An R -motivic v 1 − self-map of periodicity 1
AAN R -MOTIVIC v –SELF-MAP OF PERIODICITY P. BHATTACHARYA, B. GUILLOU, AND A. LI
Abstract.
We consider a nontrivial action of C on the type 1 spectrum Y := M (1) ∧ C( η ), which is well-known for admitting a 1-periodic v –self-map. The resultant finite C -equivariant spectrum Y C can also be viewed asthe complex points of a finite R -motivic spectrum Y R . In this paper, we showthat one of the 1-periodic v –self-maps of Y can be lifted to a self-map of Y C as well as Y R . Further, the cofiber of the self-map of Y R is a realization of thesubalgebra A R (1) of the R -motivic Steenrod algebra. We also show that theC -equivariant self-map is nilpotent on the geometric fixed-points of Y C . Introduction
In classical stable homotopy theory, the interest in periodic v n –self-maps of finitespectra lies in the fact that one can associate to each v n –self-map an infinite familyin the chromatic layer n stable homotopy groups of spheres. Therefore, interestlies in constructing type n spectra and finding v n –self-maps of lowest possible pe-riodicity on a given type n spectrum. This, in general, is a difficult problem,though progress has been made sporadically throughout the history of the subject[T, DM, BP, BHHM, N, BEM, BE]. With the modern development of motivic sta-ble homotopy theory, one may ask if there are similar periodic self-maps of finitemotivic spectra.Classically any non-contractible finite p -local spectrum admits a periodic v n –self-map for some n ≥
0. This is a consequence of the thick-subcategory theorem[HS, Theorem 7], aided by a vanishing line argument [HS, § Sp p, fin (the homotopy category of finite p -localspectra) are also prime (in the sense of [B]). The thick tensor-ideals of the homotopycategory of cellular motivic spectra over C or R are not completely known (but see[HO]). However, one can gather some knowledge about the prime thick tensor-ideals in Ho( Sp R , fin ) (the homotopy category of 2-local cellular R -motivic spectra)through the Betti realization functor β : Ho( Sp R , fin ) Ho( Sp C , fin )using the complete knowledge of prime thick subcategories of Ho( Sp C , fin ) [BS].The prime thick tensor-ideals of Ho( Sp C , fin ) are essentially the pull-back of theclassical thick subcategories along the two functors, the geometric fix point func-tor Φ C : Ho( Sp C , fin ) Ho( Sp , fin ) B. Guillou and A. Li were supported by NSF grants DMS-1710379 and DMS-2003204. a r X i v : . [ m a t h . A T ] N ov P. BHATTACHARYA, B. GUILLOU, AND A. LI and the forgetful functorΦ e : Ho( Sp C , fin ) Ho( Sp , fin ) . Let C n denote the thick subcategory of Ho( Sp , fin ) consisting of spectra of type atleast n . The prime thick subcategories, C ( e, n ) = (Φ e ) − ( C n ) and C (C , n ) = (Φ C ) − ( C n ) , are the only prime thick subcategories of Ho( Sp C , fin ). Definition 1.1.
We say a spectrum X ∈ Ho( Sp C , fin ) is of type ( n, m ) iff Φ e ( X ) isof type n and Φ C ( X ) is of type m .For a type ( n, m ) spectrum X , a self-map f : X → X is periodic if and only if atleast one of { Φ e ( f ) , Φ C ( f ) } are periodic (see [BGH, Proposition 3.17]). Definition 1.2.
Let X ∈ Ho( Sp C , fin ) be of type ( n, m ). We say a self-map f : X → X is(i) a v ( n,m ) –self-map of mixed periodicity ( i, j ) if Φ e ( f ) is a v n –self-map ofperiodicity i and Φ C ( f ) is a v m –self-map of periodicity j ,(ii) a v ( n, nil) –self-map of periodicity i if Φ e ( f ) is a v n –self-map of periodicity i and Φ C ( f ) is nilpotent, and,(iii) a v (nil ,m ) –self-map of periodicity j if Φ e ( f ) is a nilpotent self-map andΦ C ( f ) is a v m –self-map of periodicity j . Example 1.3.
The sphere spectrum S C is of type (0 , v (0 , –self-map. In general, if we consider the v n –self-map of a type n spectrum withtrivial action of C , then the resultant equivariant self-map is a v ( n,n ) –self-map. Example 1.4.
