TOWARD A FUNDAMENTAL GROUPOID FOR THE STABLE
HOMOTOPY CATEGORY
JACK MORAVA
Abstract.This very speculative talk suggests that a theory of fundamental
groupoids for tensor triangulated categories could be used to describe t*
*he ring
of integers as the singular fiber in a family of ring-spectra parametriz*
*ed by a
structure space for the stable homotopy category, and that Bousfield loc*
*aliza-
tion might be part of a theory of `nearby' cycles for stacks or orbifold*
*s.
x0, Introduction
One of the motivations for this talk comes from John Rognes' Galois theory for
structured ring spectra. His paper [36] ends with some very interesting remarks
about analogies between classical primes in algebraic number fields and the non-
Euclidean primes of the stable homotopy category, and I try here to develop a
language in which these analogies can be restated as the assertion that Waldhau*
*sen's
unfolding
specZ ! specS
of the integers in the category of brave new rings (or E1 ring-spectra, or com*
*mu-
tative S-algebras) leads to the existence of commutative diagrams of the form
speco__Qp___//specLMUK(n)MU
|Gal(__Qp=Qp)| |Dx|
fflffl| fflffl|
specZp ______//specLK(n)S .
The vertical arrow on the left is the Galois cover defined by the ring of integ*
*ers
in an algebraic closure of the p-adic rationals, but to make sense of the right-
hand side would require, among other things, a good theory of structure objects
(analogous to the prime ideal spectra of commutative algebra) for some general
class of tensor triangulated categories. In this direction I have mostly hopes *
*and
analogies, summarized in x3.
The main result of the first section below, however, is that local classfield t*
*heory
implies the existence of an interesting system of group homomorphisms
__ x
ae : W(Q p=Qp) ! D
which can plausibly be interpreted as the maps induced on the fundamental groups
of these hypothetical structure objects. [The group on the left is Weil's techn*
*ical
____________
Date: 6 August 2005.
1991 Mathematics Subject Classification. 11G, 19F, 57R, 81T.
The author was supported in part by the NSF.
1
2 JACK MORAVA
variant of the Galois group; its topology [30] is slightly subtler than the usu*
*al one.]
This example is local: it depends on a choice of the prime p. But Rognes [36 x1*
*2.2.1]
has global results as well; in particular, he identifies the ringspectrum S[BU]*
* as the
Hopf algebra of functions on an analog Galhotof a Galois group for MU, regarded
as an (inseparable) algebraic closure of S. This seems to be in striking agreem*
*ent
with work of Connes and Marcolli [8] and others [5, 12] on a remarkable `cosmic'
generalization of Galois theory, involving a certain motivic Galois group Galmo*
*t:
there is a very natural morphism
S[BU] ! HZ QSymm *
of Hopf algebra objects, QSymm * being a certain graded Hopf algebra of qua-
sisymmetric functions [19], defined by the inclusion of the symmetric in the qu*
*a-
sisymmetric functions, and in x2 I suggest that a quotient of this map defines a
representation
Galmot! Galhot
(at least, over Q) which conjecturally plays the role of the homomorphism induc*
*ed
on fundamental groups by a `geometric realization' construction [33], sending t*
*he
derived category of mixed Tate motives to some category of complex-oriented spe*
*c-
tra.
The paper ends with a very impressionistic (fauvist?) discussion of a possible *
*theory
of fundamental groupoids for (sufficiently small) tensor triangulated categorie*
*s,
which might be flexible enough to encompass both these examples.
I am deeply indebted to several mathematicians for conversations about this and
related material, eg Bill Dwyer and John Rognes, and in particular Matilde Mar-
colli; but none of them bear any responsibility for the many errors it undoubte*
*dly
contains.
x1, Some local Galois representations
This section summarizes some_classical local number theory. The first two subse*
*c-
tions define a system W(Q p=Qp) ! Dx of representations of certain Galois-like
groups in the units of suitable p-adic division algebras; then_x1.3_sketches a *
*con-
jecture about the structure of the absolute inertia group Gal(Q p=Qnrp), which *
*says,
roughly, that these systems are nicely compatible.
1.0 A local field is a commutative field, with a nontrivial topology in which it
is locally compact. The reals or complexes are examples, but I will be concerned
here mostly with totally disconnected cases, in particular the fields of charac*
*teristic
zero obtained as non-Archimedean completions of algebraic number fields. These
are finite extensions L of Qp, for some Euclidean prime p; the topology defines
a natural equivalence class of valuations, with the elements algebraic over Zp *
*as
(local) valuation ring.
__
The Galois group Gal(Q =Q) of an algebraic closure of the rationals acts on the*
* set
of prime ideals in the ring of algebraic integers in Q, with orbits correspondi*
*ng to
the classical primes; the corresponding isotropy groups can be identified with *
*the
TOWARD A FUNDAMENTAL GROUPOID FOR THE STABLE HOMOTOPY CATEGORY 3
__
Galois groups Gal(Q p=Qp) of the algebraic closures of the p-adic rationals. Th*
*ese
isotropy groups preserve the valuation rings, and hence act on their residue fi*
*elds,
defining (split) exact sequences
__ __ __
1 ! I(Q p=Qp) ! Gal(Q p=Qp) ! Gal(Fp=Fp) ~=^Z! 0
of profinite topological groups. The cokernel is the closure_of a dense subgrou*
*p Z
generated by the Frobenius automorphism oe : x 7! xp of Fp, and Weil observed
that in some contexts it is more natural to work with the pullback extension
__ __
1 ! I(Q p=Qp) ! W(Q p=Qp) ! Z ! 0 ,
the so-called Weil group, which is now only locally compact. [These groups are
defined much more generally [40] but we won't need that here.]
