aa r X i v : . [ m a t h . A T ] J un OPERATIONS ON INTEGRAL LIFTS OF K ( n ) JACK MORAVA
Abstract.
This very rough sketch is a sequel to [27]; it presents evi-dence that operations on lifts of the functors K ( n ) to cohomology the-ories with values in modules over valuation rings o L of local numberfields, indexed by Lubin-Tate groups of such fields, are extensions of thegroups of automorphisms of the associated group laws, by the exterioralgebras on the normal bundle to the orbits of the group laws in thespace of lifts. Introduction0.1
In a symmetric monoidal category, e . g . of schemes or structured spectra,the morphisms defining an action of a monoid M on an object X can bepresented as a cosimplicial object; for example [24] if M = M U is theThom spectrum for complex cobordism ( i . e . the universal complex-oriented S -algebra), then( S / / ) M U / / / / M U ∧ S M U / / / / / / · · · is a kind of M U-free Adams-Mahowald-Novikov resolution of S . Its homo-topy groups define a cosimplicial commutative algebra resolution π ∗ S / / π ∗ M U = M U ∗ / / / / π ∗ ( M U ∧ S M U) = M U ∗ M U / / / / / / · · · of the stable homotopy algebra. Regarding these algebras as affine schemesover Spec Z , this diagram becomes a presentation for a groupoid-schemeSpec M U ∗ M U / / / / Spec M U ∗ which, by work of Quillen [29], can be identified with a moduli stack forone-dimensional commutative formal groups.Using a great deal of work on Lubin-Tate spectra by others, we constructin § p -adically complete, where p > A ∞ periodic M U-algebra spectra K ( L ), indexed by Lubin-Tate formal group laws LT L Date : 21 June 2019. The terms in this display are graded, but it is convenient to regard them as Z -gradedcomodules over the multiplicative groupscheme G m = Z [ t ± ], with coaction M k ∋ x x ⊗ t k , thus providing an excuse for often supressing this grading. for local number fields L ⊃ Q p , Galois of degree [ L : Q p ] = n with valuationrings o L . These spectra have homotopy groups π ∗ K ( L ) = K ( L ) ∗ ∼ = o L ∗ [ v ± ]( | v | = 2), and in § K ( L ) ∗ K ( L ) / / / / Spec K ( L ) ∗ of homological co-operations in terms of the isotropy or stabilizer groups of LT L , as objects in the Quillen-Lazard moduli stack. These automorphismgroups are by now well-understood, almost classical in local arithmetic ge-ometry, and the first section below summarizes some of that knowledge; itwill serve as a model for our applications to algebraic topology.Perhaps the point of this paper is to explain that, in spite of the notation, ourconstruction of the spectra K ( L ) is not functorial in L ; this note is instead aplea for a natural construction. The third section below contains preliminaryresults toward an identification of the endomorphisms or (co)operations oftheir associated cohomology theories, and argues that these have close con-nections with the Weil group of L [25]: or, more precisely, with the Galoisgroup of a maximal totally ramified abelian extension L trab of L , over Q p .Our partial results can perhaps be read as evidence toward an interpreta-tion of the spectra K ( L ) as something like a K -theory spectrum associatedto the (topological, perfectoid) completion L ∞ of L trab [28]. For example,our K ( Q p ) can be (non-canonically) identified with the p -adically completedalgebraic K -theory spectrum of the completion Q p ∞ of the field of p -powerroots of unity over Q p , and thus with the p -adic completion of Atiyah’stopological K -theory of C .To return to the organization of this paper: its second section uses thetheory of highly structured spectra to define, following the original work ofSullivan and Baas [4,38,39], the spectra K ( L ) as Koszul quotients of spectra E (Φ L ) associated to Lubin-Tate formal group laws [14,31]. The resultingconstructions are integral lifts of the ‘extraordinary’ spectra K ( n ) [44], inthat smashing with a mod p Moore spectrum defines natural isomorphisms K ( L ) ∗ ( X ∧ M ( p, ∼ = K ( n ) ∗ ( X, F p ) ⊗ o L /p o L , where o L /p o L ∼ = F q [ π ] / ( π e ) (with n = ef and q = p f , see § L is unramified then e = 1 , q = p n , and the mod p reduction of K ( L )agrees with K ( n ) ⊗ F q . In some sense the K ( n ) are indexed by the finitefields, while the K ( L ) are indexed by finite Galois extensions of Q p . § I Notation and recollections1.1 If A is a commutative ring, let FG ( A ) ⊂ A [[ X, Y ]] be the set of power se-ries F ( X, Y ) = X + Y + . . . satisfying the standard axioms for a commutativeformal group law over A , and let Γ( A ) ⊂ A [[ T ]] be the group of invertible PERATIONS ON INTEGRAL LIFTS OF K ( n ) 3 power series t ( T ) = t T + . . . ( i . e . with t ∈ A × ) under composition; thenthe group Γ acts on the set FG byΓ( A ) × FG ( A ) ∋ t, F F t ( X, Y ) = t − ( F ( t ( X ) , t ( Y ))) ∈ FG ( A ) . Both Γ and FG are co-representable functors: FG ( A ) ∼ = Hom alg ( L , A ), whereLazard’s ring L is polynomial over Z , and Γ( A ) ∼ = Hom alg ( S, A ), where S = t − Z [ t i ] i ≥ is a Hopf algebra with coproduct(∆ t )( T ) = ( t ⊗ ⊗ t )( T )) ∈ ( S ⊗ Z S )[[ T ]] . Yoneda’s lemma then implies the existence of a coproduct homomorphism ψ : L → L ⊗ Z S of rings, corepresenting the group action. These rings are implicitly gradedby the coaction of the multiplicative subgroup G m ⊂ Spec S . A group action α : G × X → X in (Sets) defines a groupoid[ X//G ] : G × X s / / t / / X with X as set of objects, G × X as set of morphisms, and s ( g, x ) = x, t ( g, x ) = α ( g, x ) as source and target maps. The usual convention in algebraic topol-ogy regards L ⊗ Z S as a two-sided L -algebra, with the obvious structure onthe left, and a right L -algebra structure( L ⊗ Z S ) ⊗ Z L ⊗ ψ / / ( L ⊗ Z S ) ⊗ L ( L ⊗ Z S ) / / ( L ⊗ Z S ) ;this is what’s meant by saying that L η L / / η R / / L ⊗ Z S is a Hopf algebroid.Following Grothendieck and Segal, a category C with set C [0] of objects and C [1] of morphisms can be presented as a simplicial set C [0] C [1] o o o o C [1] × C [0] C [1] o o o o o o · · · (where X × Z Y denotes the fiber product or equalizer of two maps X, Y → Z ). In the case of a group action as above, this is isomorphic to a simplicialobject X G × X o o o o G × G × X o o o o o o · · · which can alternatively be regarded as a bar construction. The functor A [ FG ( A ) // Γ( A )] thus defines a simplicial scheme: the moduli stack ofone-dimensional formal groups. A homomorphism A → B of commutative rings defines an extension ofscalars map FG ( A ) ∋ F F ⊗ A B ∈ FG ( B ) . JACK MORAVA