Let S , denote the C -equivariant sphere which is the one-pointcompactification of the real sign representation. The unstable twist-map (cid:15) u : S , ∧ S , S , ∧ S , stabilizes to a nonzero element (cid:15) ∈ π , ( S C ). Let h = 1 − (cid:15) ∈ π , ( S C ) be thestabilization of the map h u = 1 − (cid:15) u : S , S , . Note that on the underlying space (cid:15) is of degree −
1, while on the fixed points itis the identity. Therefore Φ e ( h ) is multiplication by 2, whereas Φ C ( h ) is trivial.Thus h is a v (0 , nil) –self-map. Thus C C ( h ) is of type (1 , Example 1.5.
The equivariant Hopf-map η , ∈ π , ( S C ) is the Betti realizationof the R -motivic Hopf-map η [M2, DI3]. Up to a unit, it is the stabilization of the N R -MOTIVIC v –SELF-MAP OF PERIODICITY 1 3 projection map η u , := π : S , (cid:39) C \ { } CP ∼ = S , , where the domain and the codomain are given the C -structure using complexconjugation. On fixed-points, the map π is the projection map π : R \ { } RP , which is a degree 2 map. From this we learn that while Φ e ( η , ) is nilpotent,Φ C ( η , ) is the periodic v –self-map. Hence, η , is a v (nil , –self-map and thecofiber C( η , ) is of type (0 , Remark 1.6.
In the C -equivariant stable homotopy groups, the usual Hopf-map(sometimes referred to as the ‘topological Hopf-map’) is different from η , of Ex-ample 1.5. The ‘topological Hopf-map’ η , ∈ π , ( S C ) should be thought of as thestabilization of the unstable Hopf-map( η , ) u : S , S , where both domain and codomain are given the trivial C -action. Definition 1.7.
We say a spectrum X ∈ Ho( Sp R , fin ) is of type ( n, m ) if β ( X ) isof type ( n, m ). We call an R -motivic self-map f : X → X a v ( n,m ) –self-map, where m and n are in N ∪ { nil } (but not both nil), if β ( f ) is aC -equivariant v ( n,m ) –self-map. Remark 1.8.
The maps ‘multiplication by 2’ (of Example 1.3), h (of Example 1.4),and η , (of Example 1.5) admit R -motivic lifts along β and provide us with ex-amples of a v (0 , –self-map, v (0 , nil) –self-map and v (nil , –self-map of the R -motivicsphere spectrum S R , respectively.A theorem of Balmer and Sanders [BS] asserts that C ( e, n ) ⊂ C (C , m ) if and onlyif n ≥ m + 1. In particular, C ( e, n ) is contained in C (C , n − n, m ) (C -equivariant or R -motivic) spectra if n ≥ m + 2. Their resultalso implies the following: Proposition 1.9.
Let X ∈ Ho( Sp C , fin ) be of type ( n + 1 , n ) for some n . Then X cannot support a v ( n +1 , nil) –self-map. The proposition holds since the cofiber of such a self-map would be of type ( n +2 , n ),contradicting the results of Balmer-Sanders. In particular, neither C C ( h ) nor C R ( h )supports a v (1 , nil) –self-map. However, it is possible that C C ( h ) as well as C R ( h )can admit a v (1 , –self-map or a v (nil , –self-map. In fact, η , ∈ π , ( S R ) and η , ∈ π , ( S C ) induce v (nil , –self-maps of C R ( h ) and C C ( h ) respectively. InSection 5, we show that: Theorem 1.10.