1.1.1 The work of Lubin and Tate defines very interesting representations of th*
*ese
groups as automorphisms of certain one-dimensional formal groups. Honda's loga-
rithm X i
logq(T ) = p-iT q
i 0
(with q = pn) defines a formal group law
Fq(X, Y ) = log-1q(logq(X) + logq(Y )) 2 Zp[[X, Y ]]
(it's not obvious that its coefficients are integral!) whose endomorphism ring *
*con-
tains, besides the elements
T 7! [a](T ) = log-1q(a logq(T ))
defined by multiplication in the formal group by a p-adic integer a, the endomo*
*r-
phism
T 7! [!](T ) = log-1q(! logq(T )) = !T
defined by multiplication by a (q - 1)st root of unity !. In fact the full rin*
*g of
endomorphisms of Fq can be identified with the Witt ring W (Fq), which can also
be described as the algebraic integers in the unramified extension field Qq obt*
*ained
from Qp by adjoining !.
*
*__
1.1.2 By reducing the coefficients of Fq modulo p we obtain a formal group law *
*Fq
over Fp, which admits
T 7! F (T ) = T p
as a further endomorphism. Honda's group law is a Lubin-Tate group [37] for the
field Qq, and it can be shown that
[p](T ) T qmod p ;
__
in other words, F n= p in the ring of endomorphisms of F q. From this it is not
hard to see that __
End _Fp(F q) = W (Fq)=(F n- p)
(the pointed brackets indicating a ring of noncommutative indeterminates, subje*
*ct
to the relation
aoeF = F a
for a 2 W (Fq), with oe 2 Gal(Fq=Fp) ~=Z=nZ being the (other!) Frobenius endo-
morphism). This is the ring oD of integers in the division algebra
D = Qq=(F n- p)
4 JACK MORAVA
with_center Qp; its group oxDof strict units is the full group of automorphisms*
* of
Fq, and it is convenient to think of
1 ! oxD! Dx ! Z ! 0
as a semidirect product, with quotient generated by F . Thus conjugation by F a*
*cts
on W (Fq)x oxDas oe 2 Gal(Fq=Fp): the Galois action is encoded in the division
algebra structure.
1.2.1 For each n 1, a deep theorem of Weil and Shafarevich defines a homomor-
phism
wsq : W(Qabq=Qp) ~=W (Fq)x n Z ! oxDn Z = Dx
of locally compact topological groups. It extends the local version of Artin's *
*reci-
procity law, which defines an isomorphism
Lx ~=W(Lab=L)
for any totally disconnected local field L (with Lab its maximal abelian extens*
*ion).
When L = Qq is unramified, the sequence
1 ! W(Qabq=Qq) ! W(Qabq=Qp) ! W(Qq=Qp) ! 0
is just the product
0 ! W (Fq)x x Z ! W (Fq)x n Z ! Z=nZ ! 0
of the elementary exact sequence
0 ! Z ! Z ! Z=nZ ! 0
with a copy of the units in W (Fq): in the semidirect product extension above, *
*the
generator 1 2 Z acts on W (Fq) by oe, so its nth power acts trivially.
1.2.2 More generally, any Galois extension L of degree n over Qp can be embedded
in a division algebra of rank n with center Qp, as a maximal commutative subfie*
*ld.
When the division algebra D has invariant 1=n in the Brauer group Q=Z of Qp (as
it does in our case: this invariant equals the class, modulo Z, of the p-order *
*of an
element (e.g. F ) generating the maximal ideal of oD ), the theorem [44, append*
*ix]
of Weil and Shafarevich defines an isomorphism of the normalizer of Lx in Dx wi*
*th
the Weil group of Lab over Qp.
[The maximal abelian extension Labis obtained by adjoining the p-torsion points*
* of
the Lubin-Tate group of L to the maximal unramified extension Lnr; the resulting
field acquires an action of the group oxLof units of L (by `complex multiplicat*
*ion'
on the Lubin-Tate group) together with an action of the automorphisms ^Zof the
algebraic closure of the residue field.
The point is that the Lubin-Tate group of L is natural in the etale topology: t*
*o be
precise, any two Lubin-Tate groups for L become isomorphic_over the completion
Lnr of the maximal unramified extension Lnr = L W(k) W (k) of L [38 x3.7].
Since an automorphism of L over Qp takes one Lubin-Tate group to another, the
resulting group of automorphisms of `the' Lubin-Tate group of L (as a completed
Hopf algebra over Zp) is an extension of Gal(L=Qp) by oxLx ^Z, i.e. the profini*
*te
completion of Lx . This extension is classified by an element of
H2(Gal(L=Qp), Lx ) ~=Z=nZ ,
TOWARD A FUNDAMENTAL GROUPOID FOR THE STABLE HOMOTOPY CATEGORY 5
which also classifies those algebras, simple with center Qp, which split after *
*ten-
soring with L. The extension in question generates this group, by a fundamental
result of local classfield theory; but the division algebra with invariant 1=n *
*also
generates this group, and the associated group extension is the normalizer of L*
*x in
Dx , QED.]
