )t-s) [X, Y ]s.
This is sometimes known as the hypercohomology spectral sequence in algebra, and
the universal coefficient spectral sequence in topology. It is conditionally co*
*nver-
gent, and the associated filtration on [X, Y ] is the ghost filtration. This sp*
*ectral
sequence is described in [Chr98, Section 4] for a general projective class, but*
* is
well-known in this case. In any case, if the right global dimension of R is n*
*, then
this spectral sequence has only n + 1 nonzero rows at E2, so of course also at *
*E1 ,
so the ghost dimension is at most n.
To prove the converse of this proposition, we note one simple source of ghost*
*s.
For any complex X, we can let X[m, 1] be the complex with X[m, 1]i = Xi if
i m and 0 if i < m, with the nonzero differentials coinciding with those of X.
Then there is an obvious map X -! X[m, 1] of complexes, and this map will be
a ghost if X has no homology in degrees m. In particular, we get the following
lemma.
Lemma 1.4. If X is the projective resolution of a module M, then gh. dim.X =
proj. dim.M. In particular, gh. dim.R is at least as big as the maximal finite *
*pro-
jective dimension of an F P1 -module.
Recall that an F P1 -module is a module which has a projective resolution by
finitely generated projectives. If the F P1 module M has finite projective dim*
*en-
sion, then there is a projective resolution of M that is a perfect complex, so *
*the
second statement of the lemma follows from the first.
4 MARK HOVEY AND KEIR LOCKRIDGE
Proof.Write X as the sequence
. .-.dk+1--!Xk dk-!Xk-1. .-.d1!X0.
Suppose proj. dim.M = n, so that Xk = 0 for k > n. Then Corollary 1.2 implies
gh. dim.X n. Now, the sequence of ghosts
X -!X[1, 1] -!X[2, 1] -!. .X.[n, 1]
cannot be 0 in D(R), since if it were it would be chain homotopic to 0, and so *
*null
homotopic, so that dn :Xn -!Xn-1 would be a split monomorphism. This would
mean that the kernel of dn-2 is projective, so that proj. dim.M n - 1. Hence
gh. dim.X n (even if n = 1).
This lemma together with Proposition 1.3 gives us the following theorem, due
to the second author [Loc06, Theorem 4.4.4].
Theorem 1.5. The maximum ghost dimension of an object in D(R) is r. gl. dim.R.
Hence gh. dim.R r. gl. dim.R.
2. Weak dimension
The object of this section is to show that gh. dim.R w. dim.R, with equality
holding when R is right coherent or when w. dim.R 1.
We first explain why the coherent case is easier.
Proposition 2.1. Suppose R is right coherent. Then gh. dim.R w. dim.R.
Proof.We will show that in the spectral sequence
Es,t2= ExtsR(H*X, H*Y )t-s) [X, Y ]s.
discussed after Proposition 1.3, E2 vanishes for s > w. dim.R if X is perfect.
Hence the same will be true for E1 . Since Es,*1represents the maps that are s-*
*fold
composites of ghost maps but not (s - 1)-fold such composites, we conclude that
every nonzero map is a composite of at most w. dim.R ghosts, giving us the desi*
*red
result.
First note that because R is right coherent, finitely presented right modules*
* are
closed under kernels and cokernels. Thus if X is a perfect complex, then HiX is*
* a
finitely presented R-module for all i. On the other hand, again because R is ri*
*ght
coherent, proj. dim.M is equal to the flat dimension of M for all finitely pres*
*ented
M. Indeed, because R is right coherent, there is a projective resolution
. .-.!Pk -!Pk-1 -!. .-.!P0 -!M
of M in which each Pk is finitely generated projective. If the flat dimension *
*of
M is n, then the kernel of Pn-1 -! Pn-2 is flat. But it is also finitely presen*
*ted,
again since R is right coherent. So it is projective, and so proj. dim.M n.
Hence we have proj. dim.M w. dim.R for all finitely presented M, completing
the proof.