Definition [ iso ( F )]( B ) is the groupoid with the orbit O Γ ( B )( F ) = { ( F ⊗ A B ) g | g ∈ Γ( B ) } (of F ⊗ A B under coordinate changes) as its set of objects, and mor iso F ( B ) ( G, G ′ ) = { h ∈ Γ( B ) | G h = G ′ } as (iso)morphisms of G with G ′ ; thus [ iso ( F )]( B ) = [ O Γ ( B )( F ) // Γ( B )] . This groupoid maps fully and faithfully to its skeleton (which has one object)and the group Aut B ( F ) ⊂ Γ( B ) (of automorphisms of F ⊗ A B as a formalgroup law over B ) as its morphisms. The homomorphism F : L → A classifying F thus defines a Hopf A - algebroid[ iso ( F )] : A / / / / A ⊗ L ( L ⊗ Z S ) ⊗ L A equivalent to a simplicial groupoid-scheme ( A − alg) ∋ B [ iso ( F )]( B ) overSpec A . It is nonstandard, but it will be convenient below to write q = p n and let Q q denote the quotient field of the ring W ( F q ) of Witt vectors, i . e . the degree n unramified extension of Q p . Following Ravenel [30 § LT Q q for this field can be defined over the p -adic integers Z p byHonda’s logarithm log Q q ( T ) = X k ≥ p − k T p nk ;this has, as its mod p reduction, a formal group law H( n ) of height n over F p , associated to the cohomology theory K ( n ). The resulting left and right F p -algebra structures on F p ⊗ H( n ) ( L ⊗ Z S ) ⊗ H( n ) F p = C ( o D × , F q ) Gal( F q / F p ) − inv = Σ( n )coincide, representing [ iso (H( n ))] by the algebra of functions on a certainpro-algebraic group scheme over Spec F p .In more detail [24], a finite field k = F q has a local domain W ( k ) of Wittvectors, with maximal ideal generated by p and a canonical isomorphism W ( k ) /pW ( k ) → k ; its quotient field W ( k ) ⊗ Z Q = Q q is the extension of Q p obtained by lifting the roots F × q of unity to Q p . This construction isfunctorial, and a generator σ of the cyclic group Gal( Q q / Q p ) ∼ = Gal( F q / F p )sends a root ω of unity to σ ( ω ) = ω p . Let D = Q q h F i / ( F n = p )be the noncommutative division algebra obtained from Q q by adjoining an n th root F of p satisfying, for any a ∈ Q q , the relation σ ( a ) · F = F · a .The valuation on Q q (normalized so ord( p ) = 1) extends to D to define asemidirect product extension1 / / o D × / / D × ord / / n Z / / PERATIONS ON INTEGRAL LIFTS OF K ( n ) 5 with a generator of the infinite cyclic group on the right acting on an el-ement u of the compact kernel o D × as F -conjugation. This kernel thusacquires an action of the cyclic group of order n , which may be identifiedwith Gal( F q / F p ), making o D × the group of points of a pro-´etale groupschemeover F p . It is represented by the F p -algebra of (Galois equivariant, conti-nous) F q -valued functions h on o D × satisfying σ ( h ( u )) = h ( F u F − ). Moreconcisely, [ iso (H( n ))] ≃ [ ∗ // o D × ]as groupoid-valued functors. Similar results [21] hold for Lubin-Tate groups of local number fields( i . e . extensions L of Q p with [ L : Q p ] = n < ∞ ); I will assume here that thisextension is Galois. Such a field has a local valuation ring o L with finiteresidue field k L ∼ = F q , where now q = p f ; moreover L contains a maximalunramified extension L = W ( k L ) ⊗ Z Q ⊃ Q p , such that [ L : L ] = e = f − n . The maximal ideal m L = ( π ) ⊂ o L is principal, and we will choose agenerator π ; it satisfies some Eisenstein equation E L ( π ) = π e + X ≤ i
JACK MORAVA over the generic point of Spec o L its fiber is the groupoid [ ∗ // o L × ], whileover the closed point it is [ ∗ // o D × ]. Note that any degree n extension of Q p embeds in D as a maximal commutative subfield, so the maximal toruses of D × in some sense parametrize Lubin-Tate groups of degree n extensions of Q p . The Weyl groups of these toruses are then Galois groups Gal( L/ Q p ), andthe normalizers of these toruses are essentially the Weil groups W ( L trab / Q p )associated to maximal totally ramified abelian extensions of L [41,43]; the(cohomology classes of the) group extensions defining them are the ‘funda-mental classes’ of local classfield theory.When L = Q q is unramified [27] we can assume that[ p ] L ( T ) = pT + L v q − T q i . e . that (a graded version of) LT Q q is p -typical, defined by a homomorphismBP ∗ = Z p [ v i ] i ≥ → Z p [ v ± ]sending Araki’s [2] generators v i to 0 when i = n , and v n to v q − . § II2.0