1.2.3 The remarks in this subsession are a digression, but they will be useful *
*in
x3.3: Since the normalizer of Lx acts as generalized automorphisms of a Lubin-T*
*ate
group for L, then for every g 2 W(Lab=Qp) there is a power series
[g](T ) 2 oLnr[[T ]]
satisfying [g0]([g1](T )) = [g0g1](T ), compatible with the natural action of t*
*he Weil
group W(Lab=Qp) on the ring of integers oLnr in Lnr. It follows that [g](t) =
ff(g)T + . .d.efines a crossed homomorphism
ff : W(Lab=Qp) ! (Lnr)x ,
ie a map satisfying ff(g0g1) = ff(g0) . ff(g1)g0, the superscript denoting the *
*action of
the Weil group on Lnr through its quotient W(Lnr=Qp). When L = Qq this implies
the existence of an extension of the identity homomorphism from Qxqto itself, to
a crossed homomorphism from Qxqn Z to (Lnr)x . A corollary is the existence of a
representation Qnrp(1) of W(Qabq=Qp) on the completion Qnrpextending the action*
* of
W (Fq)x by multiplication. [The completions in this construction are cumbersome,
and might be unnecessary. Experts may know how to do without them, but I don't.]
1.3 The representations promised in the introduction are the compositions
__ ab x
aeq : W(Q p=Qp) ! W(Qq =Qp) ! D
with the second arrow coming from the Weil-Shafarevich theorem.
In the remainder of this section I will sketch a conjecture about the relations*
* be-
tween these representations. The argument goes back to Serre's Cohomologie Ga-
loisienne, and I believe that many people have thought along the lines below, b*
*ut
I don't know of any place in the literature where this is spelled out. It is b*
*ased
on a p-adic analog of a conjecture of Deligne, related to an older conjecture of
Shafarevich [33; cf. also 14].
1.3.1 We need some basic facts:
i) in order that a pro-p-group be free, it is necessary and sufficient that its*
* coho-
mological dimension be 1 [37 I x4.2, corollary 2], and
ii) the maximal unramified extension of a local field with perfect residue fiel*
*d has
cohomological dimension 1 [II x3.3 ex c].
I will follow Serre's notation, which is very similar to that used above. K is*
* a
field complete with respect to a discrete valuation, with residue field k, e.g.*
* a
finite extension of Qp. Knr will denote its maximal unramified extension, Ks its
(separable) algebraic closure, and Ktr will be the union of the tamely ramified
Galois extensions (i.e. with Galois group of order prime to p) of K in Ks; thus
Ks Ktr Knr K .
6 JACK MORAVA
Eventually K will be the quotient field Qq of the ring W (k) of Witt vectors for
some k = Fq with q = pf elements, but for the moment we can be more general.
We have Y
Gal(Ktr=Knr) ~=lim-Fxpn~= Zi= ^Z(1)(:p)
l6=p
__
[II x4 ex 2a]; this is a module over Gal(k=k) ~=^Z, in which 1 2 Z acts as mult*
*ipli-
cation by q = #(k) [II x5.6 ex 1]. The kernel in the extension
1 ! Gal(Ks=Ktr) = P ! Gal(Ks=Knr) ! Gal(Ktr=Knr) ! 1
is a pro-p-group (ibid 2b), closed in a group of cohomological dimension one (by
assertion (ii)), hence itself of cohomological dimension one [I x3.3 propositio*
*n 14],
hence free by assertion (i).
__
The extension in question is the inertia group of K; it has a natural Gal(k=k)-
action.
1.3.2 From now on I will assume that K = Qq is unramified over Qp.
Conjecture I, This extension splits: Gal(Ks=Knr) ~=P o ^Z(1)(:p) is a semidirect
product.
[Since the kernel and quotient have relatively prime order, this would be obvio*
*us if
either were finite. This may be known to the experts.]
Let W (k)x0= (1 + pW (k))x be the group of those units of W (k) congruent to 1
mod p: the logarithm defines an exact sequence
1 ! kx ! W (k)x ! W (k) ! k ! 0 ,
so taking W (k)x0isomorphically to pW (k). Let
__x 0 x 0 __
W(k)0 = lim-{W (k )0 | k finite k}
__x
be the limit under norms; it is a compactification of W (k)0 .
Conjecture_II: The topological abelianization of P is naturally isomorphic to
W(k)x0.
Alternately: Lazard's group ring
__
Zp[[P ]] ~=lim-{T^(W (k0)x0) | k0finite k}
of P of is isomorphic to the limit (under norms) of the system of completed ten*
*sor
algebras of pW (k0) ~=W (k0)x0. It is thus (hypothetically) a kind of noncommut*
*ative
Iwasawa algebra [I x1.5 proposition 7 (p. 8)].
If both these conjectures are true, then I can think of no natural way for ^Z(1*
*)(:p)
to act on P , so I will go the rest of the way and conjecture as well that this*
* action
is trivial. I will also abbreviate the sum of these conjectures as the assertio*
*n that
the absolute inertia group I_of_Qp is the_product of ^Z(1)(:p) with the pro-free
pro-p-group generated by W (k)x0~=pW (k).
TOWARD A FUNDAMENTAL GROUPOID FOR THE STABLE HOMOTOPY CATEGORY 7
1.3.3 These conjectures seem to be compatible with other known facts of classfi*
*eld
theory. In particular, they would imply that
__
Gal(Q p=Qq) ~=I n ^Z
with 1 2 Z acting as oe(x) = xq on k. This abelianizes to W (k)x x ^Z, agreeing*
* with
Artin's reciprocity law, and if q1 = qm0this would yield an exact sequence
__ __
1 ! Gal(Q p=Qq1) ~=I n ^Z! Gal(Q p=Qq0) ~=I n ^Z! Gal(Fq1=Fq0) ~=Z=mZ ! 0 .
Finally, the composition
__x x
I ! Iab ~=W(k)0 x ^Z(1)(:p) ! W (Fq)
defines a compatible system of candidates for the quotient maps
__ x ab
aeq : Gal(Q p=Qp) ~=I n ^Z! W (Fq) n ^Z~=Gal(Qq =Qp) .
x2, Some more global representations
This section is concerned with two very global objects, each of which has some
unfamiliar features. The first subsection is concerned with the suspension spec*
*trum
S[BU] = 1 (BU+ )
of the classifying space for the infinite unitary group, and its interpretation*
* (follow-
ing [36]) as a Hopf algebra object in the category of spectra. The second subse*
*ction
reviews some properties of the Hopf algebra of quasisymmetric functions, follow*
*ing
[5, 23]. A remarkable number of the properties of the Hopf algebra of symmetric
functions generalize to this context [20], and I will try to keep this fact in *
*focus.
