aa r X i v : . [ m a t h . A T ] A ug A THEORY OF BASE MOTIVES
JACK MORAVA
Abstract.
When A is a commutative local ring with residue field k ,the derived tensor product − ⊗ LA k : D ( A ) → ( k − Mod)lifts to a functor taking values in a category of modules over the ‘Tatecohomology’ R Hom ∗ A ( k, k ), which is the universal enveloping algebra ofa certain Lie algebra. Under reasonable conditions this lift satisfies aspectral sequence of Adams (or Bockstein) type.In a suitable category of ring-spectra, replacing A → k by A ( ∗ ) → S orTC( S ) → S yields interesting Hopf objects, with Lie algebras free aftertensoring with Q , analogous to those of motivic groups studied recentlyby Deligne, Connes and Marcolli, and others. [This is a sequel to and continuation of a talk at last summer’s conferencein Bonn honoring Haynes Miller [23]. I owe many mathematicians thanksfor helpful conversations and encouragement, but want to single out JohnRognes particularly, and thank him as well for organizing this wonderfulconference.] § Historically, the first part of the stable homotopy ring to be systematicallyunderstood was the image of the J -homomorphism J : π k − O = KO k ( ∗ ) → lim n →∞ π n + k − ( S n ) = π Sk − ( ∗ ) , defined on homotopy groups by the mapO n → O → lim n →∞ Ω n − S n − := Q ( S ) ;it factors through KO k = Z → ζ (1 − k ) · Z / Z ⊂ π S k − ( ∗ )(at least, away from two).In more geometric terms, a real vector bundle over S k defines a stablecofiber sequence S k − α / / S / / cof α / / S k · · · Date : 20 August 2009.This research was supported by DARPA and the NSF. and hence an extension[0 → KO ( S k ) → · · · → KO ( S ) → Adams ( KO ( S ) , KO ( S k ) ∼ = H c (ˆ Z × , ˆ KO ( S k )) , where the Adams operation ψ α , α ∈ ˆ Z × acts on ˆ KO ( S k ) by ψ α ( b k ) = α k b k .This (essentially Galois) cohomology can be evaluated, via von Staudt’stheorem, in terms of Bernoulli numbers.In the arithmetic-geometric context, Deligne and Goncharov [[11]; cf [25]for a more homotopy-theoretic account] have constructed an abelian tensor Q -linear category MTM of mixed Tate motives over Z , generated by ob-jects Q ( n ) satisfying a (small, ie trivial when ∗ >
1) Adams-style spectralsequence Ext ∗ MTM ( Q (0) , Q ( n )) ⇒ K n −∗ ( Z ) ⊗ Q The groups on the right have rank one in degree 4 k + 1, with generatorscorresponding (via Borel regulators) to ζ (1 + 2 k ). These same zeta-values appear in differential topology [18] in the classi-fication of smooth (‘Euclidean’) cell bundles over the 4 k + 2-sphere. There,both even and odd zeta-values can be seen as having a common origin,summarized by a diagram (where, implicitly, n → ∞ ) X x x (cid:15) (cid:15) ' ' PPPPPPP B O n (cid:15) (cid:15) / / B Diff( E n ) / / B Diff c ( R n ) (cid:15) (cid:15) BQ ( S ) ΩWh( ∗ ) . The space Wh( ∗ ) on the bottom right is Waldhausen’s smooth pseudoisotopyspace, which appears in K ( S ) = A ( ∗ ) = S ∨ Wh( ∗ ) . The shift by a double suspension in the cell versus vector bundle story isexplained by the factor B on the lower left, and Ω on the lower right. Theodd zeta-values appear in both geometry and topology because the naturalmap K ( Z ) → K ( S )is a rational equivalence. THEORY OF BASE MOTIVES 3
This suggests that some of the ideas of differential topology might be usefullyreformulated in terms of a category of ‘motives over S ’ analogous to thearithmetic geometers’ motives over Z , with the algebraic K -spectrum of theintegers replaced by Waldhausen’s A -theory: these zeta-values might thenprovide a trail of breadcrumbs leading us to some deeper insights.In the following section we recall some machinery from homological algebra,regarding K ( S ) = A ( ∗ ) = S ∨ Wh( ∗ ) → S and TC( S ) ∼ S ∨ Σ C P ∞− → S (mod completions) as analogs of local rings over S , with the appropriate tracemaps interpreted as quotients by maximal ideals. Note that the algebraic K -theory spectrum of Z lacks such an augmentation.Tannakian formalism identifies the category MTM of mixed Tate motives asrepresentations of a certain pro-affine Q -groupscheme with free graded Liealgebra, conjecturally related to other areas of mathematics such as algebrasof multiple zeta-values and renormalization theory [8, 10]. In the contextproposed here, a similar group objectSpec S ∧ A S appears as derived automorphisms of A . § A -theoretic analogs, conjecturally identify-ing these arithmetic and geometric motivic groups. § I’ll start with work on commutative local rings, eg A → k with maxi-mal ideal I , with roots in the very beginnings [27] of homological algebra.Eventually A will be graded, or a DGA.The functors H ∗ ( A, − ) := Tor A ∗ ( k, − )and H ∗ ( A, − ) := Ext ∗ A ( k, − )appear in Cartan-Eilenberg; the first is covariant, and the second is con-travariant, in A . I’ll be concerned mostly with H ∗ ( A, k ) = Tor A ∗ ( k, k ) and H ∗ ( A, k ) = Ext ∗ A ( k, k ) . Under reasonable finiteness conditions, these are dual k -vector spaces [theassociativity sseqs [7 XVI §
4] degenerate]: in fact they are dual Hopf alge-bras, with H ∗ ( A, k ) being the universal enveloping algebra of a graded Liealgebra [2].