The spectrum C R ( h ) does not admit a v (1 , –self-map. P. BHATTACHARYA, B. GUILLOU, AND A. LI
However, it is possible that C C ( h ) admits a v (1 , –self-map (for details see Re-mark 5.6). In contrast to the classical case, there is no guarantee that a finiteC -equivariant or R -motivic spectrum will admit any periodic self-map, or at leastnothing concrete is known yet. This question must be studied!The goal of this paper is rather modest. We consider the classical spectrum Y := M (1) ∧ C( η )that admits, up to homotopy, 8 different v –self-maps of periodicity 1 [DM, Sec-tion 2] (see also [BEM]). We ask ourselves if the v –self-maps can preserve symme-tries upon providing Y with interesting C -equivariant structures. We will considerfour C -equivariant lifts of the spectrum Y ,(i) Y C triv , where the action of C is trivial,(ii) Y C (2 , := C C (2) ∧ C C ( η , ), with Φ C ( Y C (2 , ) = M (1) ∧ M (1),(iii) Y C ( h , := C C ( h ) ∧ C C ( η , ), with Φ C ( Y C ( h , ) = ΣC( η ) ∨ C( η ), and,(iv) Y C ( h , := C C ( h ) ∧ C C ( η , ), with Φ C ( Y C ( h , ) = ΣM (1) ∨ M (1).The C -spectra Y C triv , Y C (2 , and Y C ( h , are of type (1 , Y C ( h , is of type (1 , R -motivic lifts of the classes 2, h , η , , and η , , and therefore wehave unique R -motivic lifts of Y C triv , Y C (2 , , Y C ( h , , and Y C ( h , which we will simplydenote by Y R triv , Y R (2 , , Y R ( h , , and Y R ( h , , respectively. In this paper we prove: Theorem 1.11.
The R -motivic spectrum Y R ( h , admits a v , nil –self-map v : Σ , Y R ( h , Y R ( h , of periodicity . By applying the Betti realization functor we get:
Corollary 1.12.
The C -equivariant spectrum Y C ( h , admits a -periodic v , nil –self-map β ( v ) : Σ , Y C ( h , Y C ( h , . Corollary 1.13.
The geometric fixed-point spectrum of the telescope β ( v ) − Y C ( h , is contractible. Classically, the cofiber of a v –self-map on Y is a realization of the finite subalgebra A (1) of the Steenrod algebra A . We see a very similar phenomenon in the R -motivic as well as in the C -equivariant cases. The C -equivariant Steenrod algebra A C as well as the R -motivic Steenrod algebra A R admit subalgebras analogous to N R -MOTIVIC v –SELF-MAP OF PERIODICITY 1 5 A (1) (generated by Sq and Sq ) [H, R2], which we denote by A C (1) and A R (1),respectively. We observe that: Theorem 1.14.
The spectrum C R ( v ) := Cof( v : Σ , Y R ( h , → Y R ( h , ) is a type (2 , complex whose bigraded cohomology is a free A R (1) -module on one generator. Corollary 1.15.
The bigraded cohomology of the C -equivariant spectrum C C ( β ( v )) ∼ = β (C R ( v )) is a free A C (1) -module on one generator. Our last main result in this paper is the following.
Theorem 1.16.
The spectrum Y R ( h , does not admit a v (1 , –self-map. The above results immediately raise some obvious questions. Pertaining to self-maps one may ask: Does Y R (2 , admit a v , nil –self-map? Does Y R (2 , or Y R ( h , admit a v (1 , –self-map? Does Y R triv , Y R (2 , or Y R ( h , admit v (nil , –self-map? Ormore generally, how many different homotopy types of each kind of periodic self-maps exist? Related to A R (1), one may inquire: How many different A R -modulestructures can be given to A R (1)? Can those A R -modules be realized as a spectrum?Are the realizations of A R (1) equivalent to cofibers of periodic self-maps of Y R ( i,j ) ?We hope to address most, if not all, of the above questions in our upcoming work(see Remark 3.21, Remark 4.19 and Remark 5.6).1.1. Outline of our method.
We first construct a spectrum A R which realizesthe algebra A R (1) using a method of Smith (outlined in [R1, Appendix C]) whichconstructs new finite spectra (potentially with larger number of cells) from knownones. The idea is as follows. If X is a p -local finite spectrum then the permutationgroup Σ n acts on X ∧ n . One may then use an idempotent e ∈ Z ( p ) [Σ n ] to obtaina split summand of the spectrum X ∧ n . As explained in [R1, Appendix C], Youngtableaux provide a rich source of such idempotents. For a judicious choice of e and X , the spectrum e ( X ∧ n ) can be interesting.We exploit the relation that h · η , = 0 in π ∗ , ∗ ( S R ) [M2] to construct an R -motivicanalogue of the question mark complex. The cell-diagram of the question markcomplex is as described in the picture below. For a choice of idempotent element e Q R = h η , . Figure 1.17.
