aa r X i v : . [ m a t h . K T ] J a n E ∞ AUTOMORPHISMS OF MOTIVIC MORAVA E -THEORIES AARON MAZEL-GEEAbstract.
We apply Goerss–Hopkins obstruction theory for motivic spectra to study the motivic Morava E -theories. We find that they always admit E ∞ structures, but that these may admit “exotic” E ∞ auto-morphisms not coming from the usual Morava stabilizer group. Introduction
Overview.
In this short note, we equip the motivic Morava E -theory spectra with canonical E ∞ structures, and compute their automorphisms as E ∞ ring spectra. We find that these automorphism groupsare (homotopically) discrete, but that they are apparently distinct from the usual Morava stabilizer group.We refer the reader to Theorem 1.1 for a precise statement of our main result, and to Remark 1.4 for adiscussion of these automorphism groups. In Remark 1.5, we explain the precise relationship between ourwork and that of Naumann–Spitzweck–Østvær [NSØ15] on motivic algebraic K-theory (i.e. in the height-1case).Our proof is patterned directly on that of Goerss–Hopkins [GH04, GH] for the ordinary (i.e. non-motivic)Morava E -theory spectra (which is based on much prior work, notably that of Hopkins–Miller [Rez98]).Whereas their proof is based in Goerss–Hopkins obstruction theory for ordinary spectra, our proof uses ourgeneralization [MG] of Goerss–Hopkins obstruction theory to an arbitrary presentably symmetric monoidalstable ∞ -category.The most immediate consequence of the present work is that it endows the motivic cohomology theoriesrepresented by the motivic Morava E -theories with the rich algebraic structure of power operations. However,we also view it as a first step towards a moduli-theoretic construction of a motivic spectrum mmf of motivicmodular forms , in analogy with the ordinary spectrum tmf of topological modular forms [DFHH14]. Asthe construction of tmf has been highly influential in chromatic homotopy theory, so would the constructionof mmf significantly advance the chromatic approach to motivic homotopy theory, which is a highly activearea of research [Voe98, HK01, Vez01, Bor03, Hor, LM07, PPR08, NSØ09b, NSØ09a, Bal10, Isa09, Isa, And,Hoy15a, Hor18, Joa, HO, Ghe].There has been much recent interest in “genuine” operadic structures, e.g. genuine G -spectra with multipli-cations indexed by maps of finite G -sets (instead of just finite sets) [BH15, HH, BHb, BHc, Rub, BP, GW18],as well as analogous structures in motivic homotopy theory [BHa]. We do not contend with such structureshere. However, we are optimistic that our generalization of Goerss–Hopkins obstruction theory admits afairly direct enhancement to one that would handle them in a formally analogous way. Thereafter, it seemsquite plausible that the present work would admit a straightforward extension to give “motivically genuine” E ∞ structures on the motivic Morava E -theory spectra.0.2. Conventions. • We write S p mot for the (presentably symmetric monoidal stable) ∞ -category of motivic spectra. This comes equipped with a distinguished group of invertible objects G = { S i,j } i,j ∈ Z ∼ = Z × Z , Date : January 18, 2019. The works [Ric, GIKR] take a different approach, producing motivic spectra over R and C whose cohomologies coincidewith that expected of mmf (in analogy with tmf ). These constructions are indirect, and relatively specific to the chosen basefields; in particular, the resulting motivic spectra are not manifestly related to any theory of elliptic motivic spectra. We implicitly work over a regular noetherian base scheme of finite Krull dimension, but this is only in order to employthe results of [NSØ09b]. We will additionally use a result of [GS09], which requires a (not necessarily