JACK MORAVA If A = Z p → F p is the residue map then H ∗ ( A, k ) = E ∗ ( Q )is an exterior algebra on a Bockstein element of degree one. If A = k [ ǫ ] / ( ǫ )then H ∗ ( A, k ) = k [ x ] ( | x | = 2) is the Hopf algebra of the additive group.These are the first manifestations of Koszul duality. For local rings this homology is closely related to Hochschildtheory [7 X § The homological functor M H ∗ ( M ⊗ L Z p F p ) := M : D ( Z p − Mod) → F p − Modlifts to the category of E ( Q )-comodules. There is a Bockstein spectralsequence Ext ∗ E ( Q ) − Comod ( M , N ) ⇒ Hom D ( Z p − Mod) ( M, N ) . G ( A ) := Spec H ∗ ( A, k ) is an affine (super) k -groupscheme;its grading is encoded by an action of the multiplicative group G m = Spec k [ β ± ] , and ˜ G A ) := G ( A ) ⋊ G m . The Bockstein spectral sequence generalizes: if M ∈ D ( A − Mod), let M = H ∗ ( M ⊗ LA k ) = H ∗ ( M ⊗ A A ) ∈ ( k − Mod) , where A → A → k is a factorization of the quotient map through a cofibra-tion and a weak equivalence (ie A is a resolution of k , eg0 / / Z p p / / Z p / / . Proposition:
The functor M → M lifts to a homological functor D ( A − Mod) → ( ˜ G ( A ) − reps) , and there is an ‘ascent’ sseqExt ∗ ˜ G ( A ) − reps ( M , N ) ⇒ Hom D ( A − Mod) ( M, N )of Adams (alt: Bockstein) type . . .The
Proof is as in Adams’ Chicago notes [1], replacing the map S → MUwith A → k : thus M U ∗ ( X ) becomes an M U ∗ M U -comodule by taking ho-motopy groups of the composition X ∧ MU = X ∧ S ∧ MU → X ∧ MU ∧ MU= ( X ∧ MU) ∧ MU (MU ∧ MU) , THEORY OF BASE MOTIVES 5 yielding
M U ∗ ( X ) → M U ∗ ( X ) ⊗ MU ∗ M U ∗ M U .
In the present context the comodule structure map comes from taking thehomology of the composition M ⊗ A A = M ⊗ A A ⊗ A A → M ⊗ A A ⊗ A A = ( M ⊗ A A ) ⊗ A ( A ⊗ A A ) , resulting in M → M ⊗ k H ∗ ( A, k ) . (cid:3) The bar construction provides a cofibrant replacement for k , withunderlying algebra ⊕ n ≥ ⊗ n I [1]and a suitable differential. When A = k ⊕ I is a singular ( I = 0) extension,the differential is trivial, and Ext ∗ A ( k, k ) is the universal enveloping algebraof the free Lie algebra on I ∗ [1]. of such generalized Adams spectral sequences is a com-plicated topic, related to extending the Tannakian formalism when theremay be inequivalent fiber functors [22]. The stable homotopy category isvery unlike that of pure motives, which is semisimple in interesting cases:instead, stable homotopy is more like the categories of F p - representations offinite p -groups, whose structure is encoded entirely through iterated exten-sions of trivial objects. Away from characteristic zero, it is often unrealisticto hope to recover the full structure of an abelian (or triangulated) monoidalcategory in terms of the automorphism group of a fiber functor; instead oneusually gets at best a spectral sequence which may allow the recovery of thegraded object associated to a filtration of some localization of the originalcategory.The generalized fiber functor defined by topological K -theory, for example,has Gal( Q ab / Q ) ∼ = ˆ Z × as (more or less) its motivic group, and the associatedspectral sequence ‘sees’ only the image of the J -homomorphism; other fiberfunctors see different parts of (some generalization [17] of) the prime idealspectrum of the stable homotopy category. One of the more interestingissues emerging from this picture is the relation of deformations of fiberfunctors (eg, taking values in categories of modules over a local ring) andtheir motivic groups. Examples closer to homotopy theory appear in recent work of Dwyer,Greenlees, and Iyengar. Suppose for example that X is a connected pointedspace (eg with finitely many cells in each dimension), and let X + = X ∨ S be X with a disjoint basepoint appended. Its Spanier-Whitehead dual X D := Maps S (Σ ∞ X + , S )is an E ∞ ring-spectrum, with augmentation X D → S given by the basepoint. JACK MORAVA
