THE MOTIVIC ADAMS VANISHING LINE OF SLOPE 1/2 5 BERTRANDJ.GUILLOUANDDANIELC.ISAKSEN 1 0 2 Abstract. We establish a motivic version of Adams’ vanishing line of slope 1/2inthecohomologyofthemotivicSteenrodalgebraoverC. n a J 3 1 1. Introduction ] Oneofthemostpowerfultoolsforcomputingstablehomotopygroupsofspheres T is the Adams spectral sequence A Exts,t(F ,F )⇒πˆ , h. A 2 2 t−s t where A denotes the Steenrod algebra of stable mod 2 cohomology operations and a m πˆ∗ arethe 2-completedstablehomotopygroups. Adams establishedthatthese Ext groups vanish above a certain line of slope 1/2, with the exception of the elements [ hk in the 0-stem [A2]. 0 1 In the motivic context over C, inspection of an Adams chart [I2] immediately v showsthattheanalogousExtgroupsdonotvanishabovethesamelineofslope1/2. 2 (However,the motivic Ext groupsdo vanishabove a line of slope 1, analogouslyto 7 8 an earlier classical vanishing result of Adams [A1].) Further inspection shows that 2 ina largerange,allelementsabovethe Adams line ofslope 1/2areh -local,inthe 1 0 sense that they are h -divisible and support infinitely many multiplications by h . 1 1 . 1 This suggests the following theorem, whose proof is the goal of this article. 0 5 Theorem 1.1. Let s>0, and let A be the motivic mod 2 Steenrod algebra over C. 1 The map v: h1 :ExtsA,f,w(M2,M2)→ExtsA+1,f+1,w+1(M2,M2) i X is an isomorphism if f ≥ 1s+2, and it is a surjection if f ≥ 1s+ 1. 2 2 2 r a In the grading (s,f,w) from Theorem 1.1, s is the stem and f is the Adams filtration so that (s,f) represent the traditional coordinates in an Adams chart. Meanwhile, w is the motivic weight, which is not relevant to the statement of the theorem. Proof. In Section 7, we will show that Exts,f,w(M ,M )∼=Exts−1,f−1,w−1(N,M ), A 2 2 A 2 Date:January14,2015. 2000 Mathematics Subject Classification. 14F42,55S10. Keywordsandphrases. cohomologyoftheSteenrodalgebra,vanishingline,motivichomotopy theory. The first author was supported by Simons Collaboration Grant 282316. The second author wassupportedbyNSFgrantDMS-1202213. 1 2 BERTRANDJ.GUILLOUANDDANIELC.ISAKSEN where N is a certain A-module that is free as a left A(0)-module. We will prove in Proposition 7.2 that multiplication by h has the desired property for such A- 1 modules. (cid:4) InspectionshowsthatTheorem1.1isoptimalinthefollowingsense. Multiplica- tionbyh isnotanisomorphismalongthelinef = 1s+3,anditisnotasurjection 1 2 2 along the line f = 1s. 2 This article is only concerned with algebraic calculations of Ext groups and does not discuss Adams spectral sequences for which these Ext groups are inputs. However,our interestin h multiplications is motivated by recentwork[AM2] [GI] 1 onthe η-localmotivic sphere,where η is the first Hopfmap that is detected by h . 1 Another possible approachto Theorem1.1 uses the methods establishedby An- drews and Miller [AM2] (see especially [AM2, Lemma 5.2.1]). However, one must be careful about the slightly strange behavior of motivic Margolis homology (see Proposition 3.2). For the techniques of [AM2] to work motivically, one needs to verify that certain quotients of the motivic Steenrod algebra are M -free. This is 2 more or less the same as the homological results of Sections 3 and 4. 