This Hopf algebra is conjectured to be closely related to a certain motivic gro*
*up
of interest in arithmetic geometry, and I have tried to say a little about that*
*, in
particular because it seems to overlap with recent work of Connes and Marcolli,
discussed in x2.3.
2.1.1 Since BU is an infinite loopspace, S[BU] becomes an E1 ringspectrum (or,
in an alternate language, a commutative S-algebra); but BU is also a space, with
a diagonal map, and this structure can be used to make this ringspectrum into a
Hopf algebra in the category of S-modules.
In complex cobordism, the complete Chern class [1]
X
ct= cI tI = OE-1stOE(1) 2 MU*(BU+ ) S*
(where S* = Z[ti| i 1] is the Landweber-Novikov algebra, with coaction st) re*
*p-
resents a morphism S[BU] ! MU ^MU of ringspectra. Indeed, algebra morphisms
from S[BU] to the Thom spectrum MU correspond to maps
CP+1 = MU(1) ! MU
of spectra, and hence to elements of MU*(CP+1). The coaction
MU ~=S ^ MU ! MU ^ MU
is also ring map, so the resulting composition
S[BU] ! MU = S ^ MU ! MU ^ MU
8 JACK MORAVA
is a morphism of ring spectra. On the other hand the Thom isomorphism
OE : MU*(BU+ ) ~=MU*(MU)
satisfies Y
OE-1stOE(1) = ct= e-1it(ei)
P i
(where t(e) = tkek+1 2 MU*(CP+1) S*, with e being the Euler, or first Chern,
class).
In fact this map is also a morphism of Hopf algebra objects:
st0( ct) = ct0Ot ct0Ot2 MU*(BU+ ^ BU+ ) (S* S*) ,
so the diagram
S[BU] _______//S[BU] ^ S[BU]_____//_(MU ^ MU) ^ (MU ^ MU)
| |
| |
fflffl| |fflffl
MU ^ MU ________________________//(MU ^ MU) ^MU (MU ^ MU)
commutes.
We can thus think of S[BU] as a kind of Galois group for the category of MU-
algebras over S, or alternately for the category of complex-oriented multiplica*
*tive
cohomology theories. In particular, if E is an MU-algebra, algebra maps from
S[BU] to E define elements of E0(CP+1); thus the group Aut(E) of multiplicative
automorphisms of E maps to algebra homomorphisms from S[BU] to E. The ad-
joint construction thus sends S[BU] to the spectrum of maps from Aut(E) (regard*
*ed
naively, as a set) to E; but these maps can be regarded as E*-valued functions *
*on
Aut(E), and hence as elements of the coalgebra E*E.
Note that S[BU] is large, so the existence of an honest dual object in the cate-
gory of spectra may be problematic. The remark above implies that this perhaps
nonexistent group object admits the etale groupschemes oxDof x1 as subgroups.
2.1.2 The Landweber-Novikov algebra represents the group of invertible power
series under composition, and it may be useful below to know that (since all on*
*e-
dimensional formal groups over the rationals are equivalent) (H*(MU, Q), H*(MU ^
MU, Q)) represents the transformation groupoid defined by this group acting on
itself by translation. On the other hand H*(BU, Q) represents the group of for-
mal power series with leading coefficient 1, under multiplication, and the indu*
*ced
morphism
spec(H*(MU, Q), H*(MU ^ MU, Q)) ! spec(H*(S, Q), H*(BU, Q))
of groupoidschemes sends the pair (g, h) of invertible series, viewed as a morp*
*hism
from h to gOh, to the translated derivative g0(h(t)). This is essentially the c*
*onstruc-
tion which assigns to a formal group, its canonical invariant differential. The*
* chain
rule ensures that this is a homomorphism: we have (g, h) O (k, g O h) = (k O g,*
* h),
while
g0(h(t)) . k0((g O h)(t)) = (k O g)0(h(t)) .
2.2.1 The Z-algebra of symmetric functions manifests itself in topology as the
integral homology of BU; it is a commutative and cocommutative Hopf algebra
TOWARD A FUNDAMENTAL GROUPOID FOR THE STABLE HOMOTOPY CATEGORY 9
(with a canonical nondegenerate inner product [20] which, from the topological
point of view, looks quite mysterious). Over Q it is the universal enveloping a*
*lgebra
of an abelian graded Lie algebra with one generator in each even degree.
The graded ring QSymm *of quasi symmetric functions is the commutative Hopf
algebra dual to the free associative algebra on (noncommutative!) generators Zk*
* of
degree 2k, with coproduct
X
Zk = Zi Zk ;
i+j=k
it is thus the universal enveloping algebra for a free graded Lie algebra fZ, w*
*ith one
generator in each (even) degree [19]. There is a natural monomorphism embedding
the symmetric functions in the quasisymmetric functions [5 x2.4], dual to the m*
*ap
on enveloping algebras defined by the homomorphism from the free Lie algebra to
its abelianization. This then defines a morphism
S[BU] ! HZ QSymm *
of (Hopf) ringspectra.