The Rothenberg-Steenrod construction [14 § X D − Mod ( S , S ) ∼ S [Ω X ]of ( A ∞ , co − E ∞ ) Hopf algebra objects in the category of spectra.If X is simply connected, there is a dual result with coefficients in theEilenberg-MacLane spectrum k = Hk of a field: then the ‘double commu-tator’ Hom k [Ω X ] − Mod ( k, k ) ∼ C ∗ ( X, k )is homotopy equivalent to the (commutative) cochain algebra of X . [Thisputs the homotopy groups π ∗ C ∗ ( X, k ) ∼ = H −∗ ( X, k )in negative dimension.] This sharpens a classical [4] analogy between thehomology of loopspaces and local rings.The (generalized Koszul duality?) functor M Hom X D ( M, S ) : ( X D − Mod) → ( S [Ω X ] − Mod) . seems worth further investigation . . . Suspensions are formal, so if k = Q and X = Σ Y then X D ⊗ Q ∼ H −∗ (Σ Y, Q )is a singular extension of Q , so Q [ΩΣ Y ] is the universal enveloping algebraof the free Lie algebra on the graded dual of ˜ H −∗− ( Y, Q )[1].Recent work of Baker and Richter [5] identifies the Hopf algebra of non-commutative symmetric functions with the integral homology H ∗ (ΩΣ C P ∞ )as the universal enveloping algebra of a free graded Lie algebra. The dualHopf algebra H ∗ (ΩΣ C P ∞ ) is the (commutative) algebra of quasi-symmetricfunctions. The topological cyclic homology TC( S ; p ) of the sphere spectrum (at p ) is an E ∞ ringspectrum, equivalent to the p -completion of S ∨ Σ C P ∞− [20];the subscript signifies a twisted desuspension of projective space by the Hopfline bundle.From now on I’ll be working over the rationals, eg with the graded algebraTC n − ( S ; Q p ) ∼ = Q p ⊕ Q p h e n − i ,n ≥ The multiplication on a ring-spectrum A defines a composition[ X, A ∧ Y ] ∧ [ Y, A ∧ Z ] → [ X, A ∧ Z ](on morphism objects in spectra) by X → A ∧ Y → A ∧ A ∧ Z → A ∧ Z .
THEORY OF BASE MOTIVES 7
The map X → A ∧ X defines a functor from the category with S -modules(eg X, Y ) as objects, andCorr A ( X, Y ) := [
X, A ∧ Y ]as morphisms, to the category of A -modules, becauseCorr A ( X, Y ) = [
X, A ∧ Y ] → [ X, [ A, A ∧ Y ] A ] ∼ = [ A ∧ X, A ∧ Y ] A . Let ( A − Corr) be the triangulated subcategory of ( A − Mod) generated bythe image of this construction. The augmentation of A defines a functorfrom ( A − Corr) to S -modules which is the identity on objects, and is givenon morphisms by [ X, A ∧ Y ] → [ X, S ∧ Y ] = [ X, Y ] . I propose to inherit the composition of this functor with rationalization asan analog of the ‘fiber functor’ in § Corollary:
This homological functor ( A − Mod) → ( Q − Mod) lifts to thecategory of ˜ G ( A ⊗ Q )-representations, yielding a spectral sequenceExt ∗ , ∗ ˜ G ( A ⊗ Q ) − reps ( X, Y ) ⇒ Corr ∗ A ( X, Y ) . Proof: ( X ∧ A ) ∧ A S = X . . . (cid:3)
A free Lie algebra has cohomological dimension one, so when A isTC( S ; p ) and X and Y are spheres, this spectral sequence degenerates toExt G (TC ⊗ Q p ) ( S n Q p , S Q p ) ∼ = TC n − ( S , Q p ) , with left-hand side isomorphic to H , ( F ( f TC ∗ [1]) , S − n Q p ) ∼ = Hom ( f TC ∗ [1] , S − n Q p ) , which is just the one-dimensional vector space Q p h e n − [1] b − n i . At a regular odd prime p (cf. [13, 24]),Wh( ∗ ) / Σcoker J ∼ Σ H P ∞ ;the cokernel of the J -homomorphism is a torsion space, so this yields aspectral sequence Ext ∗ ˜ G ( A ⊗ Q ) ( S n Q , S Q ) ⇒ A n −∗ ( ∗ ) ⊗ Q with A k +1 ( ∗ ) ⊗ Q ∼ = Q h e k +1 [1] b − k − i . JACK MORAVA § A -theoretic motives3.1 A retractive space Z over X is a diagram X Z r o o X s o o whichcomposes to the identity 1 X : it’s a space over X with a cofibration section. Z is said to be finitely dominated if some finite complex is retractive over it[15].Waldhausen showed that finitely dominated retractive spaces over X forma category with weak equivalences and cofibrations, and that the K -theoryspectrum A ( X ) of this category can be identified with K ( S [Ω X ]).More generally, Z is relatively retractive over X , with respect to a map p : X → Y , if the homotopy fiber of p ◦ r over any y ∈ Y is finitelydominated as a retractive space over the homotopy fiber of p above y . Thecategory R ( p ) of such spaces is again closed under cofibrations and weakequivalences, with an associated K -theory spectrum A ( X → Y ).Bruce Williams [28 §