1.1. Organization. We review some background in Section 2. In Sections 3 through6,weassemblevariousfactsaboutmotivichomologicalalgebra. Finally,we give the main technical result in Proposition 7.2, from which Theorem 1.1 follows immediately. 2. Preliminaries 2.1. Notation. We continue with notation from [I1] as follows: (1) M is the motivic cohomology of C with F coefficients. 2 2 (2) A isthe mod2motivic SteenrodalgebraoverC generatedbyelements Sq1 of bidegree (1,0) and Sq2n of bidegree (2n,2n−1) for n≥1. (3) A(0) is the (exterior) M -subalgebra of A generated by Sq1. 2 (4) A(1) is the M -subalgebra of A generated by Sq1 and Sq2. 2 (5) Ext is the trigraded ring Ext (M ,M ). A A 2 2 The following two theorems of Voevodsky are the starting points of our calculations. Theorem 2.1 ([V1]). M is the bigraded ring F [τ], where τ has bidegree (0,1). 2 2 It is often more convenient to work with the dual A∗,∗ =HomM2(A,M2), which was described by Voevodsky. (See also [B] for a clean description.) Theorem 2.2. [V2] [V3, Theorem12.6] The dual motivic Steenrod algebra A is ∗,∗ generated as an M -algebra by ξ ∈ A and τ ∈ A subject to 2 i 2(2i−1),2i−1 i 2i+1−1,2i−1 the relations τ2 =τξ . i i+1 Thecoproduct isgiven on thegenerators bythefollowing formulas, inwhich ξ =1: 0 ∆(τ )=τ ⊗1+ ξ2i ⊗τ k k k−i i Xi ∆(ξ )= ξ2i ⊗ξ . k k−i i Xi THE MOTIVIC ADAMS VANISHING LINE OF SLOPE 1/2 3 Remark 2.3. The quotientA∗,∗/τ =A∗,∗⊗M2F2 is analogoustothe odd-primary classical dual Steenrod algebra, in the sense that there is an infinite family of exterior generators τ and an infinite family of polynomial generators ξ . On the i i other hand, the localization A [τ−1] is analogous to the mod 2 classical dual ∗,∗ Steenrod algebra, which has only polynomial generators τ . i 2.2. Grading conventions. We follow[I1]ingradingExt accordingto(s,f,w), A where: (1) f is the Adams filtration, i.e., the homologicaldegree. (2) s+f is the internal degree, i.e., corresponds to the first coordinate in the bidegrees of A. (3) s is the stem, i.e., the internal degree minus the Adams filtration. (4) w is the weight. 3. Margolis homology Recall that Sq1Sq1 = 0, so that Sq1 acts as a differential on any A-module. We write H∗∗(M;Sq1) for the resulting Margolis homology groups [AM1]. We say that an A-module is bounded below if Mp,q = 0 for sufficiently small p. A bounded below A-module is of finite type if each Mp,∗ is a finitely generated M -module. 2 Remark3.1. WewillneedthefollowingfactaboutfinitelygeneratedM -modules. 2 Such modules are of the form (M )k⊕ (M /τri). 2 2 Mi This is a graded version of the classification of finitely generated modules over a principal ideal domain, since every homogeneous ideal of M is generated by an 2 element of the form τri. Consequently, a finitely generated M2-module is free if and only if it has no τ torsion. Proposition 3.2. Let M be a bounded below A-module of finite type. Then M is free as an A(0)-module if and only if M is free as an M -module and 2 H∗,∗(M;Sq1)=0. Proof. The forwardimplication is clear. Thus suppose that M is free over M and 2 H∗,∗(M;Sq1) = 0. Suppose that M is concentrated in degrees (p,q) with p ≥ n . 