Periodic self maps and thick ideals in the stable motivic homotopy category over \mathbb{C} at odd primes
aa r X i v : . [ m a t h . A T ] J a n Periodic self maps and thick ideals in the stablemotivic homotopy category over C at odd primes Sven-Torben StahnJanuary 25, 2021
Abstract
In this article we study thick ideals defined by periodic self maps in thestable motivic homotopy category over C . In addition, we extend someresults of Ruth Joachimi about the relation between thick ideals definedby motivic Morava K-theories and the preimages of the thick ideals in thestable homotopy category under Betti realization. Contents AK ( n ) . . . . . . . . . . . . . . 12 v n -self maps 184 Existence of a self map on X n
265 The relation of C η and C AK ( n )
366 A counterexample to a statement about thick subcategories in[Joa] 38
There are two famous results by Hopkins and Smith in [HS] that provide acomplete description of the thick subcategories in the stable homotopy category1f finite topological spectra.
Definition 1.1.
A thick subcategory of a tensor triangulated category is anonempty, full, triangulated subcategory that is closed under retracts. A thickideal is a thick subcategory that is closed under tensoring with arbitrary objects.
The thick subcategory theorem states that if we localize at a prime l thethick subcategories (in fact thick ideals) of the category SH fin ( l ) are given by achain SH fin ( l ) = C ) C ) C ) ... ) C ∞ = { } and each thick ideal C i +1 , ≤ i < ∞ , is characterized by the vanishing of the i -th Morava K-theory K ( i ), where K (0) = H Q by convention. The periodicitytheorem states that these thick subcategories can also be described by the prop-erty of admitting a special kind of periodic self map; a so called v n -self map thatinduces an isomorphism in K ( n ) and nilpotent maps in K ( m ) , m = n . Usingthe older Nilpotence theorem of Devinatz, Hopkins and Smith in [DHS], Hop-kins and Smith showed that the full subcategory C v n of finite spectra admittingsuch self maps is in fact thick, and thus equal to one of the categories C i . Foralgebraic reasons (see [Rav2, 3.3.11]) the category C v n must be nested in thefollowing way: C n +1 ⊂ C v n ⊂ C n Therefore, by the thick subcategory theorem, the existence of at least one spec-trum X n in C n admitting such a self map proves the equality C v n = C n . Using anearlier construction of Smith, they prove that there is indeed such a spectrum X n that admits a v n -self map.The fact that the motivic Hopf map η is not nilpotent suggests that the pic-ture looks very different in the motivic context, even over the complex numbers.Ruth Joachimi showed in her dissertation that algebraic Morava K-theories,originally defined by Borghesi, define a similar chain of thick subcategories C AK ( n ) for odd primes over the base field C ([Joa, 9.6.4]), but also that thereare a more thick ideals in the motivic homotopy category ([Joa, Chapter 7]).In addition, she relates the thick ideals C AK ( n ) to the thick ideals thickid( c C n )and R − ( C n ) provided by the classical thick ideals via the constant simplicialpresheaf and Betti realization functors, respectively.The purpose of this article is to explore the motivic equivalents of the con-structions by Hopkins and Smith. In Theorem 3.11, we prove that periodicmotivic self maps defined by algebraic Morava K-theory define a thick subcat-egory, but we need to make use of a conjectural weakened version of a motivicnilpotence lemma. In Theorem 4.13 we lift a construction by Hopkins andSmith in [HS] to the motivic world to show that examples of these self mapsexist. Finally, in the last two sections, we use some of our computations inthe preceding sections to settle [Joa, Conjecture 7.1.7.3]. We furthermore pro-vide a counterexample to the asserted inclusion thickid( c C ) ⊂ C AK (1) in [Joa,2hapter 9, last section] and we identify an error in [Joa, Proposition 8.7.3], onwhich the assertion is based. The counterexample also proves that the inclusion C AK (1) ⊂ R − ( C ) is actually proper and hence that the thick subcategoriesdefined by algebraic Morava K-theories are distinct from the preimages of thetopological thick ideals under Betti realization.This research was originally part of my dissertation under supervision byJens Hornbostel, to who I am grateful for his support. It was conducted inthe framework of the research training group GRK 2240: Algebro-GeometricMethods in Algebra, Arithmetic and Topology , which is funded by the DFG.
We work in the motivic stable homotopy category SH C , whose objects are P -spectra of motivic spaces over the base field C . The construction of thiscategory is due to Voevodsky and Morel (see [Voe] and [MV]) and mimicksthe construction of the topological stable homotopy category, where smoothschemes take the place of topological spaces. There are two kinds of spheresin the motivic world, a simplicial and a geometric one; therefore suspensions,homotopy, homology and cohomology are all not singly graded but bigraded;and there are two common conventions for how to grade them. We index themaccording to the following convention: Definition 2.1.
Define S , as the P -suspension spectrum of the simplicialsphere ( S , and S , as the P -suspension spectrum of ( A − , . The sus-pension spectrum of P is then equivalent to S , . Define S p,q .. = ( S , ) ∧ ( p − q ) ∧ ( S , ) q . This relates to the other common notation of S α = S , by S p,q = S p − q + qα .The motivic homotopy groups of a motivic spectrum X ∈ SH k are then definedas: π p,q ( X ) .. = [ S p,q , X ] SH k There is a topological realization functor R : SH C → SH top called Betti re-alization. There are many reviews of the construction and basic properties ofthis functor. We rely on the account in [Joa, 4.3]. Betti realization maps thesuspension spectrum of a smooth scheme over C to the suspension spectrum ofthe topological space of its complex points, endowed with the analytic topology.In particular the image of the motivic sphere S p,q under Betti realization is thetopological sphere S p . This functor is a strict symmetric monoidal left Quillenfunctor.Because Betti realization maps the motivic spheres to the topological ones,it induces maps on homotopy groups R : π pq ( X ) → π p ( R ( X ))3or every motivic spectrum X ∈ SH C . Therefore for every motivic spectrum E ∈ SH C it also induces maps R : E pq ( X ) → R ( E ) p ( R ( X ))and R : E pq ( X ) → R ( E ) p ( R ( X ))on homology and cohomology associated to that spectrum. Betti realization hasa strict symmetric monoidal right inverse c : SH top → SH C called the constant simplicial presheaf functor. It is a result of Levine(see [Lev,Theorem 1]) that c is not only faithful but also full. We intent to construct an example v n -self map v : X n → X n on the motivicequivalent of the space X n used by Hopkins and Smith. This space is constructedas a retract of a finite cell spectrum. In classical topology, a retract of a finitecell spectrum is a finite cell spectrum again, but this does not necessarily need tobe the case motivically. Therefore, we want to consider the slightly larger thickenvelope SH qfin C (defined in 2.2) of the subcategory of finite spectra in SH C inthe definition of motivic v n -self maps and for the study of thick subcategoriescharacterized by v n -self maps. In contrast to classical algebraic topology, notall motivic spectra are cellular in the following sense: Definition 2.2.
1. The category of cellular spectra SH cellk in SH k is defined(c.f. [DI2, Definition 2.1]) as the smallest full subcategory that satisfies • The spheres S p,q are contained in the subcategory SH cellk . • If a spectrum X is contained in the subcategory SH cellk , then so areall spectra which are weakly equivalent to X . • If X → Y → Z is a cofiber sequence and two of the three spectra arecontained in the subcategory SH cellk , then so is the third. • The subcategory SH cellk is closed under arbitrary colimits.2. The subcategory of finite cellular spectra SH fink in SH k is defined simi-larly as the smallest full subcategory that satisfies the first three conditions(see [DI2, Definition 8.1]).3. We define the category of quasifinite cellular spectra SH qfink as the small-est full triangulated subcategory of SH k that contains SH fink and is closedunder retracts. The spectra in SH qfink are exactly finite cell spectra andtheir retracts, since the cofiber of two retracts of finite cell spectra is aretract of a finite cell spectrum by the octahedral axiom. . By [Roen, Lemma 2.2] a motivic spectrum is cellular if and only if itadmits a cell presentation, i.e. it can be built by successively attachingcells S s,t . A motivic cell spectrum X is called of finite type if it admitsa cell presentation with the following property: there exists a k ∈ N suchthat there are no cells in dimensions satisfying s − t < k and such thatthere exist only finitely many cells in dimensions ( s + t, t ) for each s . For the sake of studying periodic self maps it is useful to consider one prime ata time, because these maps are detected by a collection of cohomology theoriescalled Morava K-theory, which are defined with regard to a specific prime. Inour case this prime will usually be odd, i.e. different from two. Topologicallyone can implement this by studying the localized or completed homotopy cat-egory via the tool of Bousfield localization at an appropiate Moore spectrum.Motivically this works as well (a discussion of this in the motivic setting can befound in [RO, Section 3]): We define the l -completed motivic homotopy cate-gory SH ∧ k,l as the Bousfield localization of the category SH k at the mod-l Moorespectrum S/l . Definition 2.3.
Let l be any prime number, and let X be a motivic spectrumin SH k . The l -completion X ∧ l of X is the Bousfield localization of X at themod- l Moore spectrum
S/l . One can also describe this completion as: X ∧ l .. = L S/l X ≃ holim ← X/l n Definition 2.4.
We define the subcategory SH ∧ ,cellk,l of l -complete cellular spec-tra in SH ∧ k,l as the full subcategory of l -completions of cellular spectra. Sim-ilarly, we define the subcategories SH ∧ ,fink,l of l -complete finite cellular spectraand SH ∧ ,qfink,l of l -complete quasifinite cellular spectra as the full subcategoriesof l -completions of spectra in SH fink and SH qfink . We are going to make use of Spanier-Whitehead duality when we study periodicself maps. The sources we want to quote use different, but equivalent defini-tions of dualizability, so we collect a number of basic definitions and facts aboutSpanier-Whitehead duality that we are going to use in one place. Our primarysource is [LMS, III.1] where categorical duality is explained with great detail.Consider a spectrum X in SH k or SH ∧ k,l . Both categories are closed symmetricmonoidal categories (see [Jar]), and therefore for an arbitrary motivic spectrum5 there exists a function spectrum F ( X, Y ). The unit and counit of the canon-ical tensor-hom adjunction are given by maps η X,Y : X → F ( Y, X ∧ Y )and by the evaluation ǫ X,Y : F ( X, Y ) ∧ X → Y and furthermore there is a natural pairing F ( X, Y ) ∧ F ( X ′ , Y ′ ) → F ( X ∧ X ′ , Y ∧ Y ′ )which provides a natural map ν X,Y : F ( X, S ) ∧ Y → F ( X, Y )by specializing to the case X ′ = Y = S and using the fact F ( S, Y ′ ) ∼ = Y ′ . Proposition 2.5.
Let X be a spectrum in SH k or SH ∧ k,l . Then the followingthree conditions are equivalent:1. The canonical map ν X,Y : F ( X, S ) ∧ Y → F ( X, Y ) is an isomorphism for all spectra Y .2. The canonical map ν X,X : F ( X, S ) ∧ X → F ( X, X ) is an isomorphism.3. There is a coevaluation map coev : S → X ∧ F ( X, S ) such that the diagram S coev / / η S,X (cid:15) (cid:15) X ∧ F ( X, S ) T (cid:15) (cid:15) F ( X, X ) F ( X, S ) ∧ X ν X,X o o commutes, where T denotes the transposition map.Proof. Clearly the first point implies the second. The second point implies thethird, because one can define coev as the composite T ◦ ν − X,X ◦ η S,X . Finally,the third point implies the first (c.f. [LMS, Proposition III.1.3(ii)]) because onecan define an inverse to ν X,Y : F ( X, S ) ∧ Y → F ( X, Y )6s the following composite: ν − X,Y : F ( X, Y ) ∼ = F ( X, Y ) ∧ S id ∧ coev −→ F ( X, Y ) ∧ X ∧ F ( X, S ) ǫ X,Y ∧ id −→−→ Y ∧ F ( X, S ) T −→ F ( X, S ) ∧ Y Definition 2.6. If X satisfies any of the preceding conditions, it is called strongly dualizable .The spectrum DX = F ( X, S ) is called the (motivic) Spanier-Whitehead dual of X . By definition, D .. = F ( − , S ) is a contravariant functor D : SH k → SH k and similarly D .. = F ( − , S ∧ l ) is a contravariant functor D : SH ∧ k,l → SH ∧ k,l on the category of l -complete spectra. In fact, the obvious map F ( − , S ) → F ( − , S ∧ l ) is a completion at l , but we will neither need nor prove it. We will need the following general facts about strongly dualizable spectra,which are proven in [LMS, Proposition III.1.3 (i, iii)]:
Lemma 2.7.