Cell-diagram of the R -motivic question mark com-plex P. BHATTACHARYA, B. GUILLOU, AND A. LI in the group ring Z (2) [Σ ], we observe that e (H ∗ , ∗ ( Q R ) ⊗ ) is a free A R (1)-module.This is the cohomology of an R -motivic spectrum ˜ e ( Q ∧ R ), which we call Σ , A R (see (3.4) for details). The observation requires us to develop a criterion that willdetect freeness for modules over certain subalgebras of A R . Writing M R for the R -motivic cohomology of a point, we prove: Theorem 1.18.
A finitely generated A R (1) -module M is free if and only if(1) M is free as an M R -module, and(2) F ⊗ M R M is a free F ⊗ M R A R (1) -module. The cohomology of A R provides an A R -module structure on A R (1), which immedi-ately gives us a short exact sequence0 → H ∗ , ∗ (Σ , Y R ( h , ) → H ∗ , ∗ ( A R ) → H ∗ , ∗ ( Y R ( h , ) → A R -modules. Thus, we get a candidate for a v , nil –self-map in the R -motivicAdams spectral sequence v ∈ Ext ∗ , ∗ , ∗A R (H ∗ , ∗ ( Y R ( h , ) , H ∗ , ∗ ( Y R ( h , )) ⇒ [ Y R ( h , , Y R ( h , ] ∗ , ∗ which survives as there is no potential target for a differential supported by v . Organization of the paper.
In Section 2, we review the R -motivic Steenrodalgebra A R , discuss the structure of its subalgebra A R ( n ), and prove Theorem 1.18.In Section 3, we construct the spectrum A R that realizes the subalgebra A R (1)and prove that it is of type (2 , Y R ( h , admits a v , nil –self-map and that its cofiberhas the same A R -module structure as that of H ∗ , ∗ ( A R ). In Section 5, we show thenon-existence of a v (1 , –self-map on C R ( h ) and Y R ( h , ; i.e., we prove Theorem 1.10and Theorem 1.16. Acknowledgement.
The authors are indebted to Nick Kuhn for explaining someof the subtle points of Smith’s work exposed in [R1][Appendix C], which is the keyidea behind Theorem 1.11. The authors also benefited from conversations withMark Behrens, Dan Isaksen, and Zhouli Xu.2.
The R -motivic Steenrod algebra and a freeness criterion We begin by reviewing the R -motivic Steenrod algebra A R following Voevodsky[V]. The algebra A R is the bigraded homotopy classes of self-maps of the R -motivicEilenberg-Mac Lane spectrum H F R : A R = [H F R , H F R ] ∗ , ∗ . The unit map S R → H F R induces a canonical projection map (cid:15) : A R −→ M R := [ S R , H F R ] ∗ , ∗ ∼ = F [ τ, ρ ] , where | τ | = (0 , −
1) and | ρ | = ( − , − F R ∧ H F R → H F R one can give A R a left M R -module structure as well as a right M R -module structure. Voevodsky shows that A R is a free left M R -module. There N R -MOTIVIC v –SELF-MAP OF PERIODICITY 1 7 is an analogue of the classical Adem basis in the motivic setting, and Voevodsy es-tablished motivic Adem relations, thereby completely describing the multiplicativestructure of A R .The motivic Steenrod algebra A R also admits a diagonal map, so that its left M R -linear dual is an algebra over F . Note that A R is an F -algebra but not an M R -algebra as τ is not a central element since(2.1) Sq ( τ ) = ρ (cid:54) = τ Sq . This complication is also reflected in the fact that the pair ( M R , hom M R ( A R , M R ))is a Hopf-algebroid, and not a Hopf-algebra like its complex counterpart. Theunderlying algebra of the dual R -motivic Steenrod algebra is given by A R ∗ ∼ = M R [ ξ i +1 , τ i : i ≥ / ( τ i = τ ξ i +1 + ρτ i +1 + ρτ ξ i +1 )where ξ i and τ i live in bidegree (2 i +1 − , i −
1) and (2 i +1 − , i − [R2] identified the quotient Hopf-algebroids of A R ∗ (see also [H]). In partic-ular, there are quotient Hopf-algebroids A R ( n ) ∗ = A R ∗ / ( ξ n , . . . , ξ n , ξ n +1 , . . . , τ n +1 , . . . , τ n , τ n +1 , . . . )which can be thought of as analogues of the quotient Hopf-algebra A ( n ) ∗ = A ∗ / ( ξ n +1 , . . . , ξ n +1 , ξ n +2 , . . . )of the classical dual Steenrod algebra A ∗ . It is an algebraic fact that τ − ( A R ( n ) ∗ / ( ρ )) ∼ = F [ τ ± ] ⊗ A ( n ) ∗ as Hopf algebras. The quotient Hopf-algebroid A R ( n ) ∗ is the M R -linear dual of thesubalgebra A R ( n ) of A R , which is generated by { τ, ρ, Sq , Sq , . . . , Sq n } .Even though τ is not in the center (2.1), ρ is in the center. We make use of thisfact to prove the following result. Lemma 2.2.
