aa r X i v : . [ m a t h . A T ] O c t REMARKS ON MOTIVIC MOORE SPECTRA
OLIVER R ¨ONDIGS
Abstract.
The term “motivic Moore spectrum” refers to a cone of an element α : Σ s,w → in the motivic stable homotopy groups of spheres. Homotopygroups, multiplicative structures, and Voevodsky’s slice spectral sequence arediscussed for motivic Moore spectra. Introduction
Let R be a ring and a ∈ R an element, generating a two-sided ideal ( a ) ⊂ R . Theprojection onto the quotient R → R/a := R/ ( a ) is then a ring homomorphism. If R is commutative, then so is R/a . In homotopy theory, the situation is more subtle.The topological sphere spectrum S is the unit in a closed symmetric monoidalcategory modeling the stable homotopy category, and in particular a commutativemonoid. Given any endomorphism a : S → S , the homotopy-theoretic quotient isalmost never a commutative monoid. The first instance occurs when a = 2: Thehomotopy-theoretic quotient S /
2, also known as the Moore spectrum for the group Z /
2, satisfies π S / ∼ = Z / S /n for n >
2, itsassociativity or commutativity in the stable homotopy category is not automatic.See [16] and the references therein for details.Within motivic or A -homotopy theory, Moore spectra have appeared for ex-ample in [18], [6], and [11]. As the structure of the endomorphisms of the motivicsphere spectrum is much richer, also the notion of Moore spectra should be moresophisticated. The degree zero part of these endomorphisms over a field is theMilnor-Witt K -theory graded by weight [14], and the weight zero part of that isthe Grothendieck-Witt ring of the field. Since neither is a principal ideal domainin general, and usually far from Noetherian, Moore spectra with respect to idealsinstead of single elements are more sensible. See for example [11, Remark 1.4]. Nev-ertheless, an elementary approach is chosen here, which still suffices to illustratea few interesting phenomena. More precisely, multiplicative structures on motivicMoore spectra – whose existence may depend on the ground field – are discussedin Section 5, based to some extent on results concerning Toda brackets listed inSection 4. These in turn rely on some preliminaries on the few first stable stems ofthe motivic sphere spectrum, to be discussed in Section 2, which follows and partlyexpands [21]. The article closes with some results on slices and slice differentialsfor special motivic Moore spectra in Section 6. These results may be used for slicespectral sequence computations of homotopy groups of motivic Moore spectra. A Date : October 3, 2019.This work was supported through DFG grants within the SPP 1786 “Homotopy theory andalgebraic geometry”, and a guest professorship at the University of Oslo. noteworthy feature in comparison with corresponding slice spectral sequence com-putations for the motivic sphere spectrum is the absence of motivic cohomologygroups with integral coefficients; motivic cohomology groups with finite coefficientsare understood much better.2.
Preliminaries on π , π , and π Determining the existence of multiplications or pairings on Moore spectra re-quires information about stable homotopy groups of motivic spheres. Let π s,w E denote the abelian group [Σ s,w , E ], where E is a motivic spectrum and is themotivic sphere spectrum. Set π s +( w ) E := π s + w,w E , and let π s +( ⋆ ) E = M w ∈ Z π s + w,w E denote the direct sum, considered as a Z -graded module over the Z -graded ring π ⋆ ) . The notation π s − ( ⋆ ) E := π s +( − ⋆ ) E will be used frequently. The strictly A -invariant sheaf obtained as the associated Nisnevich sheaf of U π s,w E U for U ∈ Sm F is denoted π s,w E , which gives rise to π s +( ⋆ ) E . See [14] for the followingstatement. Theorem 2.1 (Morel) . Let F be a field. Then π − ( ⋆ ) is the Milnor-Witt K -theoryof F . The Milnor-Witt K -theory of F is denoted K MW ( F ), or simply K MW , followingthe convention that the base field or scheme may be ignored in the notation. Itsgenerators are denoted η ∈ K MW − = π , and [ u ] ∈ K MW ( F ) = π − , − F for everyunit u ∈ F × . The abbreviations h u i := 1 + η [ u ] ∈ K MW ( F ) h u , . . . , u m i := h u i + · · · + h u m i ∈ K MW ( F ) ε := −h− i h := h , − i = 1 − ε for units u, u , . . . , u m ∈ F × will be convenient. Under the identification of K MW ( F )with the Grothendieck-Witt ring GW ( F ) of F , the element h u , . . . , u m i corre-sponds to the quadratic form given by the appropriate diagonal matrix. Milnor K -theory [12] is expressed as the quotient K M ⋆ ∼ = K MW ⋆ / ( η ). Set k M ⋆ := K M ⋆ /
2. The-orem 2.1 implies that for every motivic spectrum E and for every integer s , π s +( ⋆ ) E is a graded K MW -module. As a first instance besides the motivic sphere spectrum , consider the very effective cover kq → KQ of the motivic spectrum representinghermitian K -theory [1], [2]. Using kq instead of the effective cover f KQ → KQ leads to a slight improvement on the computation [21, Theorem 5.5]. Theorem 2.2 (R¨ondigs-Spitzweck-Østvær) . Let F be a field of exponential charac-teristic e = 2 . The unit map → kq induces an isomorphism π ⋆ ) → π ⋆ ) kq ,and a surjection π ⋆ ) → π ⋆ ) kq whose kernel coincides with K M − ⋆ / afterinverting e . Voevodsky’s slice filtration { f q E → E } q ∈ Z allows to be more precise, and inparticular to describe the K MW -module structure. Let M Z be Voevdosky’s inte-gral motivic Eilenberg-MacLane spectrum representing motivic cohomology, and EMARKS ON MOTIVIC MOORE SPECTRA 3 let M Z / Z /
2. Moreover, set for k a naturalnumber H ⋆ − k,⋆ := π k − ( ⋆ ) M Z and h ⋆ − k,⋆ := π k − ( ⋆ ) M Z /
2. Note the K MW -moduleisomorphism h ⋆ − k,⋆ ∼ = k M ⋆ − k given by multiplication with τ k , where τ = − ∈ h , = { , − } = ker (cid:0) H , x x −−−−→ H , (cid:1) is the unique nontrivial element. The K MW -module π − ( ⋆ ) kq is an extension of the K MW -module H ⋆ − ,⋆ = π − ( ⋆ ) s kq (on which η operates trivially) and the K MW -module given by the image of π − ( ⋆ ) f kq in π − ( ⋆ ) kq . Lemma 2.3.