1. If X is strongly dualizable, then DDX ∼ = X .2. If X and Y are strongly dualizable, then the natural map F ( X, S ) ∧ F ( Y, S ) → F ( X ∧ Y, S ) is an isomorphism. In particular, X ∧ Y is strongly dualizable. The spectrum DX ∧ X has the structure of a homotopy ring spectrum bythe same arguments as in[Rav2, Proof of Corollary 5.1.5]: Remark 2.8. If X is strongly dualizable, then the unit map e : S η S,X −→ F ( X, X ) ∼ = F ( X, S ) ∧ X = DX ∧ X and the multiplication map µ : DX ∧ X ∧ DX ∧ X D ( e ) −→ DX ∧ S ∧ X ∼ = DX ∧ X endow DX ∧ X with the structure of motivic homotopy ring spectrum (in factan A ∞ -structure, but we are not going to use or prove it), where we use X ∧ DX ∼ = DDX ∧ DX = D ( DX ∧ X ) in the definition of D ( e ) . emma 2.9. The functor D maps cofiber sequences to cofiber sequences, andthe full subcategory of strongly dualizable spectra in SH k is thick.Proof. For the first statement, let X → Y → Z be a cofiber sequence. Because SH k is the homotopy category of a pointed monoidal model category, the functor F ( − , A ) maps cofiber sequences to fiber sequences for any A in SH k (c.f. [Hov,6.6]). In particular this is true for D ( − ) = F ( − , S ). Because SH k is stable,fiber and cofiber sequences agree, and DZ → DY → DX is a cofiber sequenceagain.For the second statement we only need to show that a retract of a stronglydualizable spectrum is again strongly dualizable, so let A be a retract of astrongly dualizable spectrum X . Note that by the first point of 2.5 we have toshow that the canonical map F ( A, S ) ∧ Y → F ( A, Y )is an isomorphism for all motivic spectra Y , and we already now this statementis true if we replace A with X . But this follows immediately from the followingdiagram: F ( X, S ) ∧ Y id (cid:6) (cid:6) ∼ = / / (cid:15) (cid:15) F ( X, Y ) (cid:15) (cid:15) id (cid:9) (cid:9) F ( A, S ) ∧ Y / / I I F ( A, Y ) U U Lemma 2.10.
All spectra in SH qfin C are strongly dualizable in SH C , and SH qfin C is closed under taking duals.As a consequence, all spectra in SH ∧ ,qfink,l are strongly dualizable in SH C ( l ) , and SH ∧ ,qfink,l is closed under taking duals.Proof. Finite cell spectra are contained in the thick subcategory of compactspectra, and compact spectra are dualizable(See [NSO, Remark 4.1] or [Joa,5.2.7]). Therefore the thick subcategory generated by finite cell spectra is dual-izable.To show that SH qfin C is closed under taking duals, we only have to check thatthe duals of finite cell spectra and their retracts are in SH qfin C again by 2.2.This is true for finite cell spectra by cellular induction, because the duals ofsuspensions of the sphere spectrum are suspensions of the sphere spectrum. If X is a retract of a spectrum F ∈ SH qfin C such that DF ∈ SH qfin C , with maps r : F → X and s : X → F such that r ◦ s = id X , then DX is a retract of DF ∈ SH qfin C with maps Ds : DF → DX and Dr : DX → DF because Ds ◦ Dr = id DX . 8 .4 The motivic Steenrod algebra and the dual motivicSteenrod algebra One key ingredient for the Adams spectral sequence is knowlegde of the Steenrodalgebra or of the dual Steenrod algebra. Motivically, the Steenrod Algebra wasdescribed by Voevodsky for fields of characteristic zero and later by Hoyois,Kelly and Østvær in positive characteristic. While some interesting phenomenashappen at the prime two, the motivic Steenrod algebra is more closely related tothe classical topological Steenrod algebra at odd primes. To describe the motivicSteenrod algebra it is sufficient to know the coefficients of motivic coholomogywith Z /l Z -coefficients: Proposition 2.11.
For l = 2 a prime and k = C the coefficients H Z /l ∗∗ ofmotivic cohomology are given as a ring by H Z /l ∗∗ ∼ = Z /l [ τ ] with | τ | = (0 , , and the image of τ under Betti realization is nonzero.Proof. We know that H Z /l ∗∗ = 0 for q < p ((cf. [MVW, Theorem 3.6])).Let q ≥ p . Then there is an isomorphism from motivic to ´etale cohomology: H p,q ( Spec ( k ) , Z /l ) ∼ = H p ´ et ( k, µ ⊗ ql )This isomorphism respects the product structure([GL, 1.2,4.7]).The ´etale cohomology groups H p ´ et ( k, µ ⊗ ql ) can be computed as the Galois co-homology of the separable closure of the base field (in both cases the complexnumbers) with coefficients in the l -th roots of unity. The action of the abso-lute Galois group G is given by the trivial action if k = C and by complexconjugation if k = R : H p ´ et ( k, µ ⊗ ql ) ∼ = H ( G, µ ⊗ ql ( C ))For k = C , these groups all vanish for p = 0 by triviality of the Galois action,and they are Z /l in the degree p = 0 for all q ≥
0. The multiplicative structureis given by the tensor product of the modules.
Remark 2.12.
We denote the image of τ under H Z /l ∗∗ = H Z /l −∗ , −∗ with thesame name. This image has bidegree | τ | = (0 , − . The motivic mod- l Steenrod algebra over basefields of characteristic 0 hasbeen computed by Voevodsky in [Voe2]. The implications for the dual motivicSteenrod algebra are for example written down in the introduction of [HKO].In our special case it has the following shape:
Proposition 2.13.
Let k = C as above, and let l be an odd prime. The dualmotivic Steenrod algebra A ∗∗ and its Hopf algebroid structure can be describedas follows: A ∗∗ = H Z /l ∗∗ [ τ , τ , τ , ..., ξ , ξ , ... ] / ( τ i = 0)9 ere | τ i | = (2 l i − , l i − and | ξ i | = (2 l i − , l i − .The comultiplication is given by ∆( ξ n ) = n X i =0 ξ l i n − i ⊗ ξ i where ξ := 1 , and ∆( τ n ) = τ n ⊗ n X i =0 ξ l i n − i ⊗ τ i We will use the homological motivic Adams spectral sequence to compute thecoefficients of the l -completed motivic Brown-Peterson spectrum ABP ∧ l . Themotivic Adams spectral sequence was inspired by Morels computation of thezeroth motivic stable stem(c.f. [Mor]) and was used by Dugger and Isaksenfor extensive computations over C at the prime 2 (c.f. [DI]). They also useadditional information available in the MASS to deduce new information aboutthe classical Adams spectral sequence. In other work Isaksen has extendedthese computations to the base field R . Generalized motivic Adams spectralsequences can be constructed for E an arbitrary motivic ring spectrum and X a motivic spectrum. Define ¯ E as the cofiber of the unit map S → E . Smashingthe cofiber sequence ¯ E → S → E with ¯ E s ∧ X yields cofiber sequences¯ E ∧ ( s +1) ∧ X → ¯ E ∧ s ∧ X → E ∧ ¯ E ∧ s ∧ X giving rise to the following tower, called the canonical E ∗∗ -Adams resolution: ... / / (cid:15) (cid:15) ¯ E ∧ ( s +1) ∧ X / / (cid:15) (cid:15) ¯ E ∧ s ∧ X / / (cid:15) (cid:15) ... / / (cid:15) (cid:15) ¯ E ∧ X / / (cid:15) (cid:15) X (cid:15) (cid:15) ... E ∧ ¯ E ∧ ( s +1) ∧ X E ∧ ¯ E ∧ s ∧ X ... E ∧ ¯ E ∧ X E ∧ X The long exact sequences of homotopy groups associated to these cofiber se-quences forms a trigraded exact couple π ∗∗ ( ¯ E ∧∗ ∧ X ) / / π ∗∗ ( ¯ E ∧∗ ∧ X ) v v ❧❧❧❧❧❧❧❧❧❧❧❧❧ π ∗∗ ( E ∧ ¯ E ∧∗ ∧ X ) h h ❘❘❘❘❘❘❘❘❘❘❘❘❘ and thus give rise to a trigraded spectral sequence E s,t,ur ( E, X ) with differentials d r : E s,t,ur −→ E s + r,t + r − ,ur .If one furthermore assumes that E ∗∗ E is flat as a (left) module over thecoefficients E ∗∗ it is possible to identify the E -term via homological algebra.10n this case one can associate a flat Hopf algebroid to E (See [NSO, Lemma 5.1]for the statement and [Rav, Appendix 1] for the definition and basic propertiesof Hopf algebroids), and the category of comodules over this Hopf Algebroid isabelian and thus permits homological algebra. Because E ∗∗ E is flat over E ∗∗ there is also an isomorphism (see [NSO, Lemma 5.1(i)]) π ∗∗ ( E ∧ E ∧ X ) ∼ = E ∗∗ ( E ) ⊗ E ∗∗ E ∗∗ ( X )allowing us to identify the long exact sequences of homotopy groups of thecanonical E ∗∗ -Adams resolution with the (reduced) cobar complex C ∗ ( E ∗∗ ( X )).For this reason the resolution is also referred to as the geometric cobar complex.The E - page of the E -Adams spectral sequence can then be described as: E s,t,u ( E, X ) = Cotor s,t,uE ∗∗ ( E ) ( E ∗∗ , E ∗∗ ( X ))Here Cotor denotes the derived functors of the cotensor product in the cate-gory of E ∗∗ ( E )-comodules and can be computed as the homology of the cobarcomplex C ∗ ( E ∗∗ ( X )). Remark 2.14.
Assume now that k = C . Then Betti realization induces a mapof spectral sequences R E,X : E s,t,ur ( E, X ) → E s,tr ( R ( E ) , R ( X ))This can be checked by going through the definitions: Because Betti real-ization preserves cofiber sequences and smash products, we have R ( ¯ E ) = R ( E ),and the realization of the canonical E ∗∗ -Adams resolution for X is the canoni-cal R ( E ) ∗ -Adams resolution for the topological spectrum R ( X ). If we considerthe induced maps on the long exact sequences of homotopy groups defining theexact couple, we get the following commutative diagram: ... / / R (cid:15) (cid:15) π p, ∗ ( ¯ E ∧ ( s +1) ∧ X ) / / R (cid:15) (cid:15) π p, ∗ ( ¯ E ∧ s ∧ X ) / / R (cid:15) (cid:15) π p, ∗ ( E ∧ ¯ E ∧ s ∧ X ) / / R (cid:15) (cid:15) ... R (cid:15) (cid:15) ... / / π p ( R ( E ) ∧ ( s +1) ∧ R ( X )) / / π p ( R ( E ) ∧ s ∧ R ( X )) / / π p ( R ( E ) ∧ R ( E ) ∧ s ∧ R ( X )) / / ... In particular, Betti realization induces a map of exact couples and hence a mapof spectral sequences.Convergence of the spectral sequence has been studied for the case E = H Z /l by Hu, Kriz and Ormsby in [HKO, Theorem 1]. It turns out that over thecomplex numbers, the spectral sequence will just converge to the l -completion X ∧ l of X , which one can either describe as the Bousfield localization of X atthe mod- l Moore spectrum
S/l or explicitely as: X ∧ l .. = L S/l X ≃ holim ← X/l n The homotopy groups of X and its l -completion are related by the followingshort exact sequence([RO, End of section 3]):0 → Ext ( Z /l ∞ , π ∗∗ X ) → π ∗∗ X ∧ l → Hom( Z /l ∞ , π ∗− , ∗ X ) → X = ABP ∧ l , the spectralsequence actually converges strongly because of a vanishing line. AK ( n ) As before we work over the complex numbers, and the prime l will be odd. Inparticular this prime is implicit in the definition of the motivic Brown-Peterson-spectrum ABP and of the algebraic Morava-K-theory spectrum AK ( n ). In thissection we show that the algebraic Morava-K-theory spectra AK ( n ) admit thestructure of a commutative homotopy ring spectrum similar to their classicalcounterparts. These spectra were originally defined by Borghesi in [Bor]. Inaddition we rely on the description provided in [Joa, Def. 6.3.1]: Definition 2.15.
The connective n-th motivic Morava K-theory is defined as Ak ( n ) = ABP/ ( v , ..., v n − , v n +1 , v n +2 , ... ) and the n-th motivic Morava K-theory spectrum AK ( n ) is defined as: AK ( n ) = v − n ABP/ ( v , ..., v n − , v n +1 , v n +2 , ... ) In particular, both spectra are
M GL ( l ) -modules. AK ( n ) and Ak ( n ) are genuinely motivic in the sense that they are derivedfrom the spectrum representing algebraic cobordism. We will need some of theproperties of AK ( n ) proven in [Joa], namely: Remark 2.16.