The group-valued functors represented by such Hopf algebras are very interestin*
*g,
and have a large literature. In what follows I will simplify by tensoring every*
*thing
with Q, which will be general enough for anything I have to say. The formal
(Magnus) completion of the Q-algebra of noncommuting power series is dual to the
algebra of functions on a free prounipotent [10 x9] groupscheme F. In this cont*
*ext,
a grading on a Lie algebra can be reinterpreted as an action of the multiplicat*
*ive
groupscheme Gm , which sends an element x of degree d to ~dx, where ~ is a unit
in whatever ring we're working with, so we can describe the map above as defini*
*ng
a morphism
F n Gm ! Galhot
of group objects of some sort, over Q.
2.2.2 In arithmetic geometry there is currently great interest in a groupscheme
Galmot= Foddn Gm
which is conjectured to be (isomorphic to) the motivic Galois group of a certain
Tannakian category, that of mixed Tate motives over Z [10, 12]. Usually Fodd is
taken to be the prounipotent group defined by the free graded Q-Lie algebra fod*
*don
generators of degree 4k + 2 (hence `odd' according to the algebraists' conventi*
*ons),
with k 1; but there are reasons to allow k = 0 as well. We can regard Fodd as*
* a
subgroupscheme of F, by regarding fodd as a subalgebra of f.
A Tannakian category is, roughly, a suitably small k-linear abelian category wi*
*th
tensor product and duality - such as the category of finite-dimensional linear *
*rep-
resentations of a proalgebraic group over a field k. Indeed, the main theorem o*
*f the
subject [11] asserts that (when k has characteristic zero) any Tannakian catego*
*ry
is of this form; then the relevant group is called the motivic group of the cat*
*egory.
Present technology extracts mixed Tate motives from a certain triangulated cate-
gory of (pieces of) algebraic varieties, constructed as a subcategory of the te*
*nsor
triangulated category of more general motives [12, 43].
10 JACK MORAVA
There are more details in x3 below, but one of the points of this paper is that*
* the
language of such tensor categories can be quite useful in more general circumst*
*ances.
For example, the category of complex-oriented multiplicative cohomology theories
behaves very much like (a derived category of) representations, with S[BU] as i*
*ts
motivic group. Away from the prime two, the fibration
SO=SU+ ! BSU+ ! BSO+
(defined by the forgetful map from C to R) splits (even as maps of infinite loo*
*pspaces).
Over the rationals, this is almost trivial: it corresponds to the splitting of *
*the graded
abelian Lie algebra with one generator in each even degree, into a sum of two s*
*uch
Lie algebras, with generators concentrated in degrees congruent to 0 and 2 mod
4 respectively. The composition (the first arrow is the projection of the H-spa*
*ce
splitting, and the second corresponds to the abelianization of a graded free Lie
algebra).
S[BU] ! S[SO=SU] ^ S[CP 1] ! HQ QSymm odd
is the candidate, promised in the introduction, for a natural representation
Galmot! Galhot
(over Q, of course!).
2.2.3 The degree two (= 2(2 . 0 + 1)) generator in Fodd is closely related to t*
*he
S[CP 1] factor in the ring decomposition above; both correspond to exceptional
cases that deserve some explanation.
The symmetric functions are defined formally as an inverse limit of rings of sy*
*m-
metric polynomials in finitely many variables xn, n 1. There are thus in-
teresting maps from the symmetric functions to other rings, defined by assigning
interesting values to the xn; but because infinitely many variables are involve*
*d,
issues of convergence can arise. For example: if we send xn to 1=n, the nth pow*
*er
sum pn maps to i(n) 2 R . . . as long as n > 1, for s = 1 is a pole of i(s).
This is a delicate matter [5 x2.7, 23], and it turns out to be very natural to *
*send p1 to
Euler's constant fl. In fact this homomorphism extends, to define a homomorphism
from QSymm to R, whose image is sometimes called the ring MZN * of `multizeta
numbers'; it has a natural grading. It is classical that for any positive integ*
*er n,
i(2n) = -1_2B2n fl2n(2ssi) 2 Q(ss) ,
(where flk denotes the kth divided power) and in some contexts it is natural to*
* work
with the even-odd graded subring of C obtained by adjoining an invertible eleme*
*nt
(2ssi) to MZN *. It is known [18 x4] that the multizeta numbers are periods
of algebraic integrals, and that the ring generated by all such periods is a Ho*
*pf
algebra, closely related to the algebra of functions on the (strictly speaking,*
* still
hypothetical) motivic group of all motives over Q [28]; but it is thought that *
*Euler's
constant is probably not a period. Nevertheless, from the homotopy-theoretic po*
*int
of view presented here, it appears quite naturally.
2.3.1 These multizeta numbers may play some universal role in the general theory
of asymptotic expansions; in any case, they appear systematically in renormaliz*
*a-
tion theory. Connes and Marcolli, building on earlier work of Connes, Kreimer
TOWARD A FUNDAMENTAL GROUPOID FOR THE STABLE HOMOTOPY CATEGORY 11
[7], Broadhurst, and others, have put this in a Galois-theoretic framework. Th*
*is
subsection summarizes some of their work.
Classical techniques [of Bogoliubov, Parasiuk, Hepp, and Zimmerman] in physics
have achieved an impressive level of internal consistency, but to mathematicians
they lack conceptual coherence. Starting with a suitable Lagrangian density, Co*
*nnes
and Kreimer define a graded Lie algebra g* generated by a class of Feynman grap*
*hs
naturally associated to the interactions encoded by the Lagrangian. They interp*
*ret
the BPHZ dimensional regularization procedure as a Birkhoff decomposition for
loops in the associated prounipotent Lie group G, and construct a universal rep-
resentation of this group in the formal automorphisms of the line at the origin,
yielding a formula for a reparametrized coupling constant which eliminates the
divergences in the (perturbative) theory defined by the original Lagrangian.