4] (using a formalism developed in algebraic geometryby Fulton and MacPherson) shows that this functor has a rich bivariantstructure: compositions A ( X → Y ) ∧ A ( Y → Z ) → A ( X → Z ) , good behavior under products, &c. It behaves especially well on fibrations;in particular, the spectra ∀ A ( X, Y ) := A ( X × Y → X )(defined by relatively retractive spaces Z over X × Y → X ) admit goodproducts ∀ A ( X, Y ) ∧ ∀ A ( Y, Z ) → ∀ A ( X, Z ) . Let A − Corr be the triangulated envelope [6] of the symmetric monoidal ad-ditive category with finite CW complexes
X, Y as objects, and ∀ A ( X, Y ) = π ∀ A ( X, Y ) as morphisms. Composition A ( X × Y → Y ) → [ X, A ( Y )] → [ X, A ∧ Y ]of the standard assembly map with a slightly less familiar relative co-assemblymap [12 §
5] defines a monoidal stabilization functor( A − Corr ) → ( A − Corr)analogous to inverting the Tate motive, or to the introduction of desuspen-sion in classical homotopy theory. However, A -theory of spaces is a highlynonlinear functor, and might possess other interesting stabilizations. The motivic constructions of Suslin and Voevodsky [27] begin with a cat-egory whose objects are schemes of finite type over some nice base, and whosemorphism groups
SmCorr ( V, W ) of (roughly) sums of irreducible subvarieties Z of V × W which are finite with respect to the projection V × W → V ,and surjective on components of V . THEORY OF BASE MOTIVES 9
When V and W are defined over a number field (eg Q ), classical arguments[cf. eg [21]] show that Z ( C ) ∪ V ( C ) × W ( C ) → V ( C ) × W ( C )is finitely dominated relatively retractive with respect to V ( C ) × W ( C ) → V ( C ), defining a cycle class homomorphism SmCorr ( V, W ) → ∀ A ( V ( C ) , W ( C )) , and hence a functor V V ( C ) : ( SmCorr ) → ( A − Corr ) . My hope is that this will lead to an identification of the motivic group forthe category of mixed Tate motives with ˜ G ( A ⊗ Q ). It seems at least possiblethat ˜ G (TC ⊗ Q ) is the larger motivic group seen in physics [8, 9 § I don’t want to end this sketch without mentioning one last possibility.Dundas and Østvær have proposed a bivariant K-theory based on categories E ( E, F ) of suitably exact functors between the categories of (cell) modulesover (associative) ring-spectra E and F .These module categories are to be understood as categories with weak equiv-alences and cofibrations; the exact functors are to preserve these structures,and be additive in a certain sense. E ( E, F ) is again a Waldhausen category,which suggests that the category (Alg A ) with associative ring-spectra E, F as its objects, and Alg A ( E, F ) := K ( E ( E, F ))as morphisms, is an interesting analog of categories of noncommutative cor-respondences proposed by various research groups [9 §
6, 19 § A .A space W over X × Y defines an X D - Y D bimodule W D , and W Hom X D − Mod ( W D , − )is a natural candidate for an exact functor, and hence a map R ( X × Y → X ) → E ( X D , Y D ) . If so, this might define another interesting stabilization of A − Corr , relatedmore closely to the Waldhausen K -theory of Spanier-Whitehead duals thanto spherical group rings. References
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Department of Mathematics, Johns Hopkins University, Baltimore, Mary-land 21218
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