The K MW -module π − ( ⋆ ) f kq is generated by the image of η top underthe unit map u : → kq , and has the presentation K MW / (2 , η ) ∼ = π − ( ⋆ ) f kq . Proof.
The determination of the relevant part of the slice spectral sequence for kq given in [1, Proposition 27] implies that the K MW -module π − ( ⋆ ) f kq is an extensionof the K MW -module h ⋆, ⋆ = π − ( ⋆ ) s kq (on which η operates trivially) and the K MW -module h ⋆, ⋆ / Sq h ⋆ − , ⋆ (on which η operates trivially as well). The K M -module h ⋆, ⋆ = π − ( ⋆ ) s kq is generated by the image of η top ∈ π , , and the K M -module h ⋆, ⋆ / Sq h ⋆ − , ⋆ is generated by the image of ηη top ∈ π , . Since2 η top = 0, the extension describing π − ( ⋆ ) f kq splits in every degree as a short exactsequence of abelian groups. In other words, the surjection K MW → π − ( ⋆ ) kq factorsthrough a surjection K MW / → π − ( ⋆ ) f kq . The relation 0 = η u( η top ) ∈ π , f kq follows from the slice spectral sequence computation (which even gives π , kq = 0).In order to show that the obtained surjection K MW / (2 , η ) → π − ( ⋆ ) f kq is anisomorphism, observe that it fits into a natural transformation(2.1) η K MW / (2 , η ) K MW / (2 , η ) K MW / (2 , η ) h ⋆, ⋆ / Sq h ⋆ − , ⋆ π − ( ⋆ ) f kq π − ( ⋆ ) s kq of short exact sequences, where the outer vertical morphisms are isomorphisms.This implies the result. (cid:3) The difference between π − ( ⋆ ) f kq and its image f π − ( ⋆ ) kq in π − ( ⋆ ) kq is givenby the image of π − ( ⋆ ) s kq ∼ = H ⋆ − ,⋆ → π − ( ⋆ ) f kq . Since 2 η top = 0, this mapfactors over H ⋆ − ,⋆ /
2. The short exact sequence displayed in (2.1) then induces along exact sequence
Hom k M ( H ⋆ − ,⋆ / , k M ⋆ +1 /ρ ) → Hom K MW ( H ⋆ − ,⋆ / , π − ( ⋆ ) f kq ) → Hom k M ( H ⋆ − ,⋆ / , k M ) → · · · in which the first group is zero for any finite field or any number field. Here ρ is theclass of − h , . The homomorphism in question maps to the restriction of themotivic Steenrod square Sq in the abelian group Hom k M ( H ⋆ − ,⋆ / , h ⋆,⋆ +1 ∼ = k M ⋆ ),and is thereby uniquely determined for prime fields.Following a specific request, a probably well-known identification, in which O × denotes the sheaf of units, can be derived from Lemma 2.3. In principle, anysequence of strictly A -invariant sheaves which is exact when evaluated on fields isalready exact, by a theorem of Morel. This applies in particular to Theorem 2.2,Lemma 2.3, and Theorem 2.5. However, the following case can be proved directlyinstead. OLIVER R ¨ONDIGS
Proposition 2.4.
Let F be a field of characteristic not two. The sheaf π , KQ isisomorphic to the sheaf O × / × Z / .Proof. By construction, the canonical maps induce isomorphisms π , f kq ∼ = π , kq ∼ = π , KQ . This uses the vanishing π , s kq = π , s kq = 0, since s kq = M Z andmotivic cohomology of smooth schemes in weight zero is concentrated in degreezero. The homotopy cofiber sequence f kq → f kq → s kq induces, for every smooth connected F -scheme U , a short exact sequence0 = π , s kq ( U ) → π , f kq ( U ) → π , f kq ( U ) → π , s kq ( U ) = h , ( U ) = Z / → Z / π S → π , f kq . In case U is (thespectrum of) a field extension, this sequence appears in diagram (2.1). Already if U is an essentially smooth local F -scheme, then π , f kq ( U ) ∼ = π , s kq ( U ) ∼ = h , ( U )which supplies the result on the level of Nisnevich sheaves. (cid:3) Theorem 2.5.
Let ν ∈ π , be the second algebraic Hopf map, obtained by theHopf construction on SL , and let η top ∈ π , be the first topological Hopf map.The K MW -module map K MW ⋆ ⊕ K MW → π − ( ⋆ ) f sending ( a, b ) to a · ν + b · η top induces an isomorphism K MW ⋆ { ν } ⊕ K MW { η top } / ( ην, η top , η η top − ν ) ∼ = π − ( ⋆ ) f after inverting the exponential characteristic.Proof. Observe first that ν naturally lifts to π , f , hence defines also an element ν ∈ π , f . The element η top has image 0 ∈ π , s = π , M Z = H − , , and hencealso lifts to π , f – even uniquely, since π , s = π , M Z = H − , = 0. Thusthe K MW -module map K MW ⋆ ⊕ K MW → π − ( ⋆ ) f sending ( a, b ) to a · ν + b · η top is well-defined over any base scheme in which motiviccohomology in weight zero vanishes in degree −
1. Certain relations hold beforeinverting the exponential characteristic. The relation ην = 0 holds by [4] overSpec( Z ). Also 2 η top = 0, again over any base scheme. If 2 is invertible in the basescheme, u( ν ) = 0, where u : → kq is the unit map figuring in Lemma 2.3.Assuming that the exponential characteristic is now implicitly inverted, the ex-act sequence from Theorem 2.2 implies that π , = Z /
24, generated by ν . Eitherby reference to complex realization or by the multiplicative structure of the slicespectral sequence (more precisely, only the effect of multiplying with the first al-gebraic Hopf map η ), the element η η top ∈ π , is the unique nontrivial elementof order two, hence η η top = 12 ν = 6 h ν . Thus there is an induced K MW -modulehomomorphism K MW ⋆ { ν } ⊕ K MW { η top } / ( ην, η top , η η top − ν ) → π − ( ⋆ ) f . The K MW -module homomorphism π − ( ⋆ ) f → π − ( ⋆ ) f kq induced by the unit mapu : → kq then factors over the isomorphism K MW / (2 , η ) ∼ = π − ( ⋆ ) f kq EMARKS ON MOTIVIC MOORE SPECTRA 5 from Lemma 2.3. There results a commutative diagram(2.2) K MW ⋆ { ν } (24 ν,ην ) K MW ⋆ { ν }⊕ K MW { η top } ( ην, η top ,η η top − ν ) K MW { η top } (2 η top ,η η top ) K M ⋆ / π − ( ⋆ ) f π − ( ⋆ ) f kq of short exact sequences, where the upper horizontal row is short exact by directcomputation. Note that the relations 2 η top = 0 and η η top = 12 ν imply 24 ν = 0.The exactness of the lower horizontal row follows from the exactness of the sequencein Theorem 2.2 and the fact that the unit u : → kq induces an isomorphism on zeroslices. The right vertical map in diagram (2.2) is an isomorphism by Lemma 2.3,and the left vertical map in diagram (2.2) is an isomorphism by direct inspection,whence the result. (cid:3) As in the case of kq , the difference between π − ( ⋆ ) f and its image f π − ( ⋆ ) in π − ( ⋆ ) is given by the image of π − ( ⋆ ) s ∼ = H ⋆ − ,⋆ → π − ( ⋆ ) f . The short exactsequence 0 → f π − ( ⋆ ) → π − ( ⋆ ) → π − ( ⋆ ) s = H ⋆ − ,⋆ → K MW -modules. In particular,the canonical map π w,w f → π w,w is an isomorphism for w > −
2. For theapplications to motivic Moore spectra, an important weight is w = 0, where a shortexact sequence(2.3) 0 → K M / → π , → Z / ⊕ K M / → K MW -modules. In particular, 24 · π , = 0,in contrast with 24 · π − , − Q = 0, which is similar to 24 · π − , − KGL Q ∼ = 24 · Z / =0. Lemma 2.6.