Dimensional regularization involves regarding the number d of space-time dimen-
sions as a special value of a complex parameter; divergences are interpreted as
poles at its physically significant value. Renormalization is thus expressed in*
* terms
much like the extraction of a residue, involving a simple closed curve encircli*
*ng the
relevant value of d (the source of the loop in G mentioned above). Connes and
Marcolli [8] reformulate such data in geometric terms, involving flat connectio*
*ns on
a Gm -equivariant G-bundle over a certain `metaphysical' (not their terminology*
*!)
base space B. From this they define a Tannakian category (of flat, equisingular
vector bundles with connection over B), and identify its motivic group as F n G*
*m .
Their constructions define a representation of the motivic group in G, and henc*
*e in
the group of reparametrizations of the coupling constant.
2.3.2 Besides the complex deformation of the space-time dimension, the base spa*
*ce
B encodes information about the mass scale. It is a (trivial, but not naturally
trivialized, cf end of x2.13) principle bundle
Gm ! B !
over a complex disk centered around the physical dimension d 2 C. [In this
disk is treated as infinitesimal but there may be some use in thinking of it as*
* the
complement of infinity in CP1.] The renormalization group equations [x2.9] are
reformulated in terms of the (mass-rescaling) Gm -action on the principal bundle
G x B (the grading on g* endows G with a natural Gm -action), leading to the
existence of a unique gauge-equivalence class of flat Gm -equivariant connection
forms ~ 2 1(B, g), which are equisingular in the sense that their restrictions*
* to
sections of B (regarded as a principal bundle over ) which agree at 0 2 are
mutually (gauge) equivalent.
A key result [8, Theorem 2.25, x2.13] characterizes such forms ~ in terms of a *
*graded
element fi* 2 g* corresponding to the beta-function of renormalization group th*
*eory.
In local coordinates (z 2 near the basepoint, u 2 Gm ) we can write
~(z, u) = ~0(z, u) . dz + ~1(z, u) . u-1du
with coefficients ~i 2 C{u, z}[z-1] allowed singularities at z = 0; but flatness
and equivariance imply that these coefficients determine each other. The former
condition [2.166] can be stated as
@z~1 = H~0 - [~0, ~1]
12 JACK MORAVA
where the grading operator
H = (u@u)|u=1
is the infinitesimal generator of the Gm -action; it sends uk to kuk. Regularit*
*y of
~0 at u = 0 implies that
~0 = H-1[@z~1 + [H-1@z~1, ~1] + . .].
is determined, at least formally, by knowledge of ~1 in the fiber direction. Co*
*nnes
and Marcolli's solution [Theorems 2.15, 2.18, eq. 2.173] of the renormalization
group equations imply that
~1(u, z) = -z-1u*(fi)
for some unique fi 2 g. This characterizes a universal (formal) flat equisingu*
*lar
connection ~(fi) on B, related to the universal singular frame of [x2.14].
2.3.3 The Lie algebra V of the group of formal diffeomorphisms of the line at t*
*he ori-
gin has canonical generators vk = uk+1@u satisfying [vk, vl] = (l - k)vk+1, k, *
*l 1,
so a graded module with an action of such operators defines a flat equisingular*
* vec-
tor bundle over B with fi* = v*. A commutative ringspectrum E with S[BU]-action
defines a multiplicative complex-oriented cohomology theory, and in particular *
*pos-
sesses an MU-module structure. On rationalized homotopy groups the morphism
E ! E ^ S ! E ^ MU
defines an S*-comodule structure map (cf. x2.1.2 above)
E*Q! E*Q MU*QMU*MUQ = E*Q S*
and thus an action of the group of formal diffeomorphisms; differentiating this*
* ac-
tion assigns E*Qan action of V. The smash product of two complex-oriented spect*
*ra
is another such thing, and the functor from S[BU]-representations to V represen*
*ta-
tions takes this product to the usual tensor product of Lie algebra representat*
*ions.
This, together with the universal connection constructed above, defines a monoi*
*dal
functor from S[BU]-representations to the Tannakian category of flat equisingul*
*ar
vector bundles over B of [x2.16], and thus a (rational) representation of its m*
*otivic
group in S[BU].
Connes and Marcolli note that their motivic group is isomorphic to the motivic
group for mixed Tate motives over the Gaussian integers (which has generators in
all degrees, unlike that for mixed Tate motives over the rationals), but they d*
*o not
try to make this isomorphism canonical. It is striking to me that their formul*
*as
(cf eq. 2.137) actually take values in the group of odd formal diffeomorphisms.
Presumably Gal(Q(i)=Q) acts naturally on the motivic group of mixed Tate mo-
tives over the Gaussian integers, and it seems conceivable that these construct*
*ions
might actually yield a representation of Foddn Gm in the group of odd formal
diffeomorphisms [33]; but I have no hard evidence for this.
x3, Towards fundamental groupoids of tensor triangulated categories
3.0 A Tannakian category A is a k-linear abelian category, where k is a field,
possessing a coherently associative and commutative tensor product [11 x2.5];
moreover, A should be small enough: its objects should be of finite length, its
Hom-objects should be finite-dimensional over k, the endomorphism ring of the
TOWARD A FUNDAMENTAL GROUPOID FOR THE STABLE HOMOTOPY CATEGORY 13
identity object for the tensor product should be k, and it should admit a good
internal duality [x2.12]. The specifically Tannakian data, however, consists o*
*f a
nontrivial exact k-linear functor
! : A ! (k - Vect)
which is monoidal in the sense that
!(X Y ) ~=!(X) k !(Y ) .