To construct the spectra K ( L ) we work at a prime away from 6, in asymmetric monoidal category of p -adically complete spectra, e . g . S p - mod-ules. We will be concerned below with K ( n )-local spectra, and we will K ( n )-localize their smash products [17]. Recall [33] that the Gaussian inte-ger spectrum S [( − / ] is not E ∞ : the behavior of the stable homotopycategory under arithmetic ramification seems potentially very interesting.Lurie’s ´etale topology on the category of spectra [23 Def 7.5.1.4] definescommutative ring-spectra S W ( F q ) ´etale over S p (roughly, W ( F q ) ⊗ Z p S p ).Schwede’s Moore spectra (functorial away from 6 [34 § II Rem. 6.44]) canthen be used to define, for a valuation ring o L (free of rank e over W ( k L ))a p -adic A ∞ ring spectrum S o L (roughly, M ( o L , ⊗ W ( k L ) S W ( F q ) ) with π ∗ S o L ∼ = π ∗ S ⊗ Z o L . Following § o L = ⊕ ≤ i ≤ e − W ( k ) · π i (where k = k L for simplicity) is defined by classical structure constants m i,jl ∈ W ( k ) , ≤ i, j, l ≤ e −
1, such that π i · π j = X ≤ l ≤ e − m i,jl π l . The typeface is intended to distinguish these constructions from Quillen’s K -theory PERATIONS ON INTEGRAL LIFTS OF K ( n ) 7 Let S o L denote the wedge sum W ≤ i ≤ e − S W ( k ) · t i (with t i a book-keepingindeterminate), and let S o L × S o L → S o L ∧ S W ( k ) S o L → S o L be the morphism of S W ( k ) -module spectra defined component-wise, as thecomposition S W ( k ) × S W ( k ) · t i × t j / / S W ( k ) ∧ S W ( k ) S W ( k ) · t l · m i,jl / / S W ( k ) t l , (where the final map is multiplication by the structure constant). This isthe product map for a weak S W ( k ) -algebra structure on S o L , i . e . a kind of H ∞ structure making π ∗ S o L = π Hom S W ( k ) ( S ∗ W ( k ) , S o L ) ∼ = π ∗ ( S p ) ⊗ Z p o L as algebras. In particular we have a morphism o L × S o L → S o L of S W ( k ) -algebras, representing o L -multiplication on π ∗ S o L , used in § Remark If G is the Galois group of a finite extension L of Q p , a theoremof Noether implies that its valuation ring o L is projective over the groupring Z p G iff L is tamely ramified; but work of Swan [40] implies, moregenerally, that the class of o L in the Grothendieck group G ( Z p G ) (definedby splitting short exact sequences) is the image of a (not necessarily unique)class [ P ] in K ( Z p G ), perhaps analogous to Wall’s finiteness obstruction forCW complexes. Such Swan elements suggest constructing analogs of Moorespectra for o L as representing objects for functors such as X π ∗ ( X ∧ G + ) ⊗ Z p G P := π ∗ ( X ; P ) . . . Work several mathematical generations deep [10,14,31,32. . . ] asso-ciates to a one-dimensional formal group law Φ, of finite height n overa perfect field k of characteristic p >