0 Let x be a nonzero element of Mn0,∗ of smallest weight, and let y = Sq1(x) in Mn0+1,∗. The map Sq1 :Mn0,∗ →Mn0+1,∗ is injective because H∗,∗(M;Sq1)=0, so y is non-zero. Let N be the M -submodule of M generated by x and y, and let P be the 2 quotient M/N. Then N is a free A(0)-module generated by x. WewillnextarguethatP isM -free;byRemark3.1,thisisthesameasshowing 2 that P has no τ torsion. Equivalently, we will show that if τz belongs to N, then so does z. The main point is that y is not divisible by τ. Suppose for sake of contradiction that τz = y. Then τSq1(z) = Sq1(y) = 0. Since M is M -free, 2 Remark 3.1 implies that Sq1(z) = 0. On the other hand, z cannot be in the image of Sq1 because its weight is less than the weight of x. This contradicts the assumption that H∗,∗(M;Sq1) is zero. 4 BERTRANDJ.GUILLOUANDDANIELC.ISAKSEN This establishes that M is isomorphic to N ⊕P as an A(0)-module. The long exact sequence associated to the short exact sequence 0−→N −→M −→P −→0 shows that H∗,∗(P;Sq1)=0. Having split M as N ⊕P, we may now apply the same argument to P to split off another free A(0)-module. The finite type assumption on M guarantees that this process eventually splits all of M into free A(0)-modules. (cid:4) Corollary 3.3. Suppose that 0−→M −→M −→M −→0 1 2 3 is a short exact sequence of bounded below A-modules of finite type which are free over M . If any two modules in this sequence are free over A(0), then so is the 2 third. Proof. Two of the modules have no Margolis homology by Proposition 3.2. The longexactsequenceinMargolishomologyshowsthatthethirdalsohasnoMargolis homology. Then Proposition 3.2 again implies that the third module is free over A(0). (cid:4) 4. The motivic Milnor basis Let E = (ǫ ,ǫ ,ǫ ,...) be a sequence of ones and zeros, almost all zero, and 0 1 2 let R = (r ,r ,...) be a sequence of nonnegative integers, almost all zero. Then, 1 2 according to Theorem 2.2, the elements τ(E)ξ(R):= τǫi ξrj i j iY≥0 jY≥1 give an M2-basis for A∗,∗. We follow [DI] in writing P(ǫ0+2r1,ǫ1+2r2,...) for the cor- respondingelements of the dual M -basis for A. These elements formthe motivic 2 Milnor basis for A. The resulting M -basis for A(1) consists of the elements 2 P(s1,s2,0,0...) such that 0≤s1 ≤3 and 0≤s2 ≤1. Let S = (s ,s ,...). The excess of the Milnor basis element PS is defined to 1 2 be e(PS)= s . We extend this to arbitrary elements of A by taking the excess i i ofan elemenPt θ to be the maximalexcess of any Milnor basis element appearingin an expression for θ. For two sequences R = (r ,r ,...) and S = (s ,s ,...), we write R+S for the 1 2 1 2 termwise sum (r +s ,r +s ,...). 1 1 2 2 Lemma 4.1. Let R=(r ,r ,...) and S =(s ,s ,...). Then 1 2 1 2 r PR·PS = i ·PR+S + terms of lower excess. (cid:18)s (cid:19) Yi i Proof. Thisfollowsfromthe descriptionofthe dualmotivic Steenrodalgebra,sim- ilarly to the classical case [M]. (cid:4) WewillusetheMilnorbasistoestablishthefollowingfactabouttherightaction of A(1) on A. Proposition 4.2. The motivic Steenrod algebra A is free as a right A(1)-module. THE MOTIVIC ADAMS VANISHING LINE OF SLOPE 1/2 5 Proof. Define a filtration on A by F (A)={θ ∈A|e(θ)≤s}. Lemma 4.1 implies s that the associated graded object gr A= F (A)/F (A) ∗ s s−1 Ms inheritsthestructureofanM -algebra. LetM betheM -submoduleofAgenerated 2 2 by Milnor basis elements of the form P(4s1,2s2,s3,s4,...). Consider the composition M ⊗M2 A(1) //A // gr∗A, in which the first map is induced by the multiplication of A and the second map is an isomorphism of M -modules. 