Let Aut!(k) be the group of multiplicative automorphisms of !, ie of natural tr*
*ans-
formations of k-module-valued functors from ! to itself, which commute with the*
*se
multiplicativity isomorphisms. More generally, the multiplicative automorphisms
A 7! Aut!(A) = Aut(! k A)
of ! kA define a group-valued functor on the category of commutative k-algebra*
*s.
It sometimes happens (eg when k is of characteristic zero) that this functor is
representable by a suitable Hopf algebra H, ie
Aut (! k A) ~=Hom k-alg(H, A) ;
then ! lifts to a functor from A to the category of finite-dimensional represen*
*tations
of the affine (pro)algebraic group represented by H, and it may be that we can *
*use
this lift to identify A with such a category of representations [x7]. In any c*
*ase,
when the k-groupscheme
specH = ss1(specA, !)
exists, it is natural to think of it as a (`motivic') kind of fundamental group*
* for the
Tannakian category, with the `fiber functor' ! playing the role of basepoint.
3.1 Here are some illustrative examples, and variations on this theme:
3.1.1 The category of local systems (ie of finite-dimensional flat k-vector spa*
*ces)
over a connected, suitably locally connected topological space X is Tannakian: a
basepoint x 2 X defines an exact functor, which sends the system to its fiber o*
*ver
x, and we recover the fundamental group ss1(X, x) (or, more precisely, its `env*
*elope'
or best approximation by a proalgebraic group over k) as its automorphism object
[41].
3.1.2 The functor which assigns to a finite-dimensional Q-vector space V_,_the *
*p-adic
vector space V Q Qp, defines a Tannakian structure on Q-Vect, with Gal(Q p=Qp),
regarded as a profinite groupscheme over Qp, as its motivic fundamental group.
3.1.3 One possible variation involves fiber functors taking values in categorie*
*s more
general than vector spaces over a field. The monoidal category of even-odd grad*
*ed
(or `super') vector spaces is an important example. When this works, one gets
a Hopf algebra object (corresponding to a `super' groupscheme) in the enriched
category; the motivic group in this case is the multiplicative group ~2 of squa*
*re
roots of unity (suitably interpreted, [11 x8.19]). Perhaps in homotopy theory t*
*his
has something to do with the fact that the first stable homotopy group of the s*
*phere
spectrum is Z=2Z.
14 JACK MORAVA
3.1.4 The automorphisms of cohomology with Fp coefficients, which is defined not
on an abelian category but on a tensor triangulated one, eg finite spectra, is a
more exotic and perhaps more compelling example: the even-degree cohomology is
a representation of the groupscheme defined by the functor
X k
A 7! { akT p = a(T ) 2 A[[T ]] | a0 2 Ax } ;
k 0
its Hopf algebra is dual to the algebra defined by Steenrod's reduced pth power*
*s.
The group operation here is composition of power series. The Hopf algebra dual
to the full Steenrod algebra can be recovered by working with ~2-graded (super)
vector spaces. Note that the groupscheme defined above contains the `torus'
A 7! Ax = Hom k-alg(k[t01], A)
as the subgroup of series of the form a0T, a0 2 Ax ; this action by the multipl*
*ica-
tive groupscheme Gm allows us to recover the (even part of) the grading on the
cohomology in `intrinsic' terms. However, we're not as lucky here as in the pre*
*ced-
ing case: the stable homotopy category (at p) is not the same as the category of
modules over the Steenrod algebra. The Adams spectral sequence tells us that we
have to take higher extensions into consideration.
3.1.5 These motivic fundamental groups are in a natural sense functorial [11 x8*
*.15]:
Given a commutative diagram
j
A0 ___________//_A1
!0|| |!1|
fflffl|"j fflffl|
(k0 - Vect)____//(k1 - Vect)
of Tannakian categories Ai, fiber functors !i, and fields k0 ! k1, with j exact
k0-linear multiplicative and "j= - k0k1, there is a natural homomorphism
j* : ss1(specA1, !1) ! ss1(specA0, !0) xk0k1
of groupschemes over k1 constructed as follows: if ff : !1 ~=!1 is a multiplica*
*tive
automorphism of !1 (perhaps after some base extension which I won't record), th*
*en
ff O j : "jO !0 = !1 O j ~=!1 O j = "jO !0
is an element of ss1(specA0, !0), pulled back by "jto become a groupscheme over
k1.
As x3.1.4 suggests, it is tempting to push these constructions in various direc*
*tions;
in particular, if T is a suitably small tensor triangulated category (ie a tria*
*ngulated
category with tensor product satisfying reasonable axioms (for example that ten-
soring with a suitable object represents suspension)), and ! is a multiplicat*
*ive
homological functor (taking distinguished triangles to long exact sequences) wi*
*th
values in k-vector spaces, we can consider its multiplicative automorphisms, as*
* in
the Tannakian case; and we might hope that if T is some kind of derived category
of A, then we could try to reconstruct T in terms of [13, 34] a derived category
of representations of the automorphism group of the homological fiber functor !.
[If we ask that the automorphisms behave reasonably under suspension, then the
automorphism group will contain a torus encoding gradings, as in x3.1.4.]
TOWARD A FUNDAMENTAL GROUPOID FOR THE STABLE HOMOTOPY CATEGORY 15
3.2 What seems to be missing from this picture is a compatible understanding of
ss0. If p is a prime ideal in a commutative noetherian ring A then A=p is a dom*
*ain
with quotient field Q(A=p), and the composite
!p : A ! A=p ! Q(A=p) = k(p)
determines p, so the prime ideal spectrum specA is a set of equivalence classes*
* of
homomorphisms from A to (varying) fields k. Such a homomorphism lifts to define*
* a
triangulated functor LAk(p) from the derived category of A-modules to the deri*
*ved
category of vector spaces over k(p), taking (derived) tensor products to (deriv*
*ed)
tensor products, and (as in the case of ordinary cohomology above) we can consi*
*der
the functor of automorphisms associated to such a generalized point. G-equivari*
*ant
stable homotopy theory is another example of a category with a plentiful supply*
* of
natural `points', corresponding to conjugacy classes of closed subgoups of G.