1. The Betti realization of the (connective) motivic MoravaK-theory is the classical (connective) Morava K-theory ([Joa, Lemma 6.3.2]): R C ( AK ( n )) = K ( n ) and R C ( Ak ( n )) = k ( n )
2. By [Joa, Lemma 6.3.7] the coefficients of algebraic Morava K-theory aregiven by: AK ( n ) ∗∗ = H Z / ( l ) ∗∗ ⊗ Z / ( l ) K ∗
3. If X is a finite motivic cell spectrum such that H Z / ( l ) ∗∗ ( X ) is free over thecoefficients, then the motivic Adams spectral sequence for Y = Ak ( n ) ∧ X will converge strongly to Ak ( n ) ∗∗ ( X ) . (See [Joa, 8.3.3]) At least for odd primes, the topological spectra K ( n ) can be shown to behomotopy ring spectra. As remarked in [Joa, Remark 6.3.3(6)], it is not knownin general if the motivic Morava K-theory spectrum AK ( n ) can be endowedwith the structure of a motivic homotopy ring spectrum. In the special case k = C , l = 2 however Joachimi proved that the spectrum AP ( n ) .. = ABP/ ( v = l, v , ..., v n − ) , M GL , admits a unital homotopy associative product [Joa,9.3], and with the work done by her it is no longer difficult to do the same for AK ( n ).We want to use and extend the results in [Joa, 9.3] and follow the notationused there to make comparison easier. In particular η will not denote the mo-tivic Hopf map in this chapter, but a different map to be defined later. Theonly exception is the name of the prime l , which is referred to as p in [Joa].Let R ∈ SH k be a strictly commutative ring spectrum with multiplication map m : R ∧ R → R and unit map i : S → R . The example that we have in mind is M GL ( l ) , which is a strictly commutative motivic ring spectrum by the reason-ing given in the beginning of [Joa, 9.3].Classically one can study the products on R -modules of the form R/x anduse them to gain information about products on quotients of the form
R/X where X is a countable regular sequence of homogeneous elements. In contrastto the classical situation, the coefficients of M GL ( l ) are not known, but thecoefficients M GL ( l ) /l are. Therefore motivically one has to consider R -modulesof the form R/ ( x, y ).In the section immediately preceding [Joa, 9.3.7] and in the proof of [Joa,Lemma 9.3.8] Joachimi constructs a product on quotients of this form and provesthe following statement: Lemma 2.17.
Let y ∈ π k ′ ,l ′ ( R ) and let x ∈ π k,l ( R ) . Define the R -modules M .. = R/y and N .. = M/x and denote the structure map of M as ν M : R ∧ M → M . Write η ′ for the canonical map η ′ : R → M = R/y and η for the canonical map η : M → N = R/ ( x, y ) . If π k ′ +1 , l ′ ( M ) = 0 and π k +1 , l ( N ) = 0 , there are maps of R -modules µ M : M ∧ M → Mν M,N : M ∧ N → Mµ N : N ∧ N → N aking the following diagrams commute up to homotopy: R ∧ R η ′ ∧ η ′ / / m (cid:15) (cid:15) ∧ η ′ % % ❑❑❑❑❑❑❑❑❑ M ∧ M µ M (cid:15) (cid:15) R ∧ M ν M % % ▲▲▲▲▲▲▲▲▲▲▲ η ′ ∧ rrrrrrrrrr R η ′ / / M (2.17.1) M ∧ M η ∧ η / / µ M (cid:15) (cid:15) ∧ η % % ▲▲▲▲▲▲▲▲▲▲ N ∧ N µ N (cid:15) (cid:15) M ∧ N ν M,N % % ❑❑❑❑❑❑❑❑❑❑❑ η ∧ ssssssssss M η / / N (2.17.2) In particular, if we choose the maps η ′ ◦ i and η ◦ η ′ ◦ i as unit maps, µ M and µ N are unital products on M and N respectively. Furthermore the following result of Joachimi [Joa, Lemma 9.3.8] proves as-sociativity, and we wish to extend it to include commutativity:
Lemma 2.18. If π k ′ +1 ,l ′ ( M ) = π k ′ +2 , l ′ ( M ) = π k ′ +3 , l ′ ( M ) = 0 , then µ M ishomotopy associative.If furthermore π k +1 ,l ( N ) = π k +2 , l ( N ) = π k +3 , l ( N ) = 0 , then µ N is alsohomotopy associative. We need the following lemma of Joachimi [Joa, Lemma 9.3.3] in the proofof commutativity:
Lemma 2.19.
Let R ′ be a (homotopy) ring spectrum, M ′ a left R ′ -module, and π k,l ( M ′ ) = 0 . Then any R ′ -module map ψ : S k,l ∧ R ′ → M ′ is homotopicallytrivial. Proposition 2.20.
Let R be a homotopy ring spectrum and commutative up tohomotopy. Let M and N be quotient modules defined as in 2.17.If π k ′ +1 ,l ′ ( M ) = π k ′ +2 , l ′ ( M ) = 0 , then µ M is homotopy commutative.If furthermore the homotopy groups of N satisfy π k +1 ,l ( N ) = π k +2 , l ( N ) = 0 ,then µ N is also homotopy commutative.Proof. The R module M = R/y is defined by the following cofiber sequence:Σ k ′ ,l ′ R φ → R η ′ → M δ → Σ k ′ +1 ,l ′ R Recall that m : R ∧ R → R is the product on the ring spectrum R . To showthat the product µ M : M ∧ M → M is commutative, it suffices to show θ .. = µ M ◦ (1 − T ) : M ∧ M → M
14s homotopic to the zero map, where T is the transposition map. The map θ ′ .. = ( η ′ ∧ id M ) ◦ θ fits into the following diagram of R modules R ∧ R id R ∧ η ′ (cid:15) (cid:15) m ◦ (1 − T )=0 / / $ $ ❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍❍ R η ′ (cid:15) (cid:15) R ∧ M (cid:15) (cid:15) θ ′ / / M Σ k ′ +1 ,l ′ R ∧ R ¯ θ ′ : : ✈✈✈✈✈✈✈✈✈✈ which commutes by 2.17.1. The top horizontal map is zero up to homotopybecause m is homotopy commutative by assumption, and the first column isthe cofiber sequence defining M , smashed with R . Together, this implies theexistence of the dashed map ¯ θ ′ .Now R is a R ∧ R module via the product map m and we can consider thisdiagram as a diagram of R ∧ R modules. Then proposition 2.19, applied to thering spectrum R ∧ R , implies that ¯ θ ′ is null homotopic by our assumptions onthe homotopy groups of M . Therefore θ ′ is null homotopic as well. We then getthe following commutative diagram for θ : R ∧ M η ′ ∧ id M (cid:15) (cid:15) θ ′ =0 / / & & ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ MM ∧ M (cid:15) (cid:15) θ / / M Σ k ′ +1 ,l ′ R ∧ R id R ∧ η ′ / / Σ k ′ +1 ,l ′ R ∧ M ¯ θ qqqqqqqqqqqq / / Σ k ′ +2 , l ′ R ∧ R ˜ θ O O Once again the first column is a cofiber sequence, which implies the existenceof the dashed map. The composite ¯ θ ◦ ( id R ∧ η ′ ) is null homotopic becausethis diagram is a diagram of R ∧ R modules again, so we can use the sameargument as before. This implies the existence of the dotted map ˜ θ . This map15lso vanishes by the second condition on the homotopy groups of M, which inturn implies that ¯ θ is zero up to homotopy. Therefore θ also vanishes, so µ M ishomotopy commutative.Because we used only the fact that R is a homotopy ring spectrum and notstrict commutativity, and because diagram 2.17.2 in 2.17 commutes, we canthen repeat the same proof with M replacing R and N replacing M . Note thatthis would not have been possible if we worked over R -modules, because it isnot clear that R/x is a strictly commutative ring spectrum again.
Lemma 2.21.
Let k = C and l = 2 . The spectrum AP ( n ) admits a unital,homotopy associative and homotopy commutative product µ AP ( n ) : AP ( n ) ∧ AP ( n ) → AP ( n ) and so do the spectra A i = AP ( n ) / ( v n +1 , ..., v n + i ) .Proof. Except for the statement about commutativity, the first part of thislemma is the content of [Joa, 9.3.9]. The essential argument in the proof of thecited lemma is as follows: if one has a sequence of elements J ⊂ R ∗∗ and oneknows that A .. = R/ ( J −{ x, y } ) is a homotopy associative and commutative ringspectrum, then one can describe the product on R/J ∼ = A/ ( x, y ) ∼ = A ∧ R R/ ( x, y )by( N ∧ R A ) ∧ ( N ∧ R A ) τ −→ ( N ∧ N ) ∧ R ( A ∧ A ) id N ∧ id N ∧ µ A −→ N ∧ N ∧ R A µ N ∧ id A −→ N ∧ A and thus has to prove the vanishing of the obstruction groups to associativityonly after application of ( − ) ∧ R A to the associativity diagram.Now choose R = M GL ( l ) and A = ABP and J such that M GL ( l ) /J = AP ( n ).Then the relevant obstruction groups are trivial because for odd primes l = 2, ABP ∗∗ is concentrated in bidegrees where the first degree is divisible by 4.We can then show that there is a homotopy associative product on AP ( n ) byinduction; because AP ( n ) / ( v , ..., v n ), we only have to do finitely many steps,and we can use the fact (see [Joa, Lemma 9.3.7]) that for any sequence ( l ) ⊂ J ′ : ABP/ ( J ′ ∪ { y } ) ∼ = M GL ( l ) / ( l, y ) ∧ MGL ( l ) ABP/J ′ We can use the same argument to prove commutativity: if we apply ( − ) ∧ R A toall the relevant diagrams in 2.20, we see that the obstructions to commutativitylie in groups π i,j ( M ∧ R A ) and π i,j ( N ∧ R A ) which are trivial because 4 does notdivide i in the relevant bidegrees. Therefore the product on AP ( n ) is in facthomotopy commutative.Now consider the spectra A i . To define them, we add finitely many elements,namely v n +1 , ..., v n + i , to the sequence J . The proof of [Joa, Lemma 9.3.7]carries through verbatim and we can conclude that there is a product map A i ∧ A i → A i . Similarly, because we had to add only finitely many elements to J , we can repeat the induction argument above for the spectra A i . This shows16hat the multiplication on A i is in fact homotopy associative and homotopycommutative.By essentially classical arguments, this allows us to conclude that Ak ( n ) hasthe desired ring structure: Proposition 2.22.
Let k = C and let l be an odd prime. Then the connectivealgebraic Morava K-theory spectrum Ak ( n ) = hocolim −→ A i = ABP/ ( v , v , ...v n − , v n +1 , v n +2 , ... ) admits the structure of a homotopy associative and homotopy commutative mo-tivic ring spectrum.Proof. By [Joa, Corollary 9.3.5] the elements v i , i = n act trivially on Ak ( n ).This is in particular the case for v = l . Therefore [Str, Lemma 6.7] holds for A = M = Ak ( n ) (although Strickland considers rings in R-modules, the onlynecessary modification is replacing the map ρ ∗ by the map ρ ∗ : [ R/ ( l, x i ) ∧ R B, M ] → [ R/x i ∧ R B, M ] → [ B, M ]), and we can use the arguments of [Str,Proposition 6.8] to conclude that the constructed products on A i induce a unital,homotopy associative product on Ak ( n ). As noted in the proof of Stricklandsproposition, this product is commutative if and only if the maps A i → Ak ( n )commute with themself (see [Str, Definition 6.1] for a definition of this notion).Because the product on A i is commutative, this is the case for every map outof A i . Corollary 2.23.
Let k = C and let l be an odd prime. The algebraic MoravaK-theory spectrum AK ( n ) = v − n Ak ( n ) admits the structure of a commutativeand associative motivic homotopy ring spectrum.Proof. We have an isomorphism AK ( n ) ∼ = v − n M GL ( l ) ∧ MGL ( l ) Ak ( n ) and bothsmash factors admit a homotopy commutative and associative product([Str,Proposition 6.6]). Therefore we can endow AK ( n ) with the desired structure asin the proof of 2.21.It remains to show that this product induces the same product structure on AK ( n ) ∗∗ as one would expect from the computation of these coefficients: Lemma 2.24.
The multiplication map µ AK ( n ) : AK ( n ) ∧ AK ( n ) → AK ( n ) induces the multiplication on AK ( n ) ∗∗ given by the multiplication on K ( n ) ∗ andthe isomorphism AK ( n ) ∗∗ ∼ = HZ/l ∗∗ ⊗ Z /l K ( n ) ∗ of [Joa, Lemma 6.3.7].Proof. The proof is similar to the proof of [Joa, Lemma 9.3.10]17
Thick subcategories characterized by motivic v n -self maps Let l be an odd prime and let k = C . The aim of this section is to show thatthe existence of v n -self maps characterizes thick subcategories in SH qfin C andhence also in the motivic homotopy category. We consider only the case n > Definition 3.1.