2 Lemma 4.1 implies that the composition is an isomorphism of M -modules. 2 Therefore,thefirstmapisalsoanisomorphism. ThisshowsthatAisfreeasaright A(1)-module with basis consisting of elements of the form P(4s1,2s2,s3,s4,...). (cid:4) Remark 4.3. The reader may wonder why Proposition 4.2 is not an immediate consequence of results in [MM]. The problem is that it is not obvious that the projection A −→ A//A(1) is split as an M -map. The proof of Proposition 4.2 is 2 essentially the same as showing that the projection is in fact split. Example 4.4. Let B be the Hopf subalgebra of A that is generated over M by 2 τSq1. The projection A−→A//B is not split as an M -map, since Sq1 projects to 2 an element that is τ torsion. Moreover, A is not free as a right B-module. 5. The module A. Definition 5.1. Let A(1) be the left A(1)-moduele on two generators a and b of degrees (0,0) and (2,0) respectively, subject to the relations e Sq2a=τb, Sq1Sq2Sq1a=Sq2b. The relationSq2a=τb impliesthatSq2Sq2a=τSq2b,soτSq1Sq2Sq1a=τSq2b. However, the first relation does not imply that Sq1Sq2Sq1a = Sq2b. This explains why we need a second relation in the definition of A(1). Figure 1 represents A(1), according to the following key: e (1) Each circle represents a copy of M . e 2 (2) Each straight line represents multiplication by Sq1. (3) Each curved line represents multiplication by Sq2. (4) Each dashed line indicates that the squaring operation hits τ times an M -generator,but not the generator itself. 2 Analogous to the Milnor basis for A(1), we have a basis for A(1) consisting of the elements P(s1,s2) such that 0 ≤ s1 ≤ 3 and 0 ≤ s2 ≤ 1. Theeelements P2, P3, P2,1, and P3,1 have weight one less than the corresponding Milnor basis elements e e e for A(1). We define the excess in A(1) using this basis. e e Remark 5.2. Justlikeintheclasseicalcase[DM],theA(1)-modulesA(1)andA(1) each extend to A-modules in four different ways, determined by the action of Sq4. e Adams spectral sequence computations verify that all eight of these A-modules ariseasthecohomologyofa2-completemotivicspectrum. Theseconstructionsare the subject of work in progress on motivic v -self maps. 1 Definition 5.3. Let A be the left A-module A⊗ A(1). A(1) e e 6 BERTRANDJ.GUILLOUANDDANIELC.ISAKSEN Figure 1. A(1) e Remark 5.4. Proposition 4.2 implies that A is the left A-module generated by two elements a and b of degrees (0,0) and (2,0) subject to the relations e Sq2a=τb, Sq3Sq1a=Sq2b. Proposition 5.5. The A-module A is free as a left A(0)-module. Proof. We have a Milnor-style baesis for A consisting of elements of the form P(4r1,2r2,r3,...) ⊗Ps1,s2 such that 0 ≤ s1 ≤e3 and 0 ≤ s2 ≤ 1. We also have a filtration by excess. e Lemma 4.1 implies that P1P(4r1,2r2,r3,...) ≡P(4r1+1,2r2,r3,...) ≡P(4r1,2r2,r3,...)P1 inA,modulotermsoflowerexcess. Also, P1Ps1,s2 =Ps1+1,s2 inA(1)ifs1 iseven. Therefore, if s is even, then 1 e e e P1(P(4r1,2r2,r3,...)⊗Ps1,s2)≡P(4r1,2r2,r3,...)