0, an E ∞ p -adic complex oriented S W ( k ) -algebra spectrum E (Φ) with homotopy algebra π ∗ E (Φ) ∼ = E (Φ) ∗ ∼ = W ( k )[[ u , . . . , u n − ]][ v ± ]]of formal power series, representing Lubin and Tate’s functor [22] whichsends a complete noetherian local ring A with residue field k to the set(modulo isomorphisms which reduce to the identity over k ) of lifts of Φ to A . We will sometimes take v = 1 to suppress the grading, and to sim-plify notation we may write E F for E (Φ) for a chosen lift F of Φ to a localring ( e . g . W ( k ) or o L ) with residue field k ; we may even write E L for the S W ( k L ) = S o L -module spectrum E ( LT L ). Similarly + E Φ or + EF may de-note the associated formal group sum, and m F = [[ u ∗ ]] may signify the‘maximal ideal’ of E F ∗ over W ( k ) or o L . JACK MORAVA
Lubin and Tate show that the (pro´etale) group Aut ¯ k (Φ) ∼ = o D × of automor-phisms of Φ ⊗ k ¯ k , with its natural Gal(¯ k/k )-action, lifts to a (continuous butnot smooth) action on W (¯ k ) ⊗ W ( k ) E (Φ) ∗ ; in particular, their theorem 3.1shows that this action takes W (¯ k ) ⊗ W ( k ) m E (Φ) to itself. In the formalism of § E (Φ) ∗ // Aut ¯ k (Φ)] : E (Φ) ∗ η L / / η R / / E (Φ) ∗ E (Φ) ∼ = H Aut(Φ) ;where H Aut(Φ) [17] is a Hopf algebra of Galois-equivariant continuous func-tions from Aut(Φ) to E (Φ). Note that the two (left and right) unit ho-momorphisms send m E to m E ˆ ⊗ H Aut(Φ) , and that this Hopf algebroid isequivalent to E (Φ) ∗ η L / / η R / / E (Φ) ∗ ⊗ L ( L ⊗ Z ) ⊗ L E (Φ) ∗ . If, for example, Φ L / F p is the Lubin-Tate group law for LT Q q as in § u i to be Araki generators satisfying[ p ] E ( T ) = X E,i ≥ o v i T p i (with v = p ); the classifying homomorphism from M U ∗ then sends C P q k − to Y ≤ i ≤ k (1 − p q i − ) − · ( p − q k ) v q k − and the remaining C P l to 0 [17]. A parallel (but even more venerable) line of research, leading to themodern theory of highly structured spectra, allows us to associate to the (bydefinition, regular) sequence v ∗ = v , . . . , v n − of elements of E Q q ∗ , a choice v i : S p i − → E Q q of representatives defining, by the construction of [12 V § § p -adiccomplex-oriented A ∞ ring-spectra K ( Q q ) = E Q q / ( v ∗ )with π ∗ K ( Q q ) ∼ = W ( F q )[ v ± ], having LT Q q as formal group law. Remark
When n = 1 this construction recovers a model for Atiyah’s p -adic completion [3] of complex topological K -theory, and when n = 2 itdefines a p -adic lift of Baker’s supersingular elliptic cohomology [5]. Awayfrom the prime 6, elliptic cohomology [13 § § E , E ). A theorem of Deligne [19] identifies the modular form definedby the Eisenstein series E p − and the (Hasse) parameter v , modulo p . This PERATIONS ON INTEGRAL LIFTS OF K ( n ) 9 suggests a close relation between p -adic elliptic cohomology, mod E p − , with K ( Q p ); but understanding that would require an understanding of E p − asa polynomial in E , E , which evidently depends on the prime p . More generally, a Lubin-Tate group law for a ramified local field ofdegree n over Q p lifts its mod π reduction to a homomorphism u i u ∗ i : E (Φ L ) ∗ = W ( k L )[[ u ∗ ]] → o L of local W ( k L )-algebras. Lemma
Let u ∗ i ∈ m L = ( π ) ⊂ o L , i ≤ i ≤ n − be a sequence of elementsin the maximal ideal m L : then u i u i − u ∗ i = v i defines an isomorphism o L [[ u ∗ ]] ∼ = o L [[ v ∗ ]] of local o L -algebras. [For if w ∈ m L and o L [[ x ]] ∋ a ( x ) = P i ≥ a i x i , then a ( x ) = X k ≥ ( X l ≥ (cid:18) k + ll (cid:19) a k + l w l )˜ x k = X l ≥ ˜ a k ˜ x k , where ˜ x = x − w . The argument for multiple variables is similar.] (cid:3) Definition