Let X be a motivic spectrum in SH qfin C or SH ∧ ,qfin C ,l . A map f : Σ t,u X → X is a motivic v n self-map if it satisfies the following conditions:1. AK ( m ) ∗∗ f is nilpotent if m = n AK ( m ) ∗∗ f is given by multiplication with an invertible element of H Q ∗∗ if m = n = 0 .3. AK ( m ) ∗∗ f is an isomorphism if m = n = 0 . As mentioned before, the topological nilpotence theorem is a key ingredient inthe proof that topological finite cell spectra spectra admitting a v n -self mapform a thick subcategory: A map of finite spectra is nilpotent if and only if itinduces zero in all Morava K-theories. The motivic equivalent of this theoremdoes not hold: For example, the motivic Hopf map η is a non-nilpotent map in SH C , but induces the zero map in motivic Morava K-theory for degree reasons.It seems likely however that a weaker version of the theorem applies, where weonly consider maps of a certain bidegree. For the remainder of this subsectionwe assume that the following motivic nilpotence conjecture holds: Conjecture 3.2.
Let k = C , let l be an odd prime and n > be an integer.If X is a motivic spectrum in SH qfin C or SH ∧ ,qfin C ,l and f : Σ p,q X → X is amotivic map such that ( p, q ) is a multiple of (2 l n − , l n − , then: ∀ m ∈ N : AK ( m ) ∗∗ ( f ) = 0 = ⇒ ∃ k ∈ N : f k ≃ AK ( n ).We will show that the kernel of the map induced by Betti realization is pre-cisely the τ -primary torsion elements. To do this, we need to compare K ( n ) ∗ and AK ( n ) ∗∗ -modules, which is only possible after inverting τ . We also needthe fact that the AK ( n )-homology of a (quasi)-finite motivic cell spectrum isfinitely generated over the coefficients:18 emma 3.3. Let X be a motivic spectrum in SH qfin C or SH ∧ ,qfin C ,l . Then1. AK ( n ) ∗∗ ( X ) is finitely presented as an AK ( n ) ∗∗ -module.2. Hom AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X ) , M ) is finitely presented as an AK ( n ) ∗∗ -modulefor every finitely presented AK ( n ) ∗∗ -module M . In particular,End AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X )) is finitely presented.Proof. Note that AK ( n ) ∗∗ is a quotient of a polynomial ring in the three vari-ables v n , v − n , τ over the field F l and hence Noetherian. Therefore a AK ∗∗ -module is finitely presented if and only if it is finitely generated.1. We will show the statement for finite cell spectra by cellular inductionand then show that it also holds for retracts of finite cell spectra. Theclaim is trivially true for the sphere spectrum. If the statement holds fora spectrum, it also holds for retracts of this spectrum because AK ( n ) ∗∗ isNoetherian and submodules of finitely generated modules are again finitelygenerated. It remains to show that if the spectra X and Y in a cofibersequence X f −→ Y g −→ Z satisfy the statement, then so does Z . Considerthe long exact sequence in AK ( n )-homology ... → AK ( n ) ∗∗ ( Y ) g −→ AK ( n ) ∗∗ ( Z ) δ −→ AK ( n ) ∗− , ∗ ( X ) f −→ AK ( n ) ∗− , ∗ ( Y ) → ... associated to this cofiber sequence. We can break it up into short exactsequences in the canonical way:0 → coker( f ) ¯ g −→ AK ( n ) ∗∗ ( Z ) ¯ δ p −→ ker( f )[ − → f )[ −
1] as a submodule of a finitely generated module over a Noetherianring, and coker( f ) as a quotient of a finitely generated module. Thereforethe middle term is also finitely generated.2. By the first part of this lemma, AK ( n ) ∗∗ ( X ) is finitely generated as an AK ( n ) ∗∗ -module. Therefore there is a surjection R k → AK ( n ) ∗∗ from afree and finitely generated AK ( n ) ∗∗ -module R k onto AK ( n ) ∗∗ ( X ). ThenHom AK ( n ) ∗∗ ( R k , M ) ∼ = M k is a free and finitely generated AK ( n ) ∗∗ -module. Because AK ( n ) ∗∗ is aNoetherian ring, Hom AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X ) , M )is finitely generated as a submodule of this finitely generated module.19 emark 3.4. One can regard K ( n ) ∗ and its modules as a bigraded ring andbigraded modules concentrated in degree 0 with respect to the second bidegree.Then every AK ( n ) ∗∗ [ τ − ] -module has the structure of a bigraded K ( n ) ∗ -modulewhere v topn acts via τ l n − v n . (This of course implies that ( v topn ) − acts via τ − l n +1 v − n , so it only makes sense after inverting τ .) With this module struc-ture, AK ( n ) ∗∗ [ τ − ] is free (with basis τ k , k ∈ Z , − l n + 1 < k < l n − ) and inparticular flat as a K ( n ) ∗ -module. Likewise it is flat as an AK ( n ) ∗∗ -module,because it is a localization. We will implicitly use this in the following state-ments and sometimes write − [ τ − ] for − ⊗ AK ( n ) ∗∗ AK ( n ) ∗∗ [ τ − ] , and − [ τ, τ − ] for − ⊗ K ( n ) ∗ AK ( n ) ∗∗ [ τ − ] as an abbreviation. Lemma 3.5. (Compare [DI, 2.7 + 2.8])1. Let X be a motivic spectrum in SH qfin C or SH ∧ ,qfin C ,l .We can define a map of bigraded AK ( n ) ∗∗ [ τ − ] -modules (even a map ofbigraded algebras if X is a ring spectrum) natural in XR : AK ( n ) ∗∗ ( X ) ⊗ AK ( n ) ∗∗ AK ( n ) ∗∗ [ τ − ] → K ( n ) ∗ ( R C ( X )) ⊗ K ( n ) ∗ AK ( n ) ∗∗ [ τ − ] via the assignment x ⊗ τ k R C ( x ) ⊗ τ − q + k where q is the motivic weight of x ∈ AK ( n ) p,q ( X ) .This map is an isomorphism.2. The induced map ¯ R End,X : End AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X ))[ τ − ] → End K ( n ) ∗ ( K ( n ) ∗ ( R C ( X )))[ τ, τ − ] is an isomorphism of bigraded AK ( n ) ∗∗ [ τ − ] -algebras.3. A homogeneous element f ∈ End AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X )) maps to zero un-der the map R End,X : End AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X )) → End K ( n ) ∗ ( K ( n ) ∗ ( R C ( X ))) induced by motivic realization if and only if it is τ -primary torsion.Proof.
1. The statement about naturality and the module/algebra structurefollow from the properties of motivic realization. It remains to show thatthe map is an isomorphism for spectra X in SH qfin C or SH ∧ ,qfin C ,l . Wewill prove this using cellular induction, and then show that it remains anisomorphism under taking retracts.Consider the case of the sphere spectrum X = S : The map R : AK ( n ) ∗∗ [ τ − ] → K ( n ) ∗ ⊗ K ( n ) ∗ AK ( n ) ∗∗ [ τ − ]20ends τ to τ and v n ∈ AK ( n ) l n − ,l ( n − to v topn ⊗ τ − l n +1 = 1 ⊗ τ − l n +1 τ l n − v n = v n , so it is an isomorphism.If X is a retract of a spectrum F for which the statement holds, then AK ( n ) ∗∗ ( X ) is a direct summand of AK ( n ) ∗∗ ( F ) and all squares in thefollowing diagram commute: AK ( n ) p,q ( X )[ τ − ] AK ( n ) ∗∗ ( s ) / / R X (cid:15) (cid:15) id * * AK ( n ) p,q ( F )[ τ − ] R F ∼ = (cid:15) (cid:15) AK ( n ) ∗∗ ( r ) / / AK ( n ) p,q ( X )[ τ − ] R X (cid:15) (cid:15) K ( n ) p ( R C ( X ))[ τ, τ − ] K ( n ) ∗ ( R C ( s )) / / id K ( n ) p ( R C ( F ))[ τ, τ − ] K ( n ) ∗ ( R C ( r )) / / K ( n ) p ( R C ( X ))[ τ, τ − ]Therefore R X is surjective and injective via a simple diagram chase.Finally, suppose X → Y → Z is a cofiber sequence and the statementholds for X and Y . Then the long exact sequence for AK ( n )-homologymaps to the long exact sequence for K ( n )-homology associated to thecofiber sequence R C ( X ) → R C ( Y ) → R C ( Z ), and the five lemma tells usthat the statement also holds for Z : ... / / ∼ = R X (cid:15) (cid:15) AK ( n ) pq ( Y )[ τ − ] / / ∼ = R Y (cid:15) (cid:15) AK ( n ) pq ( Z )[ τ − ] / / R Z (cid:15) (cid:15) AK ( n ) p − ,q ( X )[ τ − ] / / ∼ = R X (cid:15) (cid:15) ... ∼ = R Y (cid:15) (cid:15) ... / / K ( n ) p ( R C ( Y ))[ τ ][ τ − ] / / K ( n ) p ( R C ( Z ))[ τ ][ τ − ] / / K ( n ) p − ( R C ( X ))[ τ ][ τ − ] / / ...
2. Let M be a finitely presented K ( n ) ∗ -module and N be an arbitrary K ( n ) ∗ -module. As noted in 3.4, AK ( n ) ∗∗ [ τ −
1] is a flat K ( n ) ∗ -module. By [Bour, § K ( n ) ∗ ( M, N )[ τ, τ − ] ∼ = −→ Hom K ( n ) ∗ [ τ,τ − ] ( M [ τ, τ − ] , N [ τ, τ − ])Likewise, let M be a finitely presented AK ( n ) ∗∗ -module and N be anarbitrary AK ( n ) ∗∗ -module. Because AK ( n ) ∗∗ [ τ − ] is a flat AK ( n ) ∗∗ -module, there is also a canonical isomorphism:Hom AK ( n ) ∗∗ ( M, N )[ τ − ] ∼ = −→ Hom AK ( n ) ∗∗ [ τ − ] ( M [ τ − ] , N [ τ − ])The module AK ( n ) ∗∗ ( X ) is finitely presented by 3.3. Specializing to thecase M = N = AK ( n ) ∗∗ ( X ), these two isomorphisms fit in the followingcommutative diagram:End AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X ))[ τ − ] ∼ = / / (cid:15) (cid:15) End AK ( n ) ∗∗ [ τ − ] ( AK ( n ) ∗∗ ( X )[ τ − ]) (cid:15) (cid:15) End K ( n ) ∗ ( K ( n ) ∗ ( R C ( X )))[ τ, τ − ] ∼ = / / End K ( n ) ∗ [ τ,τ − ] ( K ( n ) ∗ ( R C ( X )))[ τ, τ − ])21he first statement of the lemma tells us that K ( n ) ∗ [ τ, τ − ] ∼ = AK ( n ) ∗∗ [ τ − ]and K ( n ) ∗ ( R C ( X )))[ τ, τ − ] ∼ = AK ( n ) ∗∗ ( X )[ τ − ], so the right vertical mapis an isomorphism. It follows that the left vertical map is also an isomor-phism.3. Let P : End K ( n ) ∗ ( K ( n ) ∗ ( R C ( X )))[ τ, τ − ] → End K ( n ) ∗ ( K ( n ) ∗ ( R C ( X )))be the map of K ( n ) ∗∗ -algebras defined by sending τ to 1 and elements ofEnd K ( n ) ∗ ( K ( n ) ∗ ( R C ( X ))) to themselves. Then we have a commutativediagram of K ( n ) ∗ -algebras:End AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X ))[ τ − ] ¯ R End,X (cid:15) (cid:15)
End AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X )) o o R End,X (cid:15) (cid:15)
End K ( n ) ∗ ( K ( n ) ∗ ( R C ( X )))[ τ, τ − ] P / / End K ( n ) ∗ ( K ( n ) ∗ ( R C ( X )))A homogeneous element maps to zero under the top horizontal map if andonly if it is τ -primary torsion. By the second statement of this lemma,the left vertical map is an isomorphism, and there are no homogeneouselements in the kernel of P . All this together implies the desired result. Remark 3.6. If X is strongly dualizable, the map DX ∧ X = F ( X, S ) ∧ X → F ( X, X ) is a weak equivalence, and we have a corresponding isomorphism onhomotopy groups π pq ( X ∧ DX ) ∼ = End( X ) pq . With regard to motivic MoravaK-theory the situation is more complicated. Using Spanier-Whitehead dualitywe have: AK ( n ) pq ( X ∧ DX ) = [ S, AK ( n ) ∧ X ∧ DX ] pq = [ X, AK ∧ X ] pq = [ AK ∧ X, AK ∧ X ] AK,pq
The last term is related to
End AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X )) pq via the Universal co-efficient spectral sequence(c.f [DI2, Prop. 7.7]]. The E -term of this spectralsequence is given by Ext AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X ) , AK ( n ) ∗∗ ( X )) and it converges conditionally to [ AK ∧ X, AK ∧ X ] AK,pq . In particular, if AK ( n ) ∗∗ ( X ) is free or just projective as an AK ( n ) ∗∗ -module, this spectral se-quence collapses at the E -page because it is concentrated in the 0-line, and weget an isomorphism: AK ( n ) pq ( X ∧ DX ) ∼ = End AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X )) pq owever, there is no general reason why AK ( n ) ∗∗ ( X ) should be free or projec-tive. In contrast to this, all graded modules over the graded field K ( n ) ∗ are free,and therefore we always have an isomorphism K ( n ) ∗∗ ( X ∧ DX ) ∼ = End K ( n ) ∗ ( K ( n ) ∗ ( X )) for all finite topological cell spectra X . As a consequence, instead of workingwith AK ( n ) ∗∗ ( X ∧ DX ) , we will work directly with End AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X )) motivically. Every element in AK ( n ) ∗∗ induces a map in End AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X )) givenby multiplication with that element. We will denote this map by the samename as the element. We can now prove the motivic equivalent of asymptoticuniqueness: Lemma 3.7.