regular) noetherian basescheme of finite Krull dimension. AARON MAZEL-GEE the motivic sphere spectra : the unit object is = S , , its categorical suspension is Σ = S , , andthen by definition we have Σ ∞ G m = S , . In particular, it follows that S , = Σ ∞ P . • For any X ∈ S p mot , we write X ∗∗ = π ∗∗ X for its bigraded homotopy groups, i.e. X i,j = π i,j X =[ S i,j , X ] S p mot . Additionally, we write X ∗ = π ∗ X for its (2 , X i = π i X = [ S i,i , X ] S p mot . • We write S p motcell ⊂ S p mot for the coreflective subcategory of cellular motivic spectra. This is thesubcategory generated under colimits by the motivic sphere spectra. It can also be characterized asthe subcategory of colocal objects for the “bigraded homotopy groups” functor; in particular, withinthis subcategory, bigraded homotopy groups detect equivalences. • We fix a finite field k of characteristic p >
0, and we fix a formal group law G over k of finite height n . • We write E ( k, G ) for the corresponding Lubin–Tate deformation ring, we write m ⊂ E ( k, G ) forits unique maximal ideal, and we fix an isomorphism E ( k, G ) / m ∼ = k . • We fix a versal deformation G of G over E ( k, G ). To be precise, G is a formal group law over E ( k, G ), and pushes forward to G along the now-canonical map E ( k, G ) → k . Geometrically, thiscorresponds to a pullback G G Spec( k ) Spf( E ( k, G ))of formal groups (where we notationally identify formal group laws with their underlying formalgroups). • We write E top = E top k, G ∈ S pfor the (ordinary) Morava E -theory spectrum corresponding to the pair ( k, G ), coming from theLandweber exact functor theorem (see e.g. [Rez98, Theorem 6.4 and 6.9]) applied to the formalgroup law G over E ( k, G ). To be precise, we have a chosen isomorphism E top ∗ ∼ = E ( k, G )[ u ± ](with | u | = 2), and the degree-( −
2) formal group law G on E top ∗ coming from its complex orientationcorresponds to G via the unit u ∈ E top2 , considered as a degree-0 formal group law on E top ∗ . • We write E = E mot = E mot k, G ∈ S p motcell for the motivic Morava E -theory spectrum corresponding to the pair ( k, G ) coming from the motivicLandweber exact functor theorem of [NSØ09b, Theorem 8.7]. This is cellular by construction, andcomes equipped with a quasi-multiplication (i.e. a multiplication up to phantom maps). Moreover,writing M GL ∈ S p mot for the algebraic bordism spectrum and M U ∈ S p for the complex bordismspectrum, we have isomorphisms E ∗∗ ∼ = M GL ∗∗ ⊗ MU ∗ E top ∗ and E ∗∗ E ∼ = E ∗∗ ⊗ E top ∗ E top ∗ E top , and the structure maps of the Hopf algebroid ( E ∗∗ , E ∗∗ E ) are tensored up from those of ( E ∗ , E ∗ E ). This is known to be E ∞ , by [GH04, Corollary 7.6] (which is precisely the result we generalize here). ∞ AUTOMORPHISMS OF MOTIVIC MORAVA E -THEORIES 3 Acknowledgments.
David Gepner was instrumental in deducing this application of ∞ -categoricalGoerss–Hopkins obstruction theory, and it is a pleasure to acknowledge his help. We would also like toacknowledge Markus Spitzweck for his helpful correspondence, as well as the NSF graduate research fellowshipprogram (grant DGE-1106400) for financial support during the time that this research was carried out.1. E ∞ automorphisms of motivic Morava E -theories We now state the main result.
Theorem 1.1.
The motivic Morava E -theory spectrum E = E mot k, G has a unique E ∞ structure refiningthe ring structure on its bigraded homotopy groups, and as such generates a subgroupoid of CAlg( S p mot ) equivalent to B (Aut CAlg(Comod ( E ∗∗ ,E ∗∗ E ) ) ( E ∗∗ E )) . In particular, its space of automorphisms is discrete.
Lemma 1.2.
Any Landweber exact motivic spectrum satisfies Adams’s condition.Proof.