The A ∞ complex-oriented S o L -algebra spectrum E L = S L ∧ S W ( kL ) E (Φ LT L ) has π ∗ E L = E L ∗ = o L [[ v ∗ ]][ v ± ] as algebra of homotopy groups, generated by a regular sequence of elements v i = u i − u ∗ i such that v i specializes the modular lift to the chosenLubin-Tate group law of L . The definition in § K ( Q q ) uses the E ∞ structure on E (Φ LT Q q ),which is not available for nontrivially ramified fields. This issue can beavoided by reorganizing the induction in [12] (which follows Sullivan andBaas, based on iterated cofibrations) as a computation of the spectral se-quence for the homotopy groups of the geometric realization of a suitablesimplicial (Koszul) spectrum [11 § a of a commutative k -algebra A defines an (elementary) differ-ential graded A -algebraKsz A ( a ) = ( A [ e ] / ( e ) , d a ( e ) = a ) . More generally, a k -module homomorphism a ⋆ : k m → A defines the classicaldifferential graded commutative algebraKsz A ( a ⋆ ) = O A, ≤ i ≤ m Ksz A ( a i ) ∼ = A ⊗ k Λ k ( e i | ≤ i ≤ m ) or, more generally, a homogeneous element of a graded commutative A ∗ ; however thiswill be largely suppressed from our notation (with an exterior algebra denoted by Λ, to reduce the multiplicity of thingscalled E ), and differential de I = X ≤ i ≤ m ( − i +1 a ⋆ ( e i ) · e ˆ I ( i ) , where e I = ∧ l ∈ I e l is indexed by (totally ordered) subsets I of { m } = { , . . . , m } , ˆ I ( i ) is obtained from I by omitting its i th element, e I ∧ e K is 0 if I ∩ K is nonempty and equals ± e I,K if they are disjoint, with signequal to that of the permutation putting { I, K } in proper order.In the elementary case, if a is not a 0-divisor in A , this defines an A -freeresolution of the quotient algebra A/ ( a ), i . e . of the cofiber of the map of A to itself defined by a -multiplication. More generally, if a i is not a 0-divisorin the quotient ring A/ ( a , . . . , a i − ) ( i . e . a ∗ is a regular sequence), thisconstruction defines an A -free resolution of A/ ( a ⋆ ).In the context of commutative S -algebras or ring spectra, we can associateto morphisms a i : S | a i | → A a semi-simplicial ( i . e . without degeneracy operators [42 § A -algebra k K sz A ( a ⋆ )[ k ] = _ I ⊂{ m } , | I | = k A · e I with A -module face operators ∂ i e I = µ ( a i ) · e ˆ I ( i ) , where µ ( a ) : S | a | A → A is defined by multiplication by a . The fat realization [11 § § | K sz A ( a ⋆ ) | of such a semisimplicial object is canonically filtered, leading to the construc-tion of a spectral sequence computing its homotopy groups, with E page π ∗ K sz A ( a ⋆ ) = Ksz A ∗ ( a ⋆ ) ⇒ | K sz A ( a ⋆ ) | ∗ When a ⋆ is a regular sequence this complex is a resolution, and thespectral sequence collapses to an isomorphism | K sz A ( a ⋆ ) | ∗ ∼ = A ∗ / ( a ⋆ ) . Applying this as above yields a definition for K ( Q q ) = | K sz Φ( Q q ) ( v ⋆ ) | as an S W ( F q ) -algebra spectrum, essentially equivalent to the construction of [12].The ramified case is more delicate, because its building blocks are not E ∞ ;we need a Lemma
The morphisms ˜ v i , ˜ v j : S ∗ o L E L → E L ( ≤ i, j ≤ n − defined by multiplication with ˜ v i = u ∗ i ∧ W E − L ∧ W u i : S ∗ o L → S o L ∧ W ( k L ) E (Φ L ) ( = E L ) commute. PERATIONS ON INTEGRAL LIFTS OF K ( n ) 11 Proof
Define a twist isomorphism E (Φ L ) ∧ W ( k L ) o L → o L ∧ W ( k L ) E (Φ L )adjoint to the composition E (Φ L ) → Hom W ( k L ) ( o L , o L ) ∧ W ( k L ) E (Φ L ) → Hom W ( k L ) ( o L , o L ∧ W ( k L ) E (Φ L ))of S W ( k L ) -module morphisms. Since both S L and E (Φ L ) are commutative S W ( k L ) -modules, we have (with some abbreviation)˜ v i ∧ W ˜ v j = ( u ∗ i ∧ W E − L ∧ W u i ) ∧ W ( u ∗ j ∧ W E − L ∧ W u i ) = · · · = ˜ v j ∧ ˜ v i . Proposition K ( L ) = | K sz E L (˜ v ⋆ ) | is an A ∞ S o L -algebra spectrum with K ( L ) ∗ ∼ = o L [ v ± ] , complex-oriented bythe morphism M U ∗ → K ( L ) ∗ classifying the chosen Lubin-Tate group lawfor L . (cid:3) Remark