Let X be a motivic spectrum in SH qfin C or SH ∧ ,qfin C ,l and f : X → X a motivic v n -self map. Then there exist integers i and j such that: AK ( n ) ∗∗ ( f i ) = v jn Proof.
We will use the classical statement for v topn -self maps in the topologicalstable homotopy category. In addition, it is known that for any unit u in a K ( n ) ∗ -algebra that is finitely generated as a K ( n ) ∗ -module (c.f. [HS, Lemma3.2] or [Rav2, Proof of Lemma 6.1.1]) there is a power of that element such that u i = ( v topn ) j . We will deduce the motivic statement by applying these classicallemmas twice. On the one hand, one can divide out the ideal generated by τ , which yields a finitely generated K ( n ) ∗ -algebra; on the other hand, one canapply Betti realization.Our first step is to show that the map τ : AK ( n ) ∗∗ ( X ) → AK ( n ) ∗∗ ( X )can not be a unit in End AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X )):The element τ ∈ AK ( n ) ∗∗ is not a unit; if we fix the first degree p in AK ( n ) pq and vary the height q , then there is a maximum height such that AK ( n ) pq = 0for all larger heights q . If τ were a unit, all its powers τ k ∈ AK ( n ) , − k wouldneed to have an inverse τ − k ∈ AK ( n ) ,k in arbitrarily high weights, whichis a contradiction to the previous statement. By the same argument the im-age of τ cannot be a unit in any finitely generated AK ( n ) ∗∗ -module. ButEnd AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X )) was finitely generated by 3.3, so the multiplication-by- τ -map cannot be a unit. 23n the second step, we show that the statement is true modulo τ :The motivic v n -self map f induces an isomorphism in AK ( n ) ∗∗ -homology, i.e.a unit in End AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X )). In the previous step we showed that τ cannot be a unit; this implies that it cannot divide AK ( n ) ∗∗ ( f ), for if it did, τ would also be a unit.Therefore AK ( n ) ∗∗ ( f ) does not map to zero under the quotient mapEnd AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X )) → End AK ( n ) ∗∗ / ( τ ) ( AK ( n ) ∗∗ ( X ) / ( τ )))and its image AK ( n ) ∗∗ ( f ) is thus a unit in the second ring.If we forget the second bidegree, AK ( n ) ∗∗ / ( τ ) is isomorphic to K ( n ) ∗ , andEnd AK ( n ) ∗∗ / ( τ ) ( AK ( n ) ∗∗ ( X ) / ( τ )) is a finitely generated K ( n ) ∗ -algebra. In thiscase we know that there are integers i and j such that AK ( n ) ∗∗ ( f ) i = ( v topn ) j .Hence AK ( n ) ∗∗ ( f ) i = v jn + τ ˜ x for some element ˜ x ∈ AK ( n ) ∗∗ ( X ).For the last step, suppose now that ˜ x is τ -primary torsion. For the fixed prime l and any k ∈ N we can consider powers AK ( n ) ∗∗ ( f ) ikl = v jkln + ( τ ˜ x ) kl . If k issufficiently large, the second term vanishes and we are done.Suppose then that ˜ x is not τ -primary torsion. Motivic realization induces a mapEnd AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X )) → End K ( n ) ∗ ( K ( n ) ∗ ( X ))By the classical statement we know that there are integers i ′ and j ′ such that R End,X ( f ) i ′ = ( v topn ) j ′ . Replace i, i ′ and j, j ′ with their products i · i ′ and j · j ′ and call the result i and j again. Then AK ( n ) ∗∗ ( f ) i = v jn + τ ˜ x realizes to v jn , so τ ˜ x realizes to 0. Because τ realizes to 1, ˜ x realizes to 0 and by 3.5 is therefore0 itself. Lemma 3.8.
Assume that the motivic nilpotence conjecture holds. Let X ∈SH qfin C , which implies DX ∈ SH qfin C by 2.10. If f : Σ p,q X → X is a motivic v n -self map and x ∈ π p,q ( DX ∧ X ) is the element corresponding to f under motivic Spanier Whitehead duality, thenthere exists an integer i ∈ N such that x i is in the center of π p,q ( DX ∧ X ) .Proof. The proof is essentially similar to [HS, Lemma 3.5] and [Rav2, Lemma6.1.2], but we will have to use the motivic Nilpotence conjecture at one point.For all a ∈ π ∗∗ ( DX ∧ X ) there is an abstract map of rings ad ( a ) : π ∗∗ ( DX ∧ X ) → π ∗∗ ( DX ∧ X )24efined by ad ( a )( b ) = ab − ba , and the element a is central if and only if ad ( a )is the zero map. This map is realized in homotopy by the composite (here wewrite R for DX ∧ X and T for the transposition map): S p,q ∧ R a ∧ id R → R ∧ R − T → R ∧ R µ → R We also denote this composite by ad ( a ).It now suffices to show that ad ( x ) is nilpotent because of the following classicalformula (proved in [Rav2, Lemma 6.1.2]): ad ( x i )( b ) = i X j =1 (cid:18) ij (cid:19) ad j ( x )( b ) x i − j If we choose i = l N for a sufficiently large N , all summands in this formula van-ish either because of the nilpotence of ad ( x ) or because the binomial coefficientannihilates ad ( x ).Note that AK ( n ) ∗∗ ( DX ∧ X ) is a finitely generated AK ( n ) ∗∗ -algebra that mapsto K ( n ) ∗ ( DR ( X ) ∧ R ( X )) under Betti realization. It follows by the same rea-soning as in the proof of Lemma 3.7 that a suitable power of AK ( n ) ∗∗ ( x ) is givenby v in for some i ∈ N , which is in the image of AK ( n ) ∗∗ in AK ( n ) ∗∗ ( DX ∧ X )and hence central. Replace x with that power and name it x again. Then AK ( n ) ∗∗ ( ad ( x )) is zero, so ad ( x ) is nilpotent by the nilpotence conjecture. Lemma 3.9.
Let X be a motivic spectrum in SH qfin C , ( l ) or SH ∧ ,qfin C ,l . Assumethat the motivic nilpotence conjecture 3.2 holds. If f, g : X → X are two motivic v n -self maps, then there exist integers i, j ∈ N such that f i = g j .Proof. This lemma corresponds to [HS, Lemma 3.6] and [Rav2, Lemma 6.1.3].By the previous two lemmas, we can assume that f and g , after replacing themwith appropiate powers of themselves, commute with each other in regard tocomposition, and furthermore that AK ( n ) ∗∗ ( f i ′ − g j ′ ) = 0 . Using the nilpotence conjecture, we can conclude that f i ′ − g j ′ is nilpotent.Then [HS, Lemma 3.4] gives us the desired statement. Lemma 3.10.
Assume that the motivic nilpotence conjecture 3.2 holds. If f : X → X and g : Y → Y are two v n self maps of X and Y and h : X → Y isany map, then there exist integers i, j ∈ N such that h ◦ f i = g l m ◦ h .Proof. The proof is entirely similar to [Rav2, 6.1.4]
Theorem 3.11.
Let k = C and l be an odd prime. Assume that the mo-tivic nilpotence conjecture 3.2 holds. Then the full subcategories of SH qfin C , ( l ) and SH ∧ ,qfin C ,l consisting of spectra admitting motivic v n -self maps are thick. roof. First we prove that the category of spectra admitting motivic v n -selfmaps is closed under retracts:Let e : X → Y be a retract with right inverse s : Y → X and assume that thereis a v n -self map f : X → X . By 3.8 a power of f commutes with s ◦ e , so e ◦ f ◦ s is a v n -self map.Furthermore the category of spectra admitting motivic v n -self maps is closedunder cofiber sequences:Let X and Y be two spectra with motivic v n -self maps f : Σ a,b X → X and g : Σ c,d Y → Y and let h : X → Y be any map. By 3.10 we can, after replacingthe self maps with suitable powers, assume that ( a, b ) = ( c, d ) and h ◦ f = g ◦ h .Therefore there exists a map k : Σ a,b C h → C h making the following diagramcommute: X h / / Y / / C h Σ a, bX f O O h / / Σ a,b Y / / g O O Σ a,b C hk O O It follows by the five lemma and basic facts about triangulated categories that k is a v n -self map on C h as desired. X n In [HS] Hopkins and Smith used the Adams spectral sequence to prove theexistence of a self map on a spectrum X n constructed by Smith. In this sectionwe use their proof together with a suitable motivic spectrum X n constructed byJoachimi to show that at least one spectrum in SH qfin C or SH ∧ ,qfin C ,l actuallyhas a motivic v n -self map. The classical proof relies on computing K ( n ) p ( X n ∧ DX n ) ∼ = End K ( n ) ∗ ( K ( n ) ∗∗ ( X n )) p via the Adams spectral sequence, so we run into the same kind of problem as inthe previous chapter: Because motivically not all graded modules over AK ( n ) ∗∗ are free, we first have to show that AK ( n ) ∗∗ ( X n ) is in fact free. This also pro-vides us with a K¨unneth isomorphism for products involving AK ( n ) ∗∗ ( X n ).The proof of the existence of a v n -self map also relies on the approximationlemma, which relates the cohomology of the Steenrod algebra in certain degreesto the cohomology of certain subalgebras. We need the motivic analogue of thislemma. To this end we need to make two definitions: Definition 4.1.
1. Let X be a motivic spectrum. Call X k -bounded below if π m,n = 0 for m ≤ k . Similarly, call a bigraded module M m,n over themotivic Steenrod algebra k -bounded below if M m,n = 0 for m ≤ k .2. A module over the motivic Steenrod algebra has a vanishing line of slope m and intercept b if Ext s,t,uA ( M, H Z /l ∗∗ ) = 0 for s > m ( t − s ) + b . Definition 4.2.
1. Let β denote the motivic Bockstein homomorphism, and Sq i resp. P i denote the motivic Square- and Power operations as con-structed by Voevodsky in [Voe2]. If l = 2 , define A n as the subalgebra ofthe motivic Steenrod algebra generated by Sq , Sq , ..., Sq n over H Z ∗∗ l .If l = 2 , define A n as the subalgebra of the motivic Steenrod algebra gen-erated by β, P , ...P n − for n = 0 and by β for n = 0 .2. Fix the monomial Z /l -basis for the dual motivic Steenrod algebra definedby the elements τ , ξ i and τ i (if l = 2 ). The elements P st in the motivicSteenrod algebra are defined as the dual elements to ξ p s t , and the elements Q i are defined as the dual elements to τ i if l = 2 and as Q i = P i +1 in thecase l = 2 .3. Write Λ( Q n ) for the exterior algebra over the ground ring H Z /l ∗∗ in thegenerator Q n . This is a subalgebra of the motivic Steenrod algebra. We can now prove the motivic analogon to the approximation lemma (c.f. [Rav2,6.3.2]):
Proposition 4.3.