The action of GW on π ⋆ ) is determined by the following equa-tions for u ∈ F × . h u i · ν = ν ∈ π , h u i · ηη top = ηη top + [ u ] ν ∈ π , h u i · η top = η top + [ u ] ηη top ∈ π , In particular, h · η top = ρηη top , h · ηη top = 0 , and h · ν = 2 ν . If n is even, then n h acts on π ⋆ ) f as multiplication by n .Proof. This follows from Theorem 2.5, once the identification π w ) f = π w ) for w > − < (cid:3) The situation for π ⋆ ) is a bit more delicate. Nevertheless, the following canbe read off from the slice spectral sequence. Theorem 2.7.
The element ν : Σ , → induces an inclusion K MW ⋆ { ν } / ( ην , ν ) → π − ( ⋆ ) of K MW -modules, which is an isomorphism for all ⋆ < − . In particular, for all w > , the group π w,w ∼ = 0 . OLIVER R ¨ONDIGS
Proof.
Since ην = 0 [4], also ην = 0. Moreover, the ε -graded commutativity of π ∗ +( ⋆ ) implies that ν = − ν , whence a map K MW ⋆ { ν } / ( ην , ν ) → π − ( ⋆ ) of K MW -modules exists. The slice spectral sequence for π ⋆ ) shows its injectivity,as well as the isomorphism statement, using results from [21] and [17, Theorem8.3]; details are to be given in [20]. (cid:3) Elementary properties of motivic Moore spectra
Definition 3.1.
Let s ≥ w ∈ Z , and let α : Σ s,w → be an endomorphism.Choosing a homotopy cofiber sequenceΣ s,w α −→ c −→ C α d −→ Σ s +1 ,w defines the motivic Moore spectrum C α .By definition, a motivic Moore spectrum for a is unique up to equivalence. Therestriction s ≥ w is reasonable at least over a field by Morel’s connectivity theorem[15]. Proposition 3.2.
Let α : Σ s,w → be an endomorphism. The canonical mapsinduce a short exact sequence → π n +( ⋆ ) /απ n − s + w +( ⋆ − w ) c ∗ −→ π n +( ⋆ ) C α d ∗ −→ α π n − − s + w +( ⋆ − w ) → of K MW -modules, with target the submodule of elements annihilated by α .Proof. This follows from the homotopy cofiber sequenceΣ s,w α −→ c −→ C α d −→ Σ s +1 ,w defining the motivic Moore spectrum. (cid:3) In particular, Morel’s connectivity theorem implies with Proposition 3.2 that π ⋆ ) C α ∼ = K MW /α K MW if α ∈ K MW (that is, if s = w ).4. Toda brackets
Consider three composable maps D γ −→ E β −→ F α −→ G of motivic spectra such that βγ = 0 = αβ . The Toda bracket h α, β, γ i (mod α ◦ [Σ , D , F ] + [Σ , E , G ] ◦ Σ , γ )is the coset of the displayed subgroup of [Σ , D , G ] given by those compositionsΣ , D → C ( β ) → G such that the obvious diagrams commute. The base schemeor field may be indicated by a subscript. The most relevant case is where all themotivic spectra involved are appropriate suspensions of the motivic sphere spectrum . See the classical source [22], as well as [5] and [8] for interesting Toda bracketsin the motivic stable homotopy category. We consider a few examples and makeno claim to originality. Angled brackets are used to denote both Toda bracketsand quadratic forms (as in the paragraph after Theorem 2.1). The reader shouldbe aware of this possible confusion, but context will always make the meaningunambiguous. EMARKS ON MOTIVIC MOORE SPECTRA 7
Proposition 4.1.
The following equalities of subsets of π ∗ +( ⋆ ) hold. h h , η, h i = (cid:8) ηη top + [ φ ] ν : φ ∈ K M { ν } / (cid:9) (4.1) h η, h , η i = { ν, − ν } (4.2) h η, ν, η i = { ν } (4.3) h h , ν, h i R = { ρ ν } (4.4) Proof.