⊗Ps1+1,s2 in A, modulo terms of lower excesse. It follows that an A(0)-beasis for A consists of elemeents of the form P(4r1,2r2,r3,...)⊗Ps1,s2 such that s1 is even. e (cid:4) e 6. The cofiber of η Definition6.1. LetC(η)denotethecofiberofthefirstHopfmapη :S1,1 −→S0,0. We then write C forthe A-module Σ−2,−1H∗,∗(C(η)). Thus C hasa bottomcell η η in bidegrees (−2,−1) and a top cell in (0,0), connected by a Sq2 (see figure). (0,0) Sq2 (−2,−1) The followingresultimplies that, foranyA-module M,the groupsExt∗(M,C ) η may be computed using a resolution F∗ → M whose terms are of the form Fn ∼=Ar ⊕As. Propositione 6.2. Exts,f(A,C )=0 for f >0. A η e THE MOTIVIC ADAMS VANISHING LINE OF SLOPE 1/2 7 Proof. The kernel of the defining quotient A(1)⊕Σ2,0A(1)−→A(1) is isomorphic to Σ2,1A(1). It follows that we can define a periodic free A(1)-resolution e e A(1)←F0 ←F1 ←··· , where Fn ∼=A(1){xn,yn}, witeh |xn|=(2n,n), |yn|=(2n+2,n), d(x )=Sq2(x )+τ ·y , and d(y )=Sq3Sq1(x )+Sq2(y ). n n−1 n−1 n n−1 n−1 Using this resolution, it is simple to verify that Exts,f (A(1),C )=0, for f >0. A(1) η Since A is free as a right A(1)-meodule by Proposition 4.2, we get that Exts,f(A,C )∼=Exts,f (A(1),C )=0 for f >0. A η A(1) η (cid:4) e e Thenextlemmatranslatespropertiesofh multiplicationintohomologicalprop- 1 erties of the A-module C . η Lemma 6.3. Let n≥0 and let M be an A-module. The following are equivalent: (1) The map h :Exts,f,w(M,M )−→Exts+1,f+1,w+1(M,M ) 1 A 2 A 2 is injective when s < 2f −2 and f ≤ n, and it is surjective when s ≤ 2f and f 1. The bottom class of I is Sq2, which occurs in bidegree (2,1), so that we may write I ∼=Σ2,1N for a connective A-module N. It follows that for s>0, we have a commutative square Exts,f,w h1 //Exts+1,f+1,w+1 A A OO OO ∼= ∂ ∂ ∼= Exts+1,f−1,w(I,M ) // Exts+2,f,w+1(I,M ) A OO 2 h1 A OO 2 ∼= ∼= Exts−1,f−1,w−1(N,M ) // Exts,f,w(N,M ) A 2 h1 A 2 of Ext -module maps. A Lemma 7.1. The A-module N is free as a left A(0)-module. Proof. This argument is identical to the classical case. We use the admissible M -basis for A, which consists of monomials of the form 2 Sqr1Sqr2···Sqrn such that r ≥ 2r . Then A//A(0) has an M -basis consisting i i+1 2 of the admissible monomials Sqr1Sqr2···Sqrn such that r >1. n When r is even, Sq1 ·Sqr1Sqr2···Sqrn equals Sqr1+1Sqr2···Sqrn. Therefore, 1 theaugmentationidealI ofA//A(0)isafreeleftA(0)-modulewithbasisconsisting of admissible monomials of the form Sqr1Sqr2···Sqrn such that r is even and 1 r >1. (cid:4) n Proposition 7.2. Let M be an A-module that is free as an A(0)-module and concentrated in nonnegative degrees. Then the map h :Exts,f,w(M,M )−→Exts+1,f+1,w+1(M,M ) 1 A 2 A 2 is an isomorphism if s<2f −2, and it is a surjection if s≤2f. We will mimic the classical argument of Adams [A2], with some variations to account for motivic phenomena. Proof. By Lemma 6.3, it suffices to show that Exts,f,w(M,C ) vanishes when s< A η 2f. We begin by recalling from Proposition 6.4 that Exts,f,w(A(0),C ) vanishes A η when s<2f and f ≤4. An arbitrary module M can be built up iteratively as an extension 0−→ Σti,wiA(0)−→M −→M′ −→0 M ofA-modules, where Exts,f,w(M′,C ) vanishes for s<2f andf ≤4 by induction. A η The long exact sequence in Ext then shows that Exts,f,w(M,C ) also vanishes for A η s<2f and f ≤4. We have now established the proposition for f ≤ 4. The next step is to extend the result to larger values of f. As in the previous step, we start with the special case M =A(0). 