Let M be a bounded below module over the motivic Steenrodalgebra such that Ext s,tA ( M, H Z /l ∗∗ ) has a vanishing line of slope m and intercept b .For sufficiently large N the restriction mapExt s,tA ( M, H Z /l ∗∗ ) → Ext s,tA N ( M, H Z /l ∗∗ ) is an isomorphism in degrees s ≥ m ( t − s )+ b ′ , where b ′ can be chosen arbitrarilylow for sufficiently large N .Proof. Define C as the kernel of the surjective map of A -modules A ⊗ A N M → M . As an A N -module C is given by M ⊗ A//A N , where A//A N = A ⊗ A N Z / ( l ) and the bar denotes the augmentation ideal. The motivic Steenrodsquares Sq i live in bidegrees (2 i, i ) if i is even and (2 i + 1 , i ) if it is odd andthe motivic Power operations P i live in bidegrees (2 i ( l − , i ( l − A//A N will be k -bounded below, and k can be chosen arbitrarily high if N issufficiently large. Therefore C has a vanishing line of the same slope as M andarbitrarily low intercept for sufficiently large N , cf. [HS][4.4]. The short exactsequence defining C and the change-of-rings isomorphism for A N and A providethe following diagram: 27xt s − A ( C, H Z /l ∗∗ ) (cid:15) (cid:15) Ext sA ( M, H Z /l ∗∗ ) (cid:15) (cid:15) φ ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ Ext sA ( A ⊗ A N M, H Z /l ∗∗ ) (cid:15) (cid:15) ∼ = / / Ext sA N ( M, H Z /l ∗∗ )Ext sA ( C, H Z /l ∗∗ )If the upper and lower term in the diagram vanish - which is the case above thevanishing line of C - the map φ is the composite of two isomorphisms and hencean isomorphism itself.In [Joa, Theorem 8.5.12] Joachimi defined a motivic cell spectrum X n analogousto the Smith-construction spectrum X n in [HS](see also [Rav2]) by splitting offa wedge summand of a finite cell spectrum via an idempotent. We need some ofthe details of the construction of X n and its properties for the construction ofthe v n -self map, so we recall and collect all those that are relevant in one place: Definition 4.4.
The spectrum X n is defined as X n = e V ( B ∧ k V ( l ) ) = hocolim → B ∧ k V ( l ) → e V B ∧ k V ( l ) → e V ... where • B ( l ) is a motivic l -local finite cellular spectrum defined in [Joa, 8.5], im-plicitly depending on n . • V = H Z /l ∗∗ ( B ( l ) ) = H Z /l ∗∗ ( a, b ) / ( a , b l n ) , where | a | = (1 , and | b | =(2 , ([Joa, 8.5.10]) • k V is an integer dependent on V . • e V is an idempotent of the groupring Z ( l ) [Σ k V ] , which acts on B ∧ k V ( l ) bypermuting the smashfactors and adding maps. • On the level of cohomology, the effect of this idempotent is to split ofa free, nonzero H Z /l ∗∗ -submodule of V ⊗ k V . In particular, the motiviccohomology of X n is bounded below as a module over the Steenrod algebra. Furthermore Joachimi proves the following statements about X n : Theorem 4.5. AK ( s ) ∗∗ ( X n ) = 0 for s < n and AK ( n )( X n ) = 0 ([Joa,Theorem 8.5.12]) . The operation Q n acts trivially on H Z /l ∗∗ ( B ( l ) ) . This follows for degreereasons from the description of H Z /l ∗∗ ( B ∧ k V ( l ) ) in the previous remark.Since H Z /l ∗∗ ( X n ) is a H Z /l ∗∗ -submodule of this module, Q n acts triviallyon H Z /l ∗∗ ( X n ) .3. R ( X n ) = X n . ([Joa, 8.6]) By 2.10 X n is dualizable, and its dual is the retract of a finite cell spectrum.Because the spectrum X n is dualizable, it satisfies the expected relation betweenhomology and cohomology once we show that its cohomology is free: Lemma 4.6.
1. Let E be a cellular motivic ring spectrum and X be a du-alizable cellular motivic spectrum. If E ∗∗ ( X ) is a free module over thecoefficients E ∗∗ , then Hom E ∗∗ ( E ∗∗ ( X ) , E ∗∗ ) ∼ = E ∗∗ ( X ) .2. Let X be a dualizable cellular motivic spectrum such that • H Z /l ∗∗ ( X ) is free over H Z /l ∗∗ • Q n acts trivially on H Z /l ∗∗ ( X ) .Then we have an additive bigraded isomorphismExt s,t,u Λ( Q n ) ( H Z /l ∗∗ ( X ) , H Z /l ∗∗ ) ∼ = H Z /l ∗∗ ( X )[ v n ] where | v n | = (1 , l n − , l n − . Here s is the homological degree and t, u correspond to the internal bidegree. (The result also holds multiplicatively,but we are not going to need this.)Proof.
1. This is the content of [Joa, 8.1.2], using the universal coefficientspectral sequence of [DI2, 7.7] and the fact that this spectral sequencecollapses if E ∗∗ ( X ) is free over E ∗∗ . Note that the cited corollary isstated only for finite cell spectra and the case E = H Z /l , but the onlyproperties of X actually used are cellularity and dualizability, and thatthe proof also works for any cellular motivic ring spectrum E .2. This is a classical result that can be proven in the following way:Consider the following resolution of free Λ( Q n )-modules ... · Q n → Λ( Q n ) · Q n → Λ( Q n ) · Q n → Λ( Q n ) ǫ → H Z /l ∗∗ where the last map is the projection ǫ : Λ( Q n ) → H Z /l ∗∗ and apply( − ) ⊗ H Z /l ∗∗ H Z /l ∗∗ ( X ).The resulting long exact sequence ... · Q n → Λ( Q n ) ⊗ H Z /l ∗∗ H Z /l ∗∗ ( X ) · Q n → Λ( Q n ) ⊗ H Z /l ∗∗ H Z /l ∗∗ ( X ) ǫ → H Z /l ∗∗ ( X )29s a resolution of the Λ( Q n )-module H Z /l ∗∗ ( X ). Here we use the assump-tion that Q n acts trivially on this module in the claim that the last mapis a map of Λ( Q n )-modules.Now apply Hom Λ( Q n ) (( − ) , H Z /l ∗∗ ) and take cohomology. All maps arezero because the target has the trivial Λ( Q n )-module structure. Using theisomorphism from the previous part, we can rewrite degreewise:Hom Λ( Q n ) (Λ( Q n ) ⊗ H Z /l ∗∗ H Z /l ∗∗ ( X ) , H Z /l ∗∗ ) ∼ = Hom H Z /l ∗∗ ( H Z /l ∗∗ ( X ) , H Z /l ∗∗ ) ∼ = H Z /l ∗∗ ( X )Recall that the coefficient rings of the classical Morava K-theories are gradedfields in the sense that all graded modules over it are free. This is not true ofthe motivic Morava K-theories in general. The algebraic Morava K-theory ofthe spectrum X n however is free and finitely generated. To see this, we need togo through the steps of its construction. Proposition 4.7.
Let k = C and l be an odd prime. Then AK ( n ) ∗∗ ( X n ) is afree, finitely generated AK ( n ) ∗∗ -module.Proof. To prove the statement it suffices to show that • AK ( n ) ∗∗ ( X n ) is a finitely generated AK ( n ) ∗∗ -module • AK ( n ) ∗∗ ( X n ) has no τ -torsion.We are going to show both claims in three steps: First we compute Ak ( n )( B ( l ) )using the motivic Adams spectral sequence. We show that it is finitely generatedand does not have τ -torsion, which implies that AK ( n )( B ( l ) ) is finitely generatedand torsionfree. Then we use the K¨unneth theorem to show the same statementfor AK ( n )( B ∧ k V ( l ) ). Finally we use the definition of the idempotent defining X n to show that AK ( n ) ∗∗ ( X n ) satisfies both claims.We begin with the first step: The motivic Adams spectral sequence for Ak ( n ) ∧ B ( l ) converges strongly to Ak ∗∗ ( B ( l ) ) ([Joa, 8.3.3]). We claim that there are nonontrivial differentials in this spectral sequence. The E -term of this motivicAdams spectral sequence can be written asExt Λ( Q n ) ( H Z /l ∗∗ ( B ( l ) ) , H Z /l ∗∗ )by change of rings ([Joa, 8.2.3]). Recall that H Z /l ∗∗ ( B ( l ) ) = H Z /l ∗∗ ( a, b ) / ( a , b l n ).The element Q n acts trivially on this free and finitely generated H Z /l ∗∗ -module,which implies by the previous lemmaExt ∗∗∗ Λ( Q n ) ( H Z / ( l ) ∗∗ ( B ( l ) )) , H Z /l ∗∗ ) ∼ = H Z /l ∗∗ [ v n ] ⊗ H Z / ( l ) ∗∗ H Z / ( l ) ∗∗ ( B ( l ) )30he right hand side is a tensor product of polynomial algebras, and the positionof the polynomial generators and of v n in the spectral sequence imply that theycannot support a nontrivial differential at any stage. In the following sketch ofthe spectral sequence in an abuse of notation a and b denote the dual of thecohomology classes with the same name. Note that the spectral sequence to theright of the depicted area looks very similar to the displayed area - the sameelements appear in the same configuration, just multiplied by some power of v n .In the standard Adams grading the differential d r maps one entry to the leftand r entries up. Thus it is clear that no potentially nontrivial differential canhave a target different from zero.s t − s01 ... ... l n − l n − a bab b ab ... b l n − ab l n − v n av n Therefore Ak ( n )( B ( l ) ) is finitely generated over Ak ( n ) ∗∗ and does not have τ -primary torsion. For all cellular spectra X we have AK ( n ) ∗∗ ( X ) ∼ = v − n Ak ( n ) ∗∗ ( X ).Therefore AK ( n )( B ( l ) ) is free and finitely generated over Ak ( n ) ∗∗ .The second step is now easy: Since AK is a cellular spectrum and since we justproved that the cellular spectrum B ( l ) has free AK -homology over the coeffi-cients, we can apply the K¨unneth theorem (([DI2][Remark 8.7])) and obtain AK ( n ) ∗∗ ( B ∧ k V ( l ) ) ∼ = AK ( n ) ∗∗ ( B ( l ) ) ⊗ k V Therefore also AK ( n ) ∗∗ ( B ∧ k V ( l ) ) is free and finitely generated over the coefficients.For the last step, note that AK ( n ) ∗∗ ( X n ) is a finitely generated AK ( n ) ∗∗ -moduleas well, since it is a submodule of the finitely generated module AK ( n ) ∗∗ ( B ∧ k V ( l ) )over the noetherian ring AK ( n ) ∗∗ .It remains to show that no torsion occurs. The idempotent e V ∈ Z ( l ) [Σ k V ]acts on AK ( n ) ∗∗ ( B ∧ k V ( l ) ) by permutation of the tensor factors and multiplicationby integers. No τ -Torsion can occur in e V ( AK ( n ) ∗∗ ( B ∧ k V ( l ) )) = AK ( n ) ∗∗ ( X n )because the order of an element in a fixed bidegree in AK ( n ) ∗∗ ( B ∧ k V ( l ) ) is thesame as that of the τ -multiples of that element. Consequently, AK ( n ) ∗∗ ( X n ) isa free AK ( n ) ∗∗ -module. Definition 4.8.
Define R = D X n ∧ X n . It is a quasifinite cell spectrum bydefinition and by 2.8 it can be endowed with the structure of a motivic homotopy ing spectrum, with unit map e : S → D X n ∧ X n and multiplication map µ : R ∧ R → R . As a corollary of the preceding proposition we get the following:
Corollary 4.9.
Let k = C and R = D X n ∧ X n . There are K¨unneth isomor-phisms1. AK ( n ) ∗∗ ( R ) ∼ = → AK ( n ) ∗∗ ( D X n ) ⊗ AK ( n ) ∗∗ AK ( n ) ∗∗ ( X n ) AK ( n ) ∗∗ ( R ) ∼ = → AK ( n ) ∗∗ ( D X n ) ⊗ AK ( n ) ∗∗ AK ( n ) ∗∗ ( X n ) Proof.