The kernel of o L ⊗ Z M U ∗ → E L ∗ → K ( L ) ∗ is generated by an (infinite) regular sequence (which can be chosen to belongto M U ∗ in degree greater than 2( p n − § k ( L ) with K ( L ) as v n -localization. § III3.1
The spectral sequence of a geometric realization, together with theEilenberg-Moore/K¨unneth spectral sequence for the smash product of mod-ule spectra provide some understanding of the bialgebra( K ( L ) ∧ S L K ( L ) ∗ = K ( L ) ∗ K ( L ) . To begin, note that ( K ( L ) ∧ S L E L ) ∗ = | K sz E L (˜ v ⋆ ) | ∗ ( E L )is the E (Φ L ) ∗ homology of a filtered E (Φ L )-module spectrum, and that the E page of the associated spectral sequence is the Koszul algebraKsz ( E L ∧ S L E L ) ∗ (˜ v ⋆ ) ;but by [17], as in § E L ∧ S L E L ) ∗ ∼ = H Aut(Φ L ) , with deformation param-eters acting as left ˜ v ∗ -multiplication. This sequence is regular, so this is aresolution, and the spectral sequence collapses to an isomorphism( K ( L ) ∧ S L E L ) ∗ ∼ = K ( L ) ∗ ⊗ E L ∗ H Aut(Φ L ) ∼ = o L ⊗ W ( k L ) H Aut(Φ L ) of o L -algebras. Now observe that K ( L ) ∧ S L K ( L ) ≃ ( K ( L ) ∧ S L E L ) ∧ E L K ( L ) which is accessible via [12 IV Thm 6.4]. Proposition
The K¨unneth spectral sequence collapses at E to an isomor-phism K ( L ) ∗ K ( L ) ∼ = o L ⊗ E L ∗ H Aut(Φ L ) ⊗ o L Λ ∗ o L ( m L / m L ) , where the term on the right is the exterior algebra on the (free, of rank n − )tangent o L -module to the space of deformations of LT L . Proof
The E -page of this spectral sequence is again a Koszul algebra, nowof the form Ksz ( K ( L ) ∗ ⊗ E L ∗ H Aut(Φ L ) ) ( η ( v ⋆ )) , where the images η ( v i ) = X α v i,α ⊗ g i,α ∈ m L H Aut(Φ L ) of the generators v i under the right unit have coefficients v i,α in the ideal m L ([22 Thm 3.1], see § E L ∗ → K ( L ) ∗ .The homology of this DGA is therefore just its underlying graded algebra,which can be identified with the algebra of Galois-equivariant functions fromAut(Φ L ) to the exterior algebra on m L / m L . [Note that E L ∗ E L is analogous[by Kodaira-Spencer theory, cf . [15 ex 2.8.1, 20]] to an algebra of functionsfrom Aut(Φ L ) to the symmetric algebra on m L / m L .] (cid:3) It seems reasonable to conjecture that this spectral sequence collapseimplies an interpretation of K ( L ) ∗ K ( L ) as a Hopf algebroid of functions on a(super, ie nontrivially Z -graded) groupoid scheme, an extension of the auto-morphism group of LT L by an exterior algebra of deformations parametrizedby its tangent space as a point in Spf E L ∗ . However, the author feels that thisand related questions ( e . g . the possible nontriviality of such extensions, theaction of Gal( L/ Q p ) and its previously mentioned relation to Weil groups,Massey product structures [1 § L ∞ of the fields L trab ( § p -adic completion of B Q / Z (regarded as an analog of C P ∞ ), a rigid analytic analog [28] of aLubin-Tate group for L . If the spectra K ( L ) have a natural construction interms of fields like L ∞ , one might hope for the existence of a generalizedChern character or cyclotomic-like trace, mapping k ( L ) Galois-equivariantlyto THH( O L ∞ , Z p ). References
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