A finitely-generated A R ( n ) -module M is free if and only if(1) M is free as an F [ ρ ] -module, and,(2) M / ( ρ ) is a free A R ( n ) / ( ρ ) -module.Proof. The ‘only if’ part is trivial. For the ‘if’ part, choose a basis B = { b , . . . , b n } of M / ( ρ ) and let ˜ b i ∈ M be any lift of b i . Let F denote the free A R ( n )-modulegenerated by B and consider the map f : F → Mwhich sends b i (cid:55)→ ˜ b i . We show that f is an isomorphism by inductively provingthat f induces an isomorphism F / ( ρ n ) ∼ = M / ( ρ n ) for all n ≥
1. The case of n = 1is true by assumption. Ricka actually identified the quotient Hopf-algebroids of the C -equivariant dual Steenrodalgebra. However, the difference between the R -motivic Steenrod algebra and the C -equivariantSteenrod algebra lies only in the coefficient rings and results of Ricka easily identifies the quotientHopf-algebroids of the R -motivic Steenrod algebra. P. BHATTACHARYA, B. GUILLOU, AND A. LI
For the inductive argument, first note that the diagram0 F / ( ρ n − ) F / ( ρ n ) F / ( ρ ) 00 M / ( ρ n − ) M / ( ρ n ) M / ( ρ ) 0 · ρf n − f n f · ρ is a diagram of A R ( n )-modules (since ρ is in the center) where the horizontal rowsare exact. The map f is an isomorphism by assumption (2) , and f n − is anisomorphism by the inductive hypothesis; hence, f n is an isomorphism by the fivelemma. (cid:3) Proof of Theorem 1.18.
The result follows immediately from Lemma 2.2 combinedwith [HK, Theorem B] and the fact that A C ( n ) = A R ( n ) / ( ρ ) . (cid:3) The work of Adams and Margolis [AM] provides a freeness criterion for modulesover finite-dimensional subalgebras of the classical Steenrod algebra. For an A ( n )-module M and element x ∈ A ( n ) such that x = 0, one can define the Margolishomology of M with respect to x as M (M , x ) = ker( x : M → M)img( x : M → M) . Theorem 2.3. [AM, Theorem 4.4]
A finitely generated A ( n ) -module M is free ifand only if M (M , P st ) = 0 for < s < t with s + t ≤ n . Remark 2.4.
In the classical Steenrod algebra, P st is the element dual to ξ s t . Interms of the Milnor basis, this is Sq( (cid:122) (cid:125)(cid:124) (cid:123) , . . . , t − , s ). The element P t is often denotedby Q t − .Note that A R ( n ) ∗ / ( ρ, τ ) = F [ ξ , . . . , ξ n ]( ξ n , . . . , ξ n ) ⊗ Λ( τ , . . . , τ n )as a Hopf-algebra. Further, if we forget the motivic grading, we have an isomor-phism(2.5) A R ( n ) / ( ρ, τ ) ∼ = ϕ A ( n − ⊗ Λ(P , . . . , P n ) , where ϕ A ( n −
1) denotes the ‘double’ (see [M1, Chapter 15, Proposition 11]) of A ( n − st = ( ξ s t ) ∗ ∈ A R ( n ) . It can be shown that (P st ) ≡ ρ, τ )for s ≤ t . Combining (2.5), Theorem 2.3 and a similar result for primitively gener-ated exterior Hopf-algebras [AM, Theorem 2.2], we deduce: N R -MOTIVIC v –SELF-MAP OF PERIODICITY 1 9 Lemma 2.6.