The Milnor short exact sequence for X n and AK ( n )-cohomology is0 → lim ← AK ( n ) ∗− , ∗ ( B ∧ k V ( l ) ) → AK ( n ) ∗∗ ( X n ) → lim ← AK ( n ) ∗∗ ( B ∧ k V ( l ) ) → e V over which the homotopy colimit defining X n is taken isan idempotent, the system AK ( n ) ∗∗ ( B ∧ k V ( l ) ) is Mittag-Leffler, which implies thatthe lim ← -term vanishes. By the same argument, we have:lim ← AK ( n ) ∗∗ ( D X n ∧ B ∧ k V ( l ) ) ∼ = AK ( n ) ∗∗ ( D X n ∧ X n )Here the limit is taken over the maps id ∧ e V .Since B ∧ k V ( l ) is the l -localization of a finite cell spectrum and AK ( n ) ∗∗ ( B ∧ k V ( l ) ) isa free module over AK ( n ) ∗∗ , we can use the K¨unneth-isomorphism of Duggerand Isaksen [DI2, Remark 8.7] to see that AK ( n ) ∗∗ ( D X n ∧ B ∧ k V ( l ) ) ∼ = → AK ( n ) ∗∗ ( D X n ) ⊗ AK ( n ) ∗∗ AK ( n ) ∗∗ ( B ∧ k V ( l ) )It remains to rewrite the inverse limit over the right hand side: AK ( n ) ∗∗ ( D X n )is a free AK ( n ) ∗∗ -module because AK ( n ) ∗∗ ( X n ) is a free AK ( n ) ∗∗ -module, sousing the earlier isomorphism we get:lim ← (cid:0) AK ( n ) ∗∗ ( D X n ) ⊗ AK ( n ) ∗∗ AK ( n ) ∗∗ ( B ∧ k V ( l ) ) (cid:1) ∼ = AK ( n ) ∗∗ ( D X n ) ⊗ AK ( n ) ∗∗ AK ( n ) ∗∗ ( X n )The K¨unneth-isomorphism in AK ( n )-homology can either be derived from theone in cohomology or from the K¨unneth-isomorphism of the l -local finite cellspectra B ∧ k V ( l ) and the fact that homology commutes with direct limits.We also need the following vanishing line:32 emma 4.10. Let l be odd and R = D X n ∧ X n as before. The A ∗∗ -module Ext A ∗∗ ( H Z /l ∗∗ ( R ) , H Z /l ∗∗ ) has a vanishing line of slope / l n − .Proof. Over odd primes, the motivic Steenrod-algebra is just the classical Steen-rod algebra (where the generators are understood to live in the appropiatemotivic bidegrees) base changed to H Z /l ∗∗ . Similarly, H Z /l ∗∗ ( D X n ∧ X n ) cor-responds to H Z /l ∗ ( DX n ∧ X n ) basechanged to H Z /l ∗∗ , where the generatorsare once again understood to live in the appropiate bidegree.Consequently Ext A ∗∗ ( H Z /l ∗∗ ( D X n ∧ X n ) , H Z /l ∗∗ ), which maps to the classicalExt-term Ext A ∗ top ( H Z /l ∗ ( DX n ∧ X n ) , H Z /l ∗ ), is just that classical Ext-termbase changed to H Z /l ∗∗ and in particular does not contain τ -torsion. Theexistence of the vanishing line then follows from the existence of a vanishingline with the same slope in the classical case for the spectrum X n (see [Rav2,6.3.1]).Furthermore, we need the following duality isomorphisms: Proposition 4.11.
Let R = D X n ∧ X n as before:1. Hom H Z /l ∗∗ ( H Z /l ∗∗ ( R ) , H Z /l ∗∗ ) ∼ = H Z /l ∗∗ ( R ) AK ( n ) ∗∗ ( D X n ) ∼ = AK ( n ) ∗∗ ( X n ) ∼ = Hom AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X n ) , AK ( n ) ∗∗ ) AK ( n ) ∗∗ ( D X n ) ∼ = AK ( n ) ∗∗ ( X n ) ∼ = Hom AK ( n ) ∗∗ ( AK ( n ) ∗∗ ( X n ) , AK ( n ) ∗∗ ) Proof. R = D X n ∧ X n is a dualizable cell spectrum since X n and D X n are. Therefore we can consider the universal coefficient spectral sequenceof [DI2, 7.7]. As explained in [Joa, 8.1.2], this spectral sequence collapses if H Z /l ∗∗ ( R ) is free over H Z /l ∗∗ . (Note that the cited corollary is stated forfinite cell spectra, but the only properties actually used are cellularity anddualizability.) To show the freeness of H Z /l ∗∗ ( R ) as a H Z /l ∗∗ -module,observe that H Z /l ∗∗ ( X n ) is free by construction ([Joa, 8.5.3]). This impliesthe existence of a K¨unneth isomorphism for X n , and thus H Z /l ∗∗ ( R ) = H Z /l ∗∗ ( D X n ) ⊗ H Z /l ∗∗ H Z /l ∗∗ ( X n )is free.2. The first isomorphism follows directly from the canonical bijection. Thesecond isomorphism is proven by the same argument as in the proof ofpart 1, using the universal coefficient spectral sequence [DI2, 7.7] togetherwith the facts that AK is a cellular spectrum and that AK ( n ) ∗∗ ( X n ) isfree over the coefficients.3. This is proven just as in part 1 or part 2.33 orollary 4.12.
1. There exists a well defined coevaluation map coev : AK ∗∗ → AK ∗∗ ( X n ) ∨ ⊗ AK ∗∗ AK ∗∗ ( X n ) Here ( − ) ∨ denotes the linear dual Hom AK ∗∗ ( − , AK ∗∗ ) . It is induced bythe map T ◦ e : S → X n ∧ D X n , where e : S → D X n ∧ X n is the unit mapof R = D X n ∧ X n and T is the map that transposes the two factors.2. Under the composition AK ∗∗ → AK ∗∗ ( R ) → Hom AK ∗∗ ( AK ∗∗ ( X n ) , AK ∗∗ ( X n )) an element v ∈ AK ∗∗ maps to multiplication by that element.Proof.
1. The coevalution map of 2.5, which is the same as T ◦ e , induces theclaimed map in AK ( n )-homology, together with the identification AK ( n ) ∗∗ ( D X n ) ∼ = AK ∗∗ ( X n ) ∼ = Hom AK ∗∗ ( AK ∗∗ ( X n ) , AK ∗∗ )of the preceding proposition. Because AK ( n ) ∗∗ ( X n ) is a free and finitelygenerated AK ( n ) ∗∗ -module, there is also an algebraic coevalution definedvia choosing a basis as for a vector space, and the two maps coincide sincethey both satisfy the equivalent of the condition of the first point of 2.5for projective and finitely generated modules.2. The element 1 ∈ AK ∗∗ maps to the coevaluation of AK ∗∗ ( X n ) under thefirst map, using the identification AK ∗∗ ( R ) ∼ = AK ∗∗ ( X n ) ∨ ⊗ AK ∗∗ AK ∗∗ ( X n )implied by the K¨unneth and duality isomorphisms. Hence an element of AK ∗∗ maps to that element times the coevaluation. The coevaluationmaps to the identity under the second map. Consequently an element in AK ∗∗ times the coevaluation maps to multiplication by that element.We now have all the ingredients to use the classical proof in the motivic setting([HS, Theorem 4.12], see also [Rav2, 6.3]): Theorem 4.13.
Let k = C and l be an odd prime. The spectrum X n has amotivic v n self-map f satisfying AK ( m ) ∗ f = δ mn v p Nm n for a sufficiently large integer N m .Proof. The aim is to construct a permanent cycle v ∈ Ext A ∗∗ ( H Z /l ∗ ( R ) , H Z /l ∗ )that maps to a power of v n in Ak ( n ) ∗∗ ( R ) and to a nilpotent element in Ak ( m ) ∗∗ ( R ) if m = n . The diagram below will specify the meaning of ”maps”.34nder motivic Spanier Whitehead duality such a class corresponds to a self-mapof the described form on X n .The cohomology of the point, H Z /l ∗∗ , is concentrated in simplicial degree0. Therefore the operations Q n act trivially on this module over the motivicSteenrod algebra. They act trivially on H Z /l ∗∗ ( R ) since they act trivially on H ∗∗ ( X n ). If we write P ( v n ) for the polynomial algebra in one generator withrespect to the base ring H Z / ( l ) ∗∗ , this provides us with the following isomor-phisms of trigraded algebras:Ext ∗∗∗ Λ( Q n ) ( H Z /l ∗∗ , H Z /l ∗∗ ) ∼ = −→ P ( v n ) ⊗ H Z /l ∗∗ (4.13.1)Ext ∗∗∗ Λ( Q n ) ( H Z /l ∗∗ ( R ) , H Z /l ∗∗ ) ∼ = −→ P ( v n ) ⊗ H Z /l ∗∗ ( R ) (4.13.2)Here v n has homological degree 1 and internal bidegree (2( l n − , l n − Q n )and A N these fit into the following diagram:Ext A ∗∗ ( H Z /l ∗∗ , H Z /l ∗∗ ) i / / φ (cid:15) (cid:15) Ext A ∗∗ ( H Z /l ∗∗ ( R ) , H Z /l ∗∗ ) φ (cid:15) (cid:15) Ext A ∗∗ N ( H Z /l ∗∗ , H Z /l ∗∗ ) i / / λ (cid:15) (cid:15) Ext A ∗∗ N ( H Z /l ∗∗ ( R ) , H Z /l ∗∗ ) λ (cid:15) (cid:15) Ext Λ( Q n ) ( H Z /l ∗∗ , H Z /l ∗∗ ) i / / ∼ =(1) (cid:15) (cid:15) Ext Λ( Q n ) ( H Z /l ∗∗ ( R ) , H Z /l ∗∗ ) ∼ =(2) (cid:15) (cid:15) P ( v n ) ⊗ H Z / ( l ) ∗∗ H Z /l ∗∗ i / / (cid:15) (cid:15) P ( v n ) ⊗ H Z / ( l ) ∗∗ H Z /l ∗∗ ( R ) (cid:15) (cid:15) Ak ( n ) ∗∗ i / / Ak ( n ) ∗∗ ( R )Step 1: Consider the element f v n ∈ Ext Λ( Q n ) ( H Z /l ∗ ( R ) , H Z /l ∗ ) that corre-sponds to v n ⊗ ∈ P ( v n ) ⊗ H ∗∗ ( R ) under the isomorphism (2). Proposition 4.14. ∀ N ≥ n there is an integer t > and an element x ∈ Ext A N, ∗∗ ( H Z /l ∗ , H Z /l ∗ ) such that λ ( x ) = v tn . The image of x under i is cen-tral in Ext A N, ∗∗ ( H Z /l ∗ ( R ) , H Z /l ∗ ) , where central is meant in respect to gradedcommutativity in the first, but not in the second bidegree.Proof. This statement is a corollary of [HS, Theorem 4.12]. Since the motiviccohomology of the point H Z / ( l ) ∗∗ = Z / ( l )[ τ ] is concentrated in simplicial degree0, the action of the motivic Steenrod algebra is trivial on this module. Hencewe can basechange the statement of the cited theorem to Z / ( l )[ τ ].35tep 2: The module Ext A ∗∗ ( H Z /l ∗∗ ( R ) , H Z /l ∗∗ ) has a vanishing line of slope1 / (2 l −
2) and a fixed intercept b . By the motivic approximation lemma, themorphism φ : Ext A ∗∗ ( H Z /l ∗ ( R ) , H Z /l ∗∗ ) → Ext A ∗∗ N ( H Z /l ∗∗ ( R ) , H Z /l ∗∗ )is an isomorphism above a line with slope 1 / l n −
1) and arbitrarily low in-tercept for sufficiently large N . Since the element x (and therefore also i ( x ))has tridegree ( t, l n − , ( l n − N . Define y ∈ Ext A ∗∗ ( H Z /l ∗∗ ( R ) , H Z /l ∗∗ ) as the preimage of i ( x )under φ . Since i ( x ) is central (in the graded sense with respect to the firstbidegree but not with respect to the second) in Ext A ∗∗ N ( H Z /l ∗∗ ( R ) , H Z /l ∗∗ ), itcommutes with all elements in the image of φ , in particular with all elementsabove the line defined by the approximation lemma.Step 3: The element y and its powers, as well as the images of y and its powersunder the differentials of the motivic Adams spectral sequence all satisfy the re-quirement of the last statement, so they commute with each other. By inductionwe can assume that a power ˜ y of y survives up to the r th page. We wish to showthat ˜ y l is a r -cycle, i.e. d r (˜ y l ) = 0. This is true since d r (˜ y l ) = l · ˜ y l − d r (˜ y ) = 0.After a finite number of pages, the differential will point in the area of the spec-tral sequence above the vanishing line, and we can stop the process. We endwith a power ˜ y of y that is a permanent cycle in the motivic Adams spectralsequence and hence represents an element of Ak ( n ) ∗∗ ( R ).Step 4: The permanent cycle ˜ y represents an element ¯ y ∈ π ∗∗ ( R ). Choose m such that v mn has the same degree as ¯ y . By the exact same arguments as in[HS] we can choose a power of ¯ y such that Ak ( n ) ∗∗ (¯ y g ) = v gmn and define f as themap corresponding to that power of ¯ y under motivic Spanier Whitehead duality.Step 5: For m = n it follows just as in the topological case that the imageof v in AK ( m ) ∗∗ is nilpotent either for trivial reasons ( m < n ) or because ofa vanishing line with tighter slope in the Adams spectral sequence computing Ak ( m ) ∗∗ ( R ) ( m > n ). C η and C AK ( n )As a corollary of the K¨unneth isomorphism, we can settle one of the open con-jectures in Ruth Joachimis dissertation [Joa, Conjecture 7.1.7.3] which concernsthe relation of the thick ideal thickid( C η ) generated by the cone of the motivicHopf map C η and the thick ideals C AK ( n ) characterized by the vanishing ofmotivic Morava K-theory. Lemma 5.1.