It would be very useful if we could construct, for suitable T, a fundamental gr*
*oupoid
ss(specT) with objects corresponding to equivalence classes of multiplicative h*
*omo-
logical fiber functors !, and morphisms coming from ss1(specT, !). The lattice *
*of
Bousfield localizations [24] of a triangulated category is in some ways analogo*
*us to
the Boolean algebra of subsets of the spectrum of an abelian category [3, 21, 3*
*5],
but we do not seem to understand, in any generality, how to identify in it a su*
*blat-
tice corresponding to the open sets of a reasonable topology. Moreover, there s*
*eem
to be subtle finiteness issues surrounding this question: constructing a good s*
*s1 at
! may require identifying a suitable subcategory of !-finite or coherent object*
*s [26
x8.6], and we might hope that a reasonable topology on specT would suggest a
natural theory of adeles [27] associated to chains of specializations.
3.3 This is all quite vague; perhaps some examples will be helpful.
The structure map S ! HZ defines a monoidal pullback functor
- ^ HZ : (Spectra)! D(Z - Mod ) ;
simply regarding a Z-algebra as an S-algebra is not monoidal: H(A B) is not t*
*he
same as HA ^ HB.
3.3.1 Let ^K*p(-) = K*(-) Zp denote classical complex K-theory, regarded as a
functor on finite spectra and p-adically completed. The Adams operations, suita*
*bly
normalized, define an actiion of Zxpby stable multiplicative automorphisms [39],
and the Chern character isomorphism
^K*p(-) Q ! H*(-, Qp)
identifies its eigenspaces with the graded components defined by ordinary cohom*
*ol-
ogy.
This can be_expressed in the language developed above, as follows: the fiber fu*
*nctor
!0 = K^p Qp on finite spectra_is ordinary cohomology in disguise, and so has
automorphism group Gm n Gal(Q p=Qp), as does
__ __
!1 : M 7! H+ (M Qp) : D(Z - Mod ) ! (Q p- Vect) .
The induced morphism
__ __
Gm n Gal(Q p=Qp) ! Gm n Gal(Q p=Qp)
16 JACK MORAVA
__
of motivic fundamental groups sends (u, g) 2 Gm n Gal(Q p=Qp) to (gabu, g), whe*
*re
__ __ x
g 7! gab : Gal(Q p=Qp) ! Gal(Q p=Qp)ab ! Zp
is the p-adic cyclotomic character; in Hopf algebra terms, this corresponds to *
*the
homomorphism
u 7! 1 - t : Qp[u 1] ! Zp[[t]][~p-1] Q ~=Zp[[Zxp]] Q
of Iwasawa theory.
3.3.2 This has a conjectural generalization to the spectra obtained by speciali*
*zing
En, with
E*n(S) = Zp[[v1, . .,.vn-1]][u 1]
at vi7! 0, 1 i n - 1. [These spectra correspond to the Honda formal group of
x1.1.1; they have some kind [15] of multiplicative structure. It is convenient *
*to call
the associated cohomology functors K(n)*(-; Zp).]
It is natural [32] to expect that the group of multiplicative automorphisms of *
*this
functor maps, under reduction modulo p, to the normalizer of the maximal torus *
*Qxq
in Dx , as in x1.2.1; in other words, precisely those automorphisms of K(n)*(-;*
* Fp)
which lift to automorphisms of the formal group law of K(n)*(-; Zp) lift to mul*
*ti-
plicative automorphisms of the whole functor. It would follow from this, that
K(n)*(-; Zp) ZpQnrp~= k2ZH*(-; Qnrp(1) k) ,
__
where Qnrp(1) is the representation of Gal(Qabq=Qp) (and thus of Gal(Q p=Qp)) c*
*on-
structed in x1.2.3. The results of the preceding section would then generalize,*
* with
the role of the cyclotomic character now being played by the crossed homomorphi*
*sm
ff.
3.4 This suggests that the remarks above about ss0 might be slightly naive: ss0
might make better sense as a small diagram category than as a topological space.
Equivariant stable homotopy theory provides further evidence for this, as do de*
*rived
categories of representations of quivers in groups.
The conjecture above asserts that the functor
A 7! Aut(K(n)*(-; Zp) ZpA) : (flatA - Mod ) ! (Groups)
is i) representable ii) by Gal(Qabq=Qp). Enlarging the useable class of fiber f*
*unctors
in this way, allowing values in (say) modules over discrete valuation rings, wo*
*uld
lead to the existence of specialization homomorphisms such as
ss1(specZp) ! ss1(specS, K(n)*(-; Zp)) ! ss1(specS, K(n)*(-; Fp)) .
This paper is so much a wish list, that while we're at it we might as well ask *
*for a way
to associate automorphism groups to suitable open subsets or subdiagrams of our
hypothetical ss0, together with homomorphisms between them defined by inclusion*
*s.
One of the beauties of Grothendieck's original account of the fundamental group
is his theory of specialization [16 X x2.3] which, as he notes, has no immediate
analog in pure algebraic topology. There is evidence [2, 4 x6, 22, 42 x4] that *
*some
generalization of his theory to a derived category context could accomodate a v*
*ery
general form of the theory of vanishing cycles [17, 9].
TOWARD A FUNDAMENTAL GROUPOID FOR THE STABLE HOMOTOPY CATEGORY 17
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Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
E-mail address: jack@math.jhu.edu