The proof is almost identical to that of [Rez98, Proposition 15.3]. First of all, the general statementfollows from the universal case of
M GL . In turn, we can present
M GL as a filtered colimit of Thom spectraover finite Grassmannians, which are then dualizable. Let us write this as
M GL ≃ colim α M GL α . So,it only remains to verify that
M GL ∗∗ (D( M GL α )) is projective as an M GL ∗∗ -module. In bidegree (0 , M GL ∗∗ (D( M GL α )) ∼ = ( M GL ∗∗ M GL α ) ∨ , so that here the claim follows from the algebrapresentation of [GS09, Proposition 2.19], which in particular implies (by inducting on the dimension of theGrassmannians) that this algebra itself is actually free as an M GL ∗∗ -module. From here, in an arbitrarybidegree ( i, j ) we then compute that M GL i,j (D(
M GL α )) ∼ = M GL , ( S − i, − j ⊗ D( M GL α )) ∼ = M GL , ( S − i, − j ) ⊗ MGL , M GL , (D( M GL α ))(using the K¨unneth theorem). (cid:3) Observation 1.3.
By definition, E ∗∗ -localization in S p mot is the localization determined by the E ∗∗ -acyclics,i.e. those objects Z such that E ∗∗ Z ∼ = 0. Note that such motivic spectra Z may not be E -acyclic, i.e. itmight still be the case that E ⊗ Z
0. On the other hand, if Z is also cellular, since E is cellular thenso is E ⊗ Z (since S p motcell is a colocalization of S p mot and the symmetric monoidal structure commutes withcolimits in each variable). Thus, when restricted to cellular motivic spectra, the localizations L E and L E ∗∗ agree. This is summarized by the diagram L E ∗∗ (CAlg( S p motcell )) L E ∗∗ (CAlg( S p mot )) L E (CAlg( S p motcell )) L E (CAlg( S p mot ))CAlg( S p motcell ) CAlg( S p mot ) ∼ of ∞ -categories. Proof of Theorem 1.1.
The proof is formally identical to that of [GH04, Corollary 7.6], only we work inthe ∞ -category S p motcell : the key pieces of input are [MG, Theorems 8.5, 8.8, and 8.9], which are respectivelygeneralizations of [GH04, Proposition 5.2, Proposition 5.5, and Theorem 5.8]. The passage from the ordinarycase to the motivic case runs as follows.First of all, a priori we only have a quasi-multiplication on E ∈ S p motcell . However, this suffices to give allthe required structure on its bigraded E -homology groups: these are by definition homotopy classes of mapsout of bigraded spheres, which by definition cannot detect phantom maps. Explicitly, D(
MGL α ) is also a Thom spectrum via the formula D( X ξ ) ≃ X − ξ . AARON MAZEL-GEE
Next, a priori, Goerss–Hopkins obstruction theory in S p motcell using the homology theory E ∗∗ computes amoduli space in L E ∗∗ (CAlg( S p motcell )). However, as explained in Observation 1.3, we have an equivalenceL E ∗∗ (CAlg( S p motcell )) ≃ L E (CAlg( S p motcell )) , and the usual proof that E is E -local then applies (see e.g. [Rav84, Proposition 1.17]). Thus we have E ∈ L E ∗∗ S p motcell , and hence the moduli space that we construct inside of CAlg(L E ∗∗ ( S p motcell )) ≃ L E ∗∗ (CAlg( S p motcell ))is that of an object whose underlying motivic spectrum is indeed E itself.Now, let us turn to the remainder of the proof of [GH04, Corollary 7.6] and its ingredients. We do not carry over the last line (which identifies the relevant automorphism group with an automorphism group in acategory of formal group laws). However, everything else used there is entirely algebraic, and works equallywell replacing ordinary gradings with bigradings. Note that the gradings appearing in [GH04, §
6] arise fromthe external simplicial direction (and the internal gradings play no real role); note too that the “Dyer–Lashofoperations” arising there arise from the algebraic theory given in [May70] (and in particular have nothingwhatsoever to do with operations in motivic homology). (cid:3)
Remark . Using various adjunctions as well as the fact that all morphisms respect bigradings, one canidentify the endomorphism monoid End