Let m ∈ N be any integer. Then the coefficients of the cone C η f η : Σ , S → S in AK ( m ) ∗∗ -homology are given by: AK ( m ) ∗∗ ( C η ) ∼ = AK ( m ) ∗∗ ⊕ AK ( m ) ∗− , ∗− In particular, they are free over AK ( m ) ∗∗ .Proof. The long exact sequence induced by the cofiber sequence S , → S , → C η → S , defining C η splits into short exact sequences0 → AK ( m ) ∗∗ → AK ( m ) ∗∗ ( C η ) → AK ( m ) ∗− , ∗− → η induces the zero map in AK ( m ) ∗∗ -homology. The sequence splitsbecause the outer terms are free AK ( m ) ∗∗ -modules, yielding the result. Corollary 5.2.
Let m ∈ N . In the case m < n we have AK ( m ) ∗∗ ( C η ∧ X n ) ∼ = 0 and in the case m = n we have: AK ( n ) ∗∗ ( C η ∧ X n ) ∼ = AK ( n ) ∗∗ ( C η ) ⊗ AK ( n ) ∗∗ AK ( n ) ∗∗ ( X n ) = 0 Proof.
By the preceding lemma the finite cell spectrum C η has free AK ( m )-homology and thus satisfies the requirements of the K¨unneth formula [DI2,Remark 8.7].Application of the K¨unneth formula yields: AK ( m ) ∗∗ ( C η ∧ X n ) ∼ = AK ( m ) ∗∗ ( C η ) ⊗ AK ( m ) ∗∗ AK ( m ) ∗∗ ( X n )If m < n the factor AK ( m ) ∗∗ ( X ) = 0 vanishes by 4.5. This implies the firstpart of the statement. If m = n the result contains AK ( n ) ∗∗ ( X n ) ⊗ AK ( n ) ∗∗ AK ( n ) ∗∗ = AK ( n ) ∗∗ ( X n ) = 0as a direct summand, so AK ( n ) ∗∗ ( C η ∧ X n ) cannot vanish. Proposition 5.3.
The spectrum X n +1 is contained in the intersection of thickideals thickid ( C η ) ∩ C AK ( n ) , but not in thickid ( C η ) ∩ C AK ( n +1) . In particular,these intersections are nonzero and distinct for all n ∈ N .Proof. Clearly C η ∧ X n +1 is in the thick ideal generated by C η . The precedingcorollary tells us on the one hand that C η ∧ X n +1 ∈ C AK ( n ) , and on the otherhand that C η ∧ X n +1 / ∈ C AK ( n +1) . 37 A counterexample to a statement about thicksubcategories in [Joa]
In this section we construct a counterexample to the inclusionthickid( c C ) ⊂ C AK (1) claimed in [Joa, Chapter 9, last section], based on an error in [Joa, Proposition8.7.3].Let l be an odd prime, and consider the topological mod- l Moore spectrum
S/l ∈ SH . We can easily compute its K (1)-homology: Lemma 6.1. K (1) ∗ ( S/l ) ∼ = K (1) ∗ ⊕ K (1) ∗− Proof.
The Moore spectrum is defined via the cofiber sequence S · l → S → S/l and the map induced by l is trivial in K (1)-homology. Therefore the long exactsequence in K (1)-homology induced by this cofiber sequence splits up into shortexact sequences, and these short exact sequences split because all graded K (1)-modules are free.In [Ada] Adams proved the existence of a non-nilpotent self map v : Σ l − S/l → S/l on the Moore spectrum which induces an isomorphism in K (1)-homology; namelymultiplication by the invertible element v top . Consequently, the K (1)-homologyof the cone C v vanishes: K (1) ∗ ( C v ) = 0, or equivalently C v ∈ C .Applying the constant simplicial presheaf functor c to the construction givesus the cofiber sequence Σ l − , S/l cv −→ S/l → C cv in SH C . The cone C cv of cv is equivalent to c ( C v ) because c is a triangulatedfunctor, and the Moore spectrum is mapped to the Moore spectrum ( cS/l = S/l because cl = l .) We can compute the AK (1)-homology of the mod- l -Moorespectrum using the same argument as in the topological case: AK (1) ∗∗ ( S/l ) ∼ = AK (1) ∗∗ ⊕ AK (1) ∗− , ∗ However, the algebraic Morava K-theory of C cv does not vanish: Lemma 6.2. AK (1)( C cv )) ∼ = AK (1) ∗∗ ( S/l ) / ( τ l − ) = 0 Proof.
The cofiber sequence
S/l cv → S/l → C cv induces a long exact sequence in AK (1)-homology: ... → AK (1) p +(2 l − ,q ( S/l ) AK (1) ∗∗ ( cv ) −→ AK (1) pq ( S/l ) → AK (1) pq ( C cv ) → ... AK (1) ∗∗ ( cv ) must be given by multiplication with τ l − v , becauseBetti realization maps AK (1) ∗∗ ( cv ) to multiplication with v top and there isonly one map realizing to this in the appropiate bidegree. This map is injectivebut, unlike the topological case, no longer an isomorphism. Hence the longexact sequence splits into short exact sequences0 → AK (1) pq ( cS/l ) · τ l − v −→ AK (1) pq ( cS/l ) → AK (1) pq ( C cv ) → v is invertible, the last term is isomorphic to AK (1) ∗∗ ( S/l ) / ( τ l − ).Because C AK (1) was defined by the vanishing of AK (1)-homology and AK (1) ∗∗ ( C cv ) =0 does not vanish, we have C cv / ∈ C AK (1) . On the other hand, we have shownthat C v ∈ C . Because R ( C cv ) = C v , this implies C cv ∈ R − ( C ). Therefore wecan conclude the following corollary from the preceding lemma: Corollary 6.3.
The inclusion C AK (1) ( R − ( C ) is proper. Furthermore we have C cv = cC v ∈ thickid( c C ). Therefore cC v is our desiredcounterexample and proves: Proposition 6.4. thickid ( c C )
6⊂ C AK (1) Remark 6.5.
The mistake on which the incorrect assertion is based occursin [Joa, Proposition 8.7.3]. This proposition states that for a finite topologicalCW spectrum Y , AK ( n ) ∗∗ ( cY ) = 0 if and only if K ( n ) ∗ ( Y ) = 0 . In the proofof this proposition Joachimi shows that the differentials in the motivic Atiyah-Hirzebruch spectral sequence are determined by the differentials of the topologicalAtiyah-Hirzebruch spectral sequence, and that the E -page of the motivic spectralsequence is given by adjoining a generator τ to each entry in the topologicalspectral sequence, where all entries are generated in motivic weight 0. Theproblem that now occurs is that the differentials in the motivic spectral sequencedo not preserve the weight, but lower it. Hence a nontrivial differential cangenerate τ -primary torsion in the spectral sequence. The above example showsthat this in fact happens. This argument can in fact be made for any topological spectrum X ∈ C n +1 \C n +2 .Any such spectrum has nontrivial K ( n )-homology and a self map v : Σ m X → X that induces multiplication by some power of v topn . We know by 3.5 that themap AK ( n ) ∗∗ ( cX ) → K ( n ) ∗ ( X ) induced by Betti realization is surjective andits kernel is exactly the τ -primary torsion elements. In particular we know that39 K ( n ) ∗∗ ( cX ) = 0 , and the self map provides us with a motivic map cv . Thismap induces multiplication by the same power of τ l − v n in AK ( n )-homology -up to a possible error term, which has to be τ -primary torsion. We can eliminatethis error term by taking sufficiently large l -fold powers of this map. We endup with a v topn -self map v ′ of X whose image cv ′ under the constant simplicialpresheaf funtor c induces multiplication by some power of τ l − v n in AK ( n )-homology. In particular, its cone has nonvanishing AK ( n )-homology by thesame argument as for our earlier counterexample and thus proves: Proposition 6.6. thickid ( c C n +1 )
6⊂ C AK ( n ) Just as before, this also proves:
Corollary 6.7.
The inclusion C AK ( n ) ( R − ( C n +1 ) is proper. References [Ada] J. F. Adams,
On the groups J(X) - IV , Topology (1966), Volume5, Issue 1, pp. 21–71[Bor] S. Borghesi,
Algebraic Morava K-theories , Inventiones Mathe-maticae (2003), Volume 151, Issue 2, pp. 381–413[Bour] Nicolas Bourbaki,
Commutative Algebra: Chapters 1-7 ,Springer(1998)[DHS] E. S. Devinatz, M. J. Hopkins, and J. H. Smith,
Nilpotence andstable homotopy theory , Annals of Mathematics (1988), Volume128, Issue 2, pp. 207–242[DI] D. Dugger and D. C. Isaksen,
The motivic Adams spectral se-quence , Geometry & Topology (2010), Volume 14, Issue 2, pp.967–1014[DI2] D. Dugger and D. C. Isaksen,
Motivic cell structures , Algebraic& Geometric Topology (2005), Volume 5, Issue 2, pp. 615-652[GL] T. Geisser and M. Levine,
The Bloch-Kato conjecture and a theo-rem of Suslin-Voevodsky , Journal fuer die reine und angewandteMath. (2001), Issue 530, pp. 55–103[HKO] P. Hu, I. Kriz, and K. M. Ormsby,
Convergence of the motivicAdams spectral sequence , J. K-Theory (2011), Volume 7, Issue3, pp. 573–596 40HS] M. Hopkins and J. Smith,
Nilpotence and Stable Homotopy The-ory II , Annals of Mathematics (1998), Volume 148, Issue 1, pp.1-49[Hor] J. Hornbostel,
Some comments on motivic nilpotence (with anappendix by M. Zibrowius) , Trans. Amer. Math. Soc. (2018),Volume 370, pp. 3001-3015[Hov] M. Hovey,
Model categories , Mathematical Surveys and Mono-graphs, Band 63, AMS[Jar] J.F. Jardine,
Motivic symmetric spectra , Documenta Mathemat-ica (2000), Volume 5, pp. 445-552[Joa] R. Joachimi,
Thick ideals in equivariant and motivic stable ho-motopy categories , Dissertation at the Bergische Universit¨atWuppertal[LMS] L. G. Jr. Lewis, J. P. May, and M. Steinberger,
Equivariant Sta-ble Homotopy Theory , Lecture Notes in Mathematics, Volume1213[Lev] M. Levine,
A comparison of motivic and classical stable homo-topy theories , Journal of Topology (2014), Volume 7, Issue 2, pp.327–362[Mor] F. Morel,
On the motivic π of the sphere spectrum , Axiomatic,enriched, and motivic homotopy theory (2004), pp. 219–260.NATO Sci. Ser. II Math. Phys. Chem., 131, Kluwer Acad.Publ.,Dordrecht[MV] F. Morel and V. Voevodsky, A -homotopy theory of schemes ,Publications Math´ematiques de l’IH´ES (1999), Volume 90, Issue1, pp. 45-143[MVW] C. Mazza, V. Voevodsky, and C. Weibel, Lecture Notes on Mo-tivic Cohomology , Clay Mathematics Monographs (2006), Vol.2[NSO] N. Naumann, M. Spitzweck and P. A. Østvær,
Motivic Landwe-ber Exactness , Documenta Math. (2009), Volume 14, pp. 551-593[RO] O. R¨ondigs and P. A. Østvær,
Rigidity in motivic homotopytheory , Mathematische Annalen (2008), Volume 341, Issue 3,pp. 651-675[Roen] O. R¨ondigs,
The η -inverted sphere , Preprint on the arxiv[Rav] D. C. Ravenel, Complex cobordism and stable homotopy groupsof spheres , Pure and Applied Mathematics (1986/2003), Volume121, Academic Press Inc.41Rav2] D. C. Ravenel,
Nilpotence and periodicity in stable homotopy the-ory , Annals of Mathematics Studies (1992), Volume 128, Prince-ton University Press[Str] N. Strickland,
Products on MU-modules , Transactions of theAmerican Mathematical Society (1999), Volume 351, pp.2569–2606[Voe] V. Voevodsky, A -homotopy theory , Documenta Mathematica(1998), Proceedings of the International Congress of Mathemati-cians, Vol. I, pp. 579–604[Voe2] V. Voevodsky, Reduced power operations in motivic cohomology ,Publ. Math. Inst. Hautes ´Etudes Sci. (2003), Volume 98, pp.1-57