A finitely generated A R ( n ) / ( ρ, τ ) -module M is free if and only if M (M , P st ) = 0 whenever ≤ s ≤ t and ≤ s + t ≤ n + 1 . We end this section by recording the following corollary, which is immediate fromTheorem 1.18 and Lemma 2.6.
Corollary 2.7.
A finitely generated A R ( n ) module M is free if and only if(1) M is free as an M R -module, and,(2) M (M ⊗ M R F , P ts ) = 0 for ≤ t ≤ s and s + t = n + 1 . A realization of A R (1)Consider the R -motivic question mark complex Q R , as introduced in Subsection 1.1.Let Σ n act on Q ∧ n R by permutation. Any element e ∈ Z (2) [Σ n ] produces a canonicalmap ˜ e : Q ∧ n R Q ∧ n R . Now let e be the idempotent e = − (1 3) − (1 3 2)3 in Z (2) [Σ ], and denote by e the resulting idempotent of F [Σ ]. We record the fol-lowing important property of e which is a special case of [R1, Theorem C.1.5]. Lemma 3.1. If V is a finite-dimensional F -vector space, then e ( V ⊗ ) = 0 if andonly if dim V ≤ . The following result, which gives the values of e on induced representations, is alsostraightforward to verify: Lemma 3.2.
Suppose that W = Ind Σ C F is induced up from the trivial repre-sentation of a cyclic 2-subgroup. Then e ( W ) ∼ = F . Moreover, for the regularrepresentation F [Σ ] = Ind Σ e F , we have dim e ( F [Σ ]) = 2 . We also record the fact that when dim F V = 2 and dim F W = 3 then(3.3) dim F e ( V ⊗ ) = 2 and dim F e ( W ⊗ ) = 8 , as we will often use this.The bottom cell of ˜ e ( Q ∧ R ) is in degree (1 , A R := Σ − , ˜ e ( Q ∧ R ) = Σ − , hocolim → ( Q ∧ R ˜ e → Q ∧ R ˜ e → . . . ) . The purpose of this section is to prove the following theorem.
Theorem 3.5.
The spectrum A R is a type (2 , complex whose bi-graded cohomol-ogy H ∗ , ∗ ( A R ) is a free A R (1) -module on one generator. A R is of type (2 , . Let A C := β ( A R ) and Q C := β ( Q R ) . Note that wehave a C -equivariant splitting Q ∧ (cid:39) ˜ e ( Q ∧ ) ∨ (1 − ˜ e )( Q ∧ )which splits the underlying spectra as well as the geometric fixed-points, as bothΦ e and Φ C are additive functors.We will identify the underlying spectrum Φ e ( A C ) by studying the A -module struc-ture of its cohomology with F -coefficients. Firstly, note thatΦ e ( A C ) (cid:39) Σ − ˜ e (Φ e ( Q ∧ )) (cid:39) Σ − ˜ e ( Q ∧ ) , where Q is the classical question mark complex, whose H F -cohomology as an A -module is well understood. It consists of three F -generators a , b , and c in internaldegrees 0, 1, and 3, such that Sq ( a ) = b and Sq ( b ) = c are the only nontrivialrelations, as displayed in Figure 3.6.H ∗ ( Q ; F ) = abc Figure 3.6.
We depict the A -structure of H ∗ ( Q ; F ) by markingSq -action by black straight lines and Sq -action by blue curvedlines between the F -generators.Because of the Kunneth isomorphism and the fact that the Steenrod algebra iscocommutative, we have an isomorphism of A -modulesH ∗ +1 (Φ e ( A C ); F ) ∼ = H ∗ (˜ e ( Q ∧ ); F ) ∼ = e (H ∗ ( Q ; F ) ⊗ ) . Lemma 3.7.
The underlying A (1) -module structure of H ∗ (Φ e ( A C ); F ) is free ona single generator.Proof. Let us denote the A -module H ∗ ( Q ; F ) by V. Since dim M (V , Q i ) = 1 for i ∈ { , } , it follows from the Kunnneth isomorphism of Q i -Margolis homologygroups, cocommutativity of the Steenrod algebra, and Lemma 3.1 that M ( e (V ⊗ ) , Q i ) = e ( M (V , Q i ) ⊗ ) = 0for i = { , } . It follows from [AM, Theorem 3.1] that H ∗ (Φ e ( A R ); F ) is free as an A (1)-module. It is singly generated because of (3.3). (cid:3) We explicitly identify the image of e : H ∗ ( Q ; F ) ⊗ −→ H ∗ ( Q ; F ) ⊗ in Figure 3.8. N R -MOTIVIC v –SELF-MAP OF PERIODICITY 1 11 baa + ababab + abbcaa + acacab + cba + bca + acb cbc + bcccac + acc cbb + bcbcab + bac + acb + abc Figure 3.8.