10 BERTRANDJ.GUILLOUANDDANIELC.ISAKSEN InordertocomputeExts,f,w(A(0),C ),wemustconstructaresolutionforA(0). A η Proposition6.2saysthatthisresolutioncanbebuiltfromcopiesofAorfromcopies of A. We construct a resolution e A(0)←R ←R ←R ←R ←··· 0 1 2 3 of A(0) in the usual way by adding a copy of A to R for each indecomposable n+1 in the kernel K of the boundary map R −→ R . However, when we find two n n n−1 indecomposable elements x and y of K such that Sq2x=τy, we add one copy of n A to R rather than two copies of A. n+1 As a result of this process, one can verify that the kernel K of the boundary e 3 map R −→ R vanishes in degrees less than 12. So we can write K = Σ12,0D, 3 2 3 where D is concentrated in non-negative degrees. Note that D is M -free since it 2 is a submodule of the M -free module Σ−12,0R . 2 3 Proposition 5.5 implies that each R is A(0)-free. Then Corollary 3.3 implies n that D is A(0)-free as well. We already know from an earlier step that Exts,f,w(D,C ) vanishes for s < 2f A η and f ≤4. We have isomorphisms Exts,f,w(D,C )∼=Exts+12,f,w(K ,C )∼=Exts+8,f+4,w(A(0),C ) A η A 3 η A η for f >0, so it follows that Exts,f,w(A(0),C ) vanishes for s<2f and f ≤8. A η As before, we can then show that when M is an arbitrary module, Exts,f,w(M,C ) vanishes for s < 2f and f ≤ 8. This establishes the proposition A η for f ≤8. We can repeat this process to establish the proposition for all f. (cid:4) References [A1] J. F. Adams, A finiteness theorem in homological algebra, Proc. Cambridge Philos. Soc. 57(1961)31–36. [A2] J.F. Adams,A periodicity theorem in homological algebra, Proc. CambridgePhilos.Soc. 62(1966)365–377. [AM1] J.F.AdamsandH.R.Margolis,Modules over the Steenrod algebra, Topology 10(1971) 271–282. [AM2] M. Andrews and H. Miller, Inverting the Hopf map in the Adams-Novikov spectral sequence,preprint,2014. [B] S. Borghesi, Cohomology operations and algebraic geometry, Proceedings of the Nishida Fest(Kinosaki2003), 75–115, Geom.Topol.Monogr.10,2007. [DI] D. Dugger and D. C. Isaksen, The motivic Adams spectral sequence, Geom. Topol. 14 (2010) 967–1014. [DM] D.DavisandM.Mahowald,v1-andv2-periodicityinstablehomotopytheory,Amer.Jour. Math.104(1981), no.3,615–659. [GI] B.GuillouandD.C.Isaksen,The η-local motivic sphere,preprint,2014. [I1] D.C.Isaksen,Stable stems,preprint,2014. [I2] D.C.Isaksen,Classical and motivic Adams charts,preprint,2014. [M] J.Milnor,The Steenrod algebra and itsdual, Ann.ofMath.67(1958)150–171. [MM] J.W.MilnorandJ.C.Moore,OnthestructureofHopfalgebras,Ann.ofMath.81(1965) 211–264. [V1] V.Voevodsky,Motiviccohomology withZ/2-coefficients,Publ.Math.Inst.HautesE´tudes Sci.No.98(2003), 59–104. [V2] V. Voevodsky, Motivic Eilenberg-Maclane spaces, Publ. Math. Inst. Hautes E´tudes Sci. No.112(2010), 1–99. [V3] V.Voevodsky,Reducedpoweroperationsinmotiviccohomology,Publ.Math.Inst.Hautes E´tudes Sci.No.98(2003), 1–57.

The motivic Adams vanishing line of slope 1/2

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