The indeterminacy in (4.1) is the subgroup h ◦ π , = 2 K M { ν } /
24, usingthe isomorphism π , ∼ = K M { ηη top } / ⊕ K M { ν } /
24 from Theorem 2.5 and theequalities [ − η ( ηη top ) = [ − η η top = [ − ν = 0 = [ − ην given in Lemma 2.6.That the Toda bracket contains ηη top follows from appropriate realization functorsand the equality h , η top , i top = { η top } in classical stable homotopy groups [22].The indeterminacy in (4.2) is η ◦ π , = { , ν } , since Theorem 2.5 implies π , ∼ = K M { ηη top } / ⊕ K M { ν } /
24, and ηηη top = 12 ν , whereas ην = 0. Complexrealization maps h η, h , η i to h η top , , η top i top = { ν top , − ν top } . The relevant group π , does not depend on the base field (at least after inverting the exponentialcharacteristic), giving the result.The indeterminacy in (4.3) is the subgroup η ◦ π , = { } , since π , ∼ = K M { ν } / ην = 0. The unique element in h η, ν, η i hasto be the unique nontrivial one in π , = Z / { } by Theorem 2.7,since (2 + 11 h ) ν = 0. Real realization sends h h , ν, h i R to h , η top , i top = { η top } , whence the previous Toda bracket has to contain the unique nonzero elementin π , R . (cid:3) The following statement, which could be formulated in greater generality, indi-cates the relevance of Toda brackets for the structure of motivic Moore spectra.The notation is as in Proposition 3.2.
Proposition 4.2.
Suppose α ∈ π s,w , β ∈ π t,x , and γ ∈ π u,y are elementswith αβ = 0 = βγ . The Toda bracket h α, β, γ i coincides with the set of elements δ ∈ π s + t + u +1 ,w + x + y for which there exists an element ˜ β ∈ π s + t +1 ,w + x C α with d ∗ ( ˜ β ) = β and c ∗ ( δ + α · π t + u +1 ,x + y ) = ˜ β · γ .Proof. The proof consists of comparing various diagrams in the motivic stable ho-motopy category and is left as an exercise to the reader. (cid:3)
Example 4.3.
Consider α = γ = η ∈ π , . Then π , C η contains an element ˜ ν with ˜ ν · η = 0. In fact, the Toda bracket h η, ν, η i = { ν } given in (4.3) containsa single nonzero element, and c ∗ : π , → π , C η is injective, since multiplicationwith η is the zero map π , = h , { ν } → h , { ν } = π , by Theorem 2.7. Example 4.4.
Let α = γ = 2 + 11 h ∈ π , R (or any formally real field).Then π , C h contains an element ˜ ν with ˜ ν · (2 + 11 h ) = 0. Indeed, theToda bracket given in (4.4) contains a nonzero element (uniquely for R ), and c ∗ : π , R → π , C h is injective, since multiplication with 2 + 11 h inducesthe zero map π , R = h , { ν } → π , R by Theorem 2.7.5. Multiplications
The “constant presheaf” functor defines a strict symmetric monoidal triangu-lated functor const : SH → SH ( S ) for any base scheme S . The motivic Moore OLIVER R ¨ONDIGS spectra C n for n ∈ N are in its image. In particular, multiplications or pairings onthese motivic Moore spectra can be transferred from the corresponding topologicalones [16]. Lemma 5.1.
Let α : Σ s,w → be an endomorphism and c : → C α the map tothe homotopy cofiber. There exists a left unital pairing C α ∧ C α → C α if and only if the identity on C α is annihilated by α .Proof. Consider the following diagram: ∧ = id (cid:15) (cid:15) c / / C α = ∧ C α c ∧ C α / / id (cid:15) (cid:15) C α ∧ C α c / / C α id / / C α There exists a left unital pairing C α ∧ C α → C α if and only if the map c ∧ C α admits a retraction. After smashing with C α , thehomotopy cofiber sequenceΣ s,w α −→ c −→ C α d −→ Σ s +1 ,w induces a long exact sequence · · · d ∗ ←− [Σ s,w C α , C α ] α ←− [ C α , C α ] c ∗ ←− [ C α ∧ C α , C α ] d ∗ ←− [Σ s +1 ,w C α , C α ] α ←− · · · which shows the desired statement. (cid:3) If a left unital pairing µ : C α ∧ C α → C α exists, then µ ◦ ( C α ∧ c ) ◦ c = c . Theshort exact sequence0 → π s +1 ,w C α d ∗ −→ [ C α , C α ] c ∗ −→ π , C α → ψ ∈ π s +1 ,w C α with ψ ◦ d = µ ◦ ( C α ∧ c ) − id C α . The shortexact sequence0 → π s +2 , w C α d ∗ −→ [Σ s +1 ,w C α , C α ] c ∗ −→ π s +1 ,w C α → θ : Σ s,w C α → C α with θ ◦ Σ s +1 ,w d = ψ . It follows that the left unitalpairing µ − θ ◦ ( d ∧ C α ) is also right unital, because (cid:0) µ − θ ◦ ( d ∧ C α ) (cid:1) ◦ ( C α ∧ c ) = µ ◦ ( C α ∧ c ) − θ ◦ ( d ∧ c ) = µ ◦ ( C α ∧ c ) − θ ◦ Σ s +1 ,w c ◦ d = id C α . Hence if a left unital pairing on C α exists, a unital pairing exists as well. In thefollowing, “multiplication” stands for “unital pairing”. Lemma 5.2.
Let α : Σ s,w → be an endomorphism. Then α · id C α = 0 .Proof. The homotopy cofiber sequence defining C α induces a short exact sequence(5.1) 0 → [Σ s +1 ,w , C α ] /α d ∗ −→ [ C α , C α ] c ∗ −→ α [ , C α ] = [ , C α ] = [ , ] /α → c ∗ (id C α ) is annihilated by α . Hence α · id C α = d ∗ ( x ) for some x ∈ [Σ s +1 ,w , C α ] /α .Thus α · x = 0, which proves the equality α · id C α = α · d ∗ ( x ) = d ∗ ( α · x ) = 0. (cid:3) EMARKS ON MOTIVIC MOORE SPECTRA 9
As a consequence, there exists a pairing C α ∧ C α → C α which in the classical case of the topological sphere spectrum and α = 2 wasdescribed by Oka in [16]. For example, using h = 2 h , there results a pairing C h ∧ C h → C h which realizes to Oka’s pairing for subfields of the complex numbers. The specificrole that squares play will be clarified by the following statement lifting a theoremof Brayton Gray [7, Theorem 10]. Theorem 5.3.