The A -module structure of H ∗ (Φ e ( A C ); F ) : Blackstraight lines, blue curved lines, and red boxed lines represent theSq -action, Sq -action, and Sq -action, respectively. Remark 3.9.
Using the Cartan formula, we can identify the action of Sq onΦ e ( A C ). We notice that its A -module structure is isomorphic to A [10] of [BEM].Since such an A -module is realized by a unique 2-local finite spectrum, we concludeΦ e ( A C ) (cid:39) A [10]and is of type 2.Our next goal is to understand the homotopy type of the geometric fixed-point spec-trum Φ C ( A C ). First observe that the geometric fixed-points of the C -equivariantquestion mark complex Q C is the exclamation mark complex E := (cid:39) S ∨ ΣM (1)!This is because Φ C ( h ) = 0 and Φ C ( η , ) = 2. Secondly,H ∗ +1 (Φ C ( A C ); F ) ∼ = H ∗ (˜ e ( E ∧ ); F ) ∼ = e (H ∗ ( E ; F ) ⊗ )is an isomorphism of A -modules. We explicitly calculate the A -module structure yxx + xyxzxx + xzx xyy + yxyzxy + xzy + yxz + xyz zyz + yzzzyy + yzyxzz + zxz xzy + zxy + zyx + yzx Figure 3.10.
The A -module structure of H ∗ (Φ C ( A C ); F ). of H ∗ (Φ C ( A C ); F ) from the above isomorphism and record it in Figure 3.10 as asubcomplex of H ∗ ( E ; F ) ⊗ , with the convention that x , y and z are generators inH ∗ ( E ; F ) in degree 0, 1 and 2 respectively. Lemma 3.11.
The finite spectrum Φ C ( A C ) is a type spectrum and equivalentto Φ C ( A C ) (cid:39) M (1) ∨ Σ (cid:16) M (1) ∧ M (1) (cid:17) ∨ Σ M (1) . Proof.
From Figure 3.10, it is clear that we have an isomorphism of A -modulesH ∗ (Φ C ( A C ); F ) ∼ = H ∗ (cid:16) M (1) ∨ Σ (cid:0) M (1) ∧ M (1) (cid:1) ∨ Σ M (1); F (cid:17) . It is possible that the A -module H ∗ (Φ C ( A C ); F ) may not realize to a unique finitespectrum (up to weak equivalence). However, other possibilities can be eliminatedfrom the fact that E ∧ splits Σ -equivariantly into four components: E ∧ (cid:39) S ∨ (cid:32) (cid:95) i =1 ΣM (1) (cid:33) ∨ (cid:32) (cid:95) i =1 Σ M (1) ∧ (cid:33) ∨ Σ M (1) ∧ . The idempotent ˜ e annihilates S ∼ = S ∧ , and Lemma 3.2 implies that˜ e (cid:32) (cid:95) i =1 ΣM (1) (cid:33) (cid:39) ΣM (1) and˜ e (cid:32) (cid:95) i =1 Σ M (1) ∧ M (1) (cid:33) (cid:39) Σ M (1) ∧ M (1) . Similarly, we see using (3.3) thatH ∗ (cid:0) ˜ e (cid:0) Σ M (1) ∧ (cid:1)(cid:1) ∼ = e (cid:16) H ∗ (ΣM (1)) ⊗ (cid:17) ∼ = H ∗ (Σ M (1)) . Hence, the result. (cid:3)
The cohomology of A R is free over A R (1) . Next, we analyze the A R -module structure of H ∗ , ∗ ( A R ). We begin by recalling some general properties ofthe cohomology of motivic spectra.If X, Y ∈ Sp R , fin such that H ∗ , ∗ ( X ) is free as a left M R -module, then we have aKunneth isomorphism [DI2, Proposition 7.7](3.12) H ∗ , ∗ ( X ∧ Y ) ∼