The general case is addressed in the following theorem.
Theorem 2.2. Suppose R is a ring. Then w. dim.R gh. dim.R. In particular,
if R is right coherent, then gh. dim.R = w. dim.R.
THE GHOST DIMENSION OF A RING 5
Proof.Let n = gh. dim.R, which we can of course assume is finite. Let M be an
R-module, and let
. .d.k+1---!Pk dk-!Pk-1 dk-1---!.d.1.-!P0 d0-!M
be a free resolution of M. Let Mk+1 be the kernel of dk, so we have short exact
sequences
0 -!Mk+1 ik+1---!Pk qk-!Mk -!0,
with M0 = M and ikqk = dk. We need to show that Mn is flat, which we will do
by applying [Lam99 , Theorem 4.23] to the short exact sequence
0 -!Mn+1 -!Pn -!Mn -!0.
So take a c 2 Mn+1. To use [Lam99 , Theorem 4.23], we need to construct a
map Pn -! Mn+1 that takes c to itself. We first build a perfect subcomplex of
P = P* that contains c (or really in+1c) by a simple induction. To start, c lie*
*s in a
finitely generated summand Qn of Pn. But then d(Qn) lies in a finitely generated
summand Qn-1 of Pn-1. Continuing in this fashion, we get a perfect subcomplex
Q of P with c 2 Qn.
Now consider the sequence of ghosts
Q -!P [1, 1] -!P [2, 1] -!. .P.[n, 1],
where the first map is the composite Q j-!P -! P [1, 1]. Since gh. dim.Q n, t*
*his
composite is nullhomotopic. Thus there are maps r :Qn -!Pn+1 and s: Qn-1 -!
Pn such that dn+1r + sdn = jn, the inclusion of Qn into Pn. Consider the map
Pn o-!Qn r-!Pn+1 qn+1---!Mn+1,
where o is a splitting of the inclusion jn. Then
in+1qn+1ro(c) = dn+1rc = in+1c - sdnc = in+1c,
since c is a cycle. This means qn+1ro(c) = c, as required to apply [Lam99 , The*
*o-
rem 4.23].
We have not been able to prove that gh. dim.R = w. dim.R for a general ring R.
However, this is true if w. dim.R = 1. To prove this, we need the following sli*
*ght
generalization of the theorem of Villamayor [Lam99 , Theorem 4.23] that we have
already used in the proof of Theorem 2.2.
Lemma 2.3. Suppose we have a short exact sequence of right modules over a ring
R
0 -!K -!P -! M -!0
where P is projective. Then the following conditions are equivalent.
(1) M is flat.
(2) For every finitely generated submodule L of K, there is a map ` :P -! K
such that `(x) = x for all x 2 L.
(3) For every element c of K, there is a map ` :P -! K such that `(c) = c.
Proof.Find a Q such that P Q is free. Then we have a short exact sequence
0 -!K Q -!P Q -!M -!0.
We can now apply [Lam99 , Theorem 4.23]. More precisely, suppose M is flat, and
L is a finitely generated submodule of K. Then, by [Lam99 , Theorem 4.23], there
6 MARK HOVEY AND KEIR LOCKRIDGE
is a map f :P Q -!K Q such that f(x, 0) = (x, 0) for all x 2 L. But then we
can define ` to be the composite
P -! P Q f-!K Q -!K
to see that (1) ) (2). It is obvious that (2) ) (3). To see that (3) ) (1), take
an element (c, q) of K Q. By assumption, there is a map ` :P -! K such that
`(c) = c. Define f :P Q -!K Q by f(x, y) = (`(x), y). Then f(c, q) = (c, q).
So [Lam99 , Theorem 4.23] implies that M is flat.
We can use this improvement to characterize rings of weak dimension 1.
Proposition 2.4. A ring R has w. dim.R 1 if and only if, for every diagram
P2 d2-!P1 d1-!P0
of finitely generated projectives with d1d2 = 0, there is a map f :P1 -! P1 with
d1f = 0 and fd2 = d2.