CAlg(Comod ( E ∗∗ ,E ∗∗ E ) ) ( E ∗∗ E )(the classifying space of whose maximal subgroup appears in the statement of Theorem 1.1) with the hom-sethom CAlg(Mod E top ∗ ) ( E top ∗ E top , M GL ∗ ⊗ MU ∗ E top ∗ ) . This appears to fall under the auspices of [Rez98, § χ via the pullback diagramSpec( E mot ∗ ) Spec( E top ∗ )Spec( M GL ∗ ) Spec( M U ∗ ) , χ then the group in question should be the group of (strict) automorphisms of the formal group law χ ∗ G over E mot ∗ = M GL ∗ ⊗ MU ∗ E top ∗ . However, we have not managed to verify this claim. If it holds, however, it would be in keeping with thegeneral philosophy that motivic homotopy theory should be thought of as a flavor of parametrized homotopytheory: the pullback of a sheaf over a small space to a larger one will generally admit more automorphismsthan the original sheaf itself.In any case, there is an evident map to this automorphism group from the Morava stabilizer group, whichtherefore acts on the object E mot ∈ CAlg( S p mot ) as well. Moreover, this map should be an inclusion wheneverthe map M U ∗ → M GL ∗ is (indeed, in certain cases the latter is even an isomorphism (see [Hoy15b])). Remark . in [NSØ15], Naumann–Spitzweck–Østvær prove that the motivic algebraic K-theory spectrum KGL (over a noetherian base scheme of finite Krull dimension) admits a unique E ∞ structure refiningthe canonical multiplication on its represented motivic cohomology theory. Meanwhile, Goerss–Hopkinsobstruction theory takes a commutative algebra in comodules and returns the moduli space of realizations.These are not directly comparable: the former addresses the question of E ∞ structures on a given object,while the latter addresses the question of the ∞ -groupoid of objects that realize some chosen algebraicdatum. Moreover, [NSØ15] addresses KGL as an integral object, whereas Theorem 1.1 only applies to E mot k, ˆ G m ≃ KGL ∧ p . However, see Remark 1.4. ∞ AUTOMORPHISMS OF MOTIVIC MORAVA E -THEORIES 5 To clarify, for a variable object X ∈ S p motcell we locate both the main theorem of [NSØ15] as well asTheorem 1.1 in the diagramhom Op (Comm , E nd S p motcell ( X ))hom Op (Comm , E nd ho( S p motcell ) ( X ))) CAlg(Comod ( E ∗∗ ,E ∗∗ E ) ) ≃ S / L E ∗∗ (CAlg( S p motcell )) E ∗∗ M ( − ) (where E nd denotes the endomorphism operad): the two downwards arrows are the settings for the respectivetheorems. • On the one hand, taking X = KGL , there is a canonical point in the set hom Op (Comm , E nd ho( S p motcell ) ( KGL ))which selects the standard multiplication on
KGL in ho( S p motcell ). The main theorem of [NSØ15] canthen be interpreted as saying that the fiber over this point is nonempty and contractible. • On the other hand, Goerss–Hopkins obstruction takes an algebraic object in CAlg(Comod ( E ∗∗ ,E ∗∗ E ) ) ≃ and provides a spectral sequence converging to the homotopy groups of its moduli space of realizations(which in our case collapses), considered as a subgroupoid of the ∞ -category L E ∗∗ (CAlg( S p motcell )).The inclusion of this subgroupoid is the target of this algebraic object under the lower vertical map.A toy example illustrating the difference between these two approaches is the difference between E ∞ structures on a fixed two-element set (of which there are four) and the moduli space of such objects inCAlg( S et) (which consists of two discrete components). These two approaches are both explored in themore sophisticated setting of algebras over an operad in [Rez96].Note that the horizontal map in this diagram may not be injective: it is a priori possible that distinctmultiplications on X in ho( S p motcell ) might induce the same commutative algebra object structure on E ∗∗ X ∈ Comod ( E ∗∗ ,E ∗∗ E ) . This represents a further obstruction to a direct comparison of these two approaches tothe realization problem. References [And] Michael Andrews,
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