Let α : Σ s,w → be any endomorphism. Then C α admits amultiplication.Proof. Lemma 5.1 implies it suffices to show that α · id C α = 0. In order toprove this, consider the following general construction for α : D → E , modeled on[22], and observing that the ε -graded ring structure on π ∗ +( ⋆ ) may be equallydefined via composition or via smash product. Let R ( α ) be the set of all maps A : Σ , D ∧ D → E ∧ E such the diagramΣ , D ∧ D E ∧ ED ∧ C α E ∧ C αA E ∧ cα ∧ C α D ∧ d commutes. If R ( α ) contains an element of the form β ∧ α , where β : Σ , D → E ,then α ∧ C α = α · id C α is the zero map. Several natural commutative diagramsshow that if α : D → E and β : F → G are two maps, then for every A ∈ R ( α ), theelementΣ , D ∧ F ∧ D ∧ F twist −−−→ Σ , D ∧ D ∧ F ∧ F A ∧ β ∧ β −−−−−→ E ∧ E ∧ G ∧ G twist −−−→ E ∧ G ∧ E ∧ G lies in R ( α ∧ β ). In particular, for α ∈ π s,w and β ∈ π t,x , the inclusion( − ( s − w )( t − x ) ε wx R ( α ) β ⊂ R ( α ∧ β ) holds, using [4, Equation (2.4)]. In thespecial case α = β , one obtains the inclusion ( − ( s − w ) ε w R ( α ) α ⊂ R ( α ∧ α ). If R ( α ) is not empty, this implies that α ∧ C α is zero. It remains to see that R ( α )is not empty for α ∈ π s,w . Since α ∧ C α ◦ Σ s,w c = c ◦ Σ s,w α = 0, there exists γ ∈ π s +1 , w C α with γ ◦ Σ s,w d = α ∧ C α . If d ◦ γ = 0, then R ( α ) contains a lift of γ along c . If d ◦ γ = 0, then the equation d ◦ γ ◦ Σ s,w d = d ◦ α ∧ C α = Σ s +1 ,w α ◦ Σ s,w d = 0supplies φ ∈ π , with φ ◦ α = d ◦ γ . Since α ◦ φ ◦ α = 0, there exists ψ ∈ π s +1 ,w C α with d ◦ ψ ◦ α = φ ◦ α It follows that ( γ − ψ ◦ α ) ◦ d = γ ◦ d = α ∧ C α and d ◦ ( γ − ψ ◦ α ) = 0. A lift of γ − ψ ◦ α ∈ π s +1 , w C α along c provides an element in R ( α ), which concludes the proof. (cid:3) Now for some negative results regarding multiplications on motivic Moore spec-tra. Recall that π , = GW identifies with the Grothendieck-Witt ring of qua-dratic forms by Theorem 2.1, which comes equipped with a dimension ring homo-morphism dim : GW → Z . Theorem 5.4.
Let α : → be an endomorphism with dim( α ) ≡ . Then C α does not admit a multiplication. Proof.
This result follows by complex realization from [16] for subfields of the com-plex numbers. In any case, consider motivic cohomology h ∗ , ∗ with coefficients in F . Since multiplication with α induces multiplication with dim( α ) on motiviccohomology, there results a split short exact sequence0 → h ∗ , ∗ → h ∗ , ∗ ( C α ) → h ∗ +1 , ∗ → x ∈ h , ( C α ) , x ∈ h , ( C α ), and notethat Sq ( x ) = x since dim( α ) ≡ → h ∗ , ∗ ( C α ) → h ∗ , ∗ ( C α ∧ C α ) → h ∗ +1 , ∗ ( C α ) → α on C α then supplies h ∗ , ∗ ( C α ∧ C α ) with basiselements x ⊗ y , x ⊗ y , x ⊗ y , x ⊗ y . Here y ∈ h , ( C α ) and y ∈ h , ( C α ) arebasis elements in the other factor of the smash product C α ∧ C α . A K¨unneth theoremsupplies the basis elements for the motivic cohomology of the smash product. TheCartan formula Sq ( x ⊗ y ) = Sq ( x ) ⊗ y + x ⊗ Sq ( y ) + τ Sq ( x ) ⊗ Sq ( y ) = τ x ⊗ y from [23, Proposition 9.7] then shows that Sq acts nontrivially on h ∗ , ∗ ( C α ∧ C α ).If α · id C α = 0, then C α ∧ C α = C α ∨ Σ , C α , and h ∗ , ∗ ( C α ∧ C α ) splits accordingly asa module over the motivic Steenrod algebra, implying Sq ( x ⊗ y ) = 0. The resultfollows. (cid:3) Lemma 5.5.
The motivic Moore spectrum C η ℓ admits a multiplication if and onlyif ℓ > .Proof. Let us prove first that C η does not admit a multiplication. Again an argu-ment via complex realization works for subfields of the complex numbers. However,one may refer to Example 4.3 and Proposition 4.2, which imply the existence of anelement ˜ ν ∈ π , C η such that ˜ ν · η = 0. In particular, η · id C η = 0.Consider now ℓ >
1. The element η ℓ · id C ηℓ lies inside the group which sits in themiddle of the short exact sequence0 → [Σ ℓ +1 , ℓ , C η ℓ ] /η ℓ d ∗ −→ [Σ ℓ,ℓ C η ℓ , C η ℓ ] c ∗ −→ η ℓ [Σ ℓ,ℓ , C η ℓ ] = [Σ ℓ,ℓ , ] /η ℓ → η ℓ : π , → π ℓ,ℓ is surjective. The initial term iscomputed by Proposition 3.2. More precisely, the group π ℓ +1 , ℓ C η ℓ sits inside theshort exact sequence0 → π ℓ +1 , ℓ /η ℓ π ℓ +1 ,ℓ c ∗ −→ π ℓ +1 , ℓ C η ℓ d ∗ −→ η ℓ π ℓ,ℓ → η ℓ π ℓ,ℓ = 0 since η ℓ : π ℓ,ℓ → π ℓ, ℓ is an isomorphism for ℓ ≥
1. The group π ℓ +1 , ℓ vanishes by Theorem 2.2 for ℓ >
1, whence so does π ℓ +1 , ℓ C η ℓ , and thusalso [Σ ℓ,ℓ C η ℓ , C η ℓ ]. (cid:3) The existence of a multiplication may depend on the base field in general. Ex-ample 4.4 and Lemma 5.1 show that the motivic Moore spectrum C h admitsno multiplication over a formally real field F . Base change to F ( √−
1) producesthe motivic Moore spectrum C h = C , which admits a multiplication by liftingfrom topology and quoting [16]. Plenty of similar examples may be constructed.A systematic study on these matters remains a project for the future. Anotherproject for the future is to enumerate possible multiplications, as well as investi-gate their qualitative properties like associativity and commutativity. Already the EMARKS ON MOTIVIC MOORE SPECTRA 11 enumeration can be challenging. For example, the group [ C h ∧ C h , C h ] receives anontrivial map from [Σ , C h , C h ], which is injective on the contribution from the h -torsion in π , (see Proposition 3.2). The latter coincides with the fundamentalideal in the Grothendieck-Witt ring, as the following statement implies. Lemma 5.6.
Let α ∈ GW ( F ) be an element with dim( α ) = 0 . Then α h GW ( F ) = dim( α ) h GW ( F ) = h GW ( F ) = I ( F ) . Proof.
Recall that h = 1 − ǫ = h , − i is a form of dimension two with the propertythat the subgroup of GW ( F ) generated by h coincides with the ideal of GW ( F )generated by h . The latter follows from the similarity h u i · h , − i = h u, − u i ∼ h , − i of quadratic forms, where u ∈ F × . Hence for every α ∈ GW ( F ), the equation α · h = dim( α ) · h follows, giving the first equation. Also the other equations followfor dim( α ) = 0, since β ∈ GW ( F ) then satisfies β · ( α · h ) = 0 if and only ifdim( β ) = 0. Here recall the short exact sequence0 → I ( F ) → GW ( F ) dim −−→ Z → GW ( F ). (cid:3) Prompted by a recent request, this section concludes with a specific example. Let n > n ε = n − P k =0 h ( − k i = h , − , . . . , ± i ∈ GW ,a quadratic form of dimension n . It turns out that the motivic Moore spectrum C n ε admits a multiplication precisely if the topological Moore spectrum S /n does.Before proving this, set e : C α d −→ Σ s +1 ,w Σ s +1 ,w c −−−−−→ Σ s +1 ,w C α for α ∈ π s,w . Fol-lowing [16], a multiplication µ : C α ∧ C α → C α is called regular if the equality e ◦ µ = (Σ s +1 ,w µ ) ◦ ( e ∧ C α + C α ∧ e ) : C α ∧ C α → Σ s +1 ,w C α holds. Lemma 5.7.
There exists a multiplication on C n ε if and only if n .Proof. Theorem 5.4 says that C n ε does not admit a multiplication if n ≡ n = 2 m + 1 is an odd natural number. Any element α ∈ GW with n ε α = 0 then satisfies dim( α ) = 0, and hence lies in the fundamental ideal.Since n ε = 1+ m h and the element h α is hyperbolic of dimension zero, one concludes α = 0. Proposition 3.2 then provides an isomorphism π , C n ε ∼ = π , /n ε π , .Lemma 2.6 implies that the map π , n ε −→ π , is surjective if n n is odd and not divisible by three, π , C n ε = 0. The short exact sequence (5.1) thenreduces to [ C n ε , C n ε ] ∼ = π , C n ε . In particular, the identity on C n ε is annihilated by n ε , which provides the existence of a multiplication by Lemma 5.1. If n is odd anddivisible by three, π , C n ε ∼ = K M /
3. In order to conclude for such odd numbers aswell, it suffices to prove that the short exact sequence(5.2) 0 → π , C n ε ∼ = K M / → [ C n ε , C n ε ] → π , C n ε → n ε = 1 + m · h acts as the identity on the Witt group, π , C n ε ∼ = Z /n for any field, generated by h . Hence if the sequence (5.2) splits over prime fields,it does so over any field, reducing the task to F = Q . In this case, [13, Theorem11.6] supplies an isomorphism K M / Q ) ∼ = L p prime ,p ≡ Z / the prime p ≡ F p ) ֒ → Spec( Z ( p ) ) ← ֓ Spec( Q ) . Since both h and the element n ε are defined over Spec( Z ), it follows that thesequence (5.2) splits.Suppose now that n = 2 r m for some odd natural number m , with r ≥
2. Then n ε = m h r . By Theorem 5.3, C h admits a multiplication µ . If µ is not regular, thedifference x = e ◦ µ − µ ◦ ( e ∧ C h + C h ∧ e ) lifts to produce a unique element y ∈ π , C h such that y ◦ ( d ∧ d ) = x . The latter equation shows that x is an element of ordertwo, because d ∧ d = − d ∧ d , hence so is y . The vanishing 0 = d ◦ y ∈ π , C h impliesthat there exists z ∈ π , with c ◦ z = y . The image of π , under compositionwith c is isomorphic, by Theorem 2.5, to K M / ⊕ k M ⊕ k M . The map λ : C h → C h induced by multiplication with h on the bottom cell does not necessarily inducethe zero homomorphism on π , . More specifically, it induces multiplication by 4as a homomorphism K M / ⊕ k M ⊕ k M → K M / ⊕ k M ⊕ k M and in particular sends the 2-torsion element y to zero. The homotopy cofibersequence C h λ −→ C h ρ −→ C h e −→ Σ , C h implies the existence of w ∈ π , C h with e ◦ w = y . Then µ − w ◦ ( d ∧ d ) is a regularmultiplication on C h . Hence a regular multiplication exists on C h .Inductively, one may lift a regular multiplication on C h ℓ to a regular multiplica-tion on C h ℓ +1 via the homotopy cofiber sequence C h λ −→ C h ℓ +1 ρ −→ C h ℓ δ −→ C h . Here λ and ρ are induced by suitable multiplications with h ℓ on the bottom celland h on the top cell, respectively, and δ is the composition C h ℓ d −→ Σ , Σ , c −−−→ C h .The homotopy cofiber sequence C h λ −→ C m ρ −→ C m h r δ −→ C h r allows to lift a regular multiplication µ on C h r to a regular multiplication on C n ε .Over quadratically closed fields, δ = 0 (as in topology, see [16, Lemma 5]), but δ = 0 for formally real fields. Nevertheless the equality δ ◦ µ ◦ ( ρ ∧ ρ ) = 0 alwaysholds. This computation follows from the fact that µ is regular and two otherinputs. One input is the result from topology that m is zero on S /m , and hencealso on its image C m in the motivic stable homotopy category. In order to applythe regularity, one uses that the composition C h r d −→ Σ , Σ , c −−−→ Σ , C m factors as C h r d −→ Σ , Σ , c −−−→ Σ , C h r Σ , φ −−−→ Σ , C m for some φ : C h r → C m . This provides aregular multiplication on C n ε for n (cid:3) Slices of motivic Moore spectra
Instead of considering slices for general motivic Moore spectra, the focus here –motivated by arguments regarding the vanishing of higher slice differentials in [21,Section 4] – is on C n h , where 0 < n ∈ N . As explained in the proof of Lemma 5.6,it suffices to consider motivic Moore spectra with respect to n h instead of α h for α ∈ GW ( F ) of dim( α ) = 0. Since h = 0 : KW → KW as an endomorphism on the EMARKS ON MOTIVIC MOORE SPECTRA 13 motivic spectrum representing higher Witt groups, the motivic spectrum KW n h splits as KW ∨ Σ , KW . The same holds for its (effective or connective) covers, andalso for the corresponding slices. In particular, the first slice differential for KW n h splits. For reference purposes, the explicit form is as follows. Theorem 6.1.
Let < n ∈ N . The restriction of the slice d -differential to thesummand Σ q + j,q M Z / of s q ( KW n h ) is given by d ( KW n h )( q, j ) = ( ( Sq Sq , , Sq ) j ≡ , Sq Sq , , Sq + ρ Sq , , τ ) j ≡ , Here the i th component of the map d ( KW n h )( q, j ) of motivic spectra is a map Σ q + j,q M Z / → Σ q + j + i,q +1 M Z / .Proof. This follows from the determination of the first slice differential for KW from[19], and the aforementioned splitting KW n h ≃ KW ∨ Σ , KW . (cid:3) Here and in the following, the notation regarding the motivic Steenrod algebrais standard; for example, ρ is the class of − h , , and Q = Sq Sq + Sq Sq .The slice computation is slightly more complicated for the motivic spectrum KQ representing hermitian K -theory. For comparison purposes, the case of kq , thevery effective cover of KQ [1], [2], is more convenient. Set kq n h := kq ∧ C n h , andsimilarly cKW n h := cKW ∧ C n h = cKW ∨ Σ , cKW . Here cKW := KW ≥ ∼ = kq [ η − ]is the connective cover of KW , not its very effective cover (which might deserve thenotation kw ). The element η acts still invertibly on cKW .The final set of notation concerns long exact sequences of motivic cohomologygroups induced by change of coefficients in cyclic groups. Given natural numbers m, n , the inclusion Z /m ֒ → Z /mn induces a homomorphism on motivic cohomologydenoted inc mmn , and the projection Z /mn ։ Z /n induces a homomorphism onmotivic cohomology denoted pr mnn . The short exact sequence0 → Z /m → Z /mn → Z /n → ∂ nm . Theorem 6.2.
Let < n, ≤ q . The slices of kq n h are given as follows: s q ( kq n h ) = Σ q, q M Z / n ∨ q − _ j =0 Σ q + j, q M Z / s q +1 ( kq n h ) = s q +1 ( kq ) ∨ Σ , s q +1 ( kq ) = q +1 _ j =0 Σ q +1+ j, q +1 M Z / The canonical map kq n h → cKW n h induces the following map on summands ofslices: Σ q, q M Z / n ( ∂ n , pr n ) −−−−−−−→ Σ q +1 , q M Z / ∨ Σ q, q M Z / q + j,q M Z / (1) −−→ Σ q + j,q M Z / j ≡ n ≡ q + j,q M Z / ( Sq , −−−−→ Σ q + j +1 ,q M Z / ∨ Σ q + j,q M Z / j ≡ n ≡ The restriction of the slice d -differential for kq n h to the summands Σ q + j,q M Z / or Σ q, q M Z / n of s q ( kq n h ) for n even is given by d ( kq n h )( q, j ) = ( ( Sq Sq , , Sq ) q − > j ≡ , Sq Sq , , Sq + ρ Sq , , τ ) q − > j ≡ , d ( kq n h )( q, q −
1) = ( ∂ n Sq Sq , , Sq ) q − ≡ Sq Sq , , Sq ) q − ≡ ∂ n Sq Sq , , Sq + ρ Sq , , τ ) q − ≡ Sq Sq , , Sq + ρ Sq , , τ ) q − ≡ . d ( kq n h )( q, q ) = ( Sq ∂ n , Sq pr n ) q ≡ n Sq Sq , Sq ) q ≡ Sq ∂ n , Sq pr n , τ ∂ n , τ pr n ) q ≡ n Sq Sq , Sq + ρ Sq , , τ ) q ≡ . The restriction of the slice d -differential for kq n h to the summands Σ q + j,q M Z / or Σ q, q M Z / n of s q ( kq n h ) for n odd is given by d ( kq n h )( q, j ) = ( Sq Sq , , Sq ) q − > j ≡ Sq Sq , Q , Sq , τ Sq ) q − > j ≡ Sq Sq , , Sq + ρ Sq , , τ ) q − > j ≡ Sq Sq , Q , Sq + ρ Sq , τ Sq + ρ, τ ) q − > j ≡ , d ( kq n h )( q, q −
1) = ( ∂ n Sq Sq , , Sq ) q − ≡ Sq Sq , Q , Sq , τ Sq ) q − ≡ ∂ n Sq Sq , , Sq + ρ Sq , , τ ) q − ≡ Sq Sq , Q , Sq + ρ Sq , τ Sq + ρ, τ ) q − ≡ . d ( kq n h )( q, q ) = ( Sq ∂ n , Sq pr n ) q ≡ n Sq Sq + ∂ n Sq , Sq , τ Sq ) q ≡ Sq ∂ n , Sq pr n , τ ∂ n , τ pr n ) q ≡ n Sq Sq + ∂ n Sq , Sq + ρ Sq , τ Sq + ρ, τ ) q ≡ . Proof.
The description of the slices follows from the fact that the slice functors aretriangulated, and [1, Theorem 3.2]. The effect of the canonical map kq n h → cKW n h on slices is readily obtained, except for the occurrence of Sq . The latter followsfrom the determination of the first slice differential for cKW n h = cKW ∨ Σ , cKW ,compared with possible first slice differentials for kq n h compatible with the firstslice differential for kq , as described in [1, Theorem 3.5]. Determining the first slicedifferential for kq n h is then essentially straightforward. (cid:3) Let 1 < n ∈ N , set kq n := kq ∧ C n , and cKW n := cKW ∧ C n . For comparison,consider the form of the first slice differential for cKW , as obtained in [9, Theorem4.3, Theorem 4.14]. Despite the abstract isomorphisms s ∗ cKW ∼ = s ∗ cKW h and s ∗ kq ∼ = s ∗ kq h , the first slice differentials differ. Note that if n is odd, then all slicesof cKW n vanish, although cKW n itself does not over formally real fields. EMARKS ON MOTIVIC MOORE SPECTRA 15
Theorem 6.3.
The restriction of the slice d -differential for cKW to the summand Σ q + j,q M Z / of s q ( cKW ) is given by d ( cKW )( q, j ) = ( Sq Sq , , Sq ) j ≡ Sq Sq , Q , Sq , ρ + τ Sq ) j ≡ Sq Sq , , Sq + ρ Sq , , τ ) j ≡ Sq Sq , Q , Sq + ρ Sq , τ Sq , τ ) j ≡ . Here the i th component of d ( cKW )( q, j ) is a map Σ q + j,q M Z / → Σ q + j + i,q +1 M Z / . Theorem 6.4.
The restriction of the slice d -differential for kq to the summand Σ q + j,q M Z / of s q ( kq ) is given by d ( kq )( q, j ) = ( Sq Sq , , Sq ) q > j ≡ Sq Sq , Q , Sq , ρ + τ Sq ) q > j ≡ Sq Sq , , Sq + ρ Sq , , τ ) q > j ≡ Sq Sq , Q , Sq + ρ Sq , τ Sq , τ ) q > j ≡ , d ( kq )( q, q ) = ( Sq Sq , Sq + ρ Sq ) q ≡ Q , Sq , ρ + τ Sq , q ≡ Sq Sq , Sq + ρ Sq , τ Sq , τ ) q ≡ Q , Sq + ρ Sq , τ Sq , τ ) q ≡ . Here the i th component of d ( kq )( q, j ) is a map Σ q + j,q M Z / → Σ q + j + i,q +1 M Z / . Note that [9, Theorems 4.24, 4.36] contain information about first slice differen-tials for kq r and cKW r , which could be used to analyze the first slice differentialsfor C r . Theorem 6.5.
Let n > , and let g := gcd(2 n, . The slices of C n h are given asfollows, up to summands of simplicial suspension higher than q + 2 for q ≥ . s ( C n h ) = s ( ) / n s ( C n h ) = s ( ) ∨ Σ , s s ( C n h ) = Σ , M Z / { α } ∨ Σ , M Z / { α } ∨ Σ , M Z /g { α } ∨ Σ , M Z /g { α } s ( C n h ) = s ( ) ∨ Σ , s s ( C n h ) = Σ , M Z / { α } ∨ Σ , M Z / { α } ∨ Σ , M Z / { α α } ∨ Σ , M Z / { ν } ∨ · · · s q ( C n h ) = Σ q,q M Z / { α q } ∨ Σ q +1 ,q M Z / { α q } ∨ Σ q +2 ,q M Z / { α q − α } ∨ · · · The unit map → kq induces a map C nh → kq nh , which induces the identity map onthe slice summands M Z / n, Σ q,q M Z / { α q } , Σ q +1 ,q M Z / { α q } , Σ q +2 ,q M Z / { α q − α } ,and Σ q +3 ,q M Z / { α q − α } , and the map Σ , M Z /g ∨ Σ , M Z /g (cid:16) ∂ g n inc g n (cid:17) −−−−−−−−−−→ Σ , M Z / n on summands of the two-slices.Proof. The form of the slices follows from [10, Section 8] or [21, Theorem 2.12], since h induces multiplication by 2 on slices, and the slice functors are triangulated. Themap C n h → kq n h is determined by the unit map → kq , whose behaviour on slicescan be read off from [21, Lemmas 2.28, 2.29]. (cid:3) Theorem 6.6.
The first slice differential has the following form on the given sum-mands: d ( C n h )( α ) = ( Sq ∂ n , Sq pr n ) d ( C n h )( α ) = ( ∂ g Sq Sq , inc g Sq Sq , , Sq ) n ≡ , inc g Sq Sq , , Sq ) n ≡ ∂ g Sq Sq , , , Sq ) n ≡ d ( C n h )( α q ) = ( Sq Sq , , Sq ) q ≥ d ( C n h )( α ) = ( (inc g Sq Sq , , Sq ) n ≡ g Sq Sq , inc g Sq , Sq , τ Sq ) n ≡ d ( C n h )( α q ) = ( ( Sq Sq , , Sq ) n ≡ Sq Sq , Q , Sq , τ Sq ) n ≡ d ( C n h )( α ) = (0 , Sq ∂ g , , τ ∂ g ) d ( C n h )( α ) = ( ( Sq ∂ g , , τ ∂ g , n ≡ Sq ∂ g , Sq pr g , τ ∂ g , τ pr g ) n d ( C n h )( α q − α ) = ( Sq Sq , , Sq + ρ Sq , , τ ) q ≥ d ( C n h )( α q − α ) = ( ( Sq Sq , , Sq + ρ Sq , , τ ) n ≡ Sq Sq , Q , Sq + ρ Sq , τ Sq + ρ, τ ) n ≡ Proof.
Based on the form of the unit map, most parts of the first slice differential aredetermined by Theorem 6.2. More precisely, the differentials s C n h → Σ , s C n h , s C n h → Σ , s C n h and the listed behaviour for q ≥ d = 0, aswell as comparison with d ( ). (cid:3) Theorem 6.6 may be used for computations of π ⋆ ) C n h and π ⋆ ) C n h . Com-pared with corresponding slice spectral sequence computations for , the absenceof integral motivic cohomology groups may be viewed as an advantage. Concretepresentations for H ⋆ − k,⋆ as K MW -modules seem to be known only in very few cases,contrary to the K MW -module h ⋆ − k,⋆ ∼ = K MW ⋆ − k / ( η, References [1] A. Ananyevskiy, O. R¨ondigs, and P. A. Østvær,
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