Proof.Suppose R has weak dimension 1, and consider the short exact sequence
0 -!kerd1 -!P1 -!im d1 -!0.
Since R has weak dimension 1, im d1 P0 is flat. Apply Lemma 2.3 to the
finitely generated submodule imd2 of kerd1 to get a map f :P1 -!kerd1 such that
f(x) = x for all x 2 imd2. This means that d1f = 0 and fd2 = d2, as required.
For the converse, it suffices to show that TorR2(N, -) = 0 for all finitely p*
*resented
N. Take a finite presentation
P1 d1-!P0 -!N -!0
of N, where P1 and P0 are free and finitely generated. This gives us the short *
*exact
sequence
0 -!K -!P1 -!im d1 -!0.
We need to show that im d1 is flat. Using [Lam99 , Theorem 4.23], it suffices *
*to
show that for any c 2 K, there is a map f :P1 -! K such that f(c) = c. But c
then gives us the complex
R c-!P1 d1-!P0
of finitely generated projectives. By hypothesis, then, there is a map f :P1 -!K
such that f(c) = c.
Theorem 2.5. If w. dim.R 1, then gh. dim.R = w. dim.R.
Proof.If w. dim.R = 0, this is proved in [HLP07 , Theorem 1.3]. Thus, in view of
Theorem 2.2, we must show that w. dim.R = 1 implies gh. dim.R 1. We do this
by showing that every perfect complex is a retract of a direct sum of complexes*
* with
only two nonzero entries. In view of Corollary 1.2, this will complete the proo*
*f.
Suppose P is a perfect complex, with maps dn :Pn -!Pn-1. Since dndn+1 = 0,
Proposition 2.4 implies that there are maps fn :Pn -!kerdn such that fndn+1 =
dn+1. For each n, choose a surjection qn :Qn -! kerdn where Qn is projective,
and choose a lift gn :Pn -! Qn so that qngn = fn. Now define a two-term chain
complex T nwith Tnn= Pn and Tnn-1= Qn-1, with differential gn-1dn.
We claim that there are chain maps
M _
P -OE! T k-! P
k
THE GHOST DIMENSION OF A RING 7
L L
exhibiting P as a retract of kT k. Note that the nth term of the complex kT*
* k
is Qn Pn, with differential Dn(x, y) = (gn-1dny, 0). Define _n :Qn Pn -! Pn
by _n(x, y) = qn(x) + y. Then
dn_n(x, y) = dny
and
_n-1Dn(x, y) = _n-1(gn-1dny, 0) = qn-1gn-1dny = fn-1dny = dny,
so _ is a chain map.
Now define OEn :Pn -!Qn Pn by OEn(x) = (gn(x), x - fnx). Then
DnOEn(x) = (gn-1dn(x - fnx), 0) = (gn-1dnx, 0)
and
OEn-1dn(x) = (gn-1dnx, dnx - fn-1dnx) = (gn-1dnx, 0).
Thus OE is a chain map, and one can readily verify that
_nOEn(x) = _n(gnx, x - fnx) = qngnx + x - fnx = x,
L
showing that P is a retract of nT n.
References
[CCM07]Sunil K. Chebolu, J. Daniel Christensen, and J'an Min'a~c, Ghosts in mod*
*ular represen-
tation theory, preprint, 2007.
[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)
[HLP07]Mark Hovey, Keir Lockridge, and Gena Puninski, The generating hypothesis*
* in the
derived category of a ring, Math. Z. 256 (2007), no. 4, 789-800. MR MR23*
*08891
[Lam99]T. Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics,*
* vol. 189,
Springer-Verlag, New York, 1999. MR MR1653294 (99i:16001)
[Loc06]Keir H. Lockridge, The generating hypothesis in general stable homotopy *
*categories,
Ph.D. thesis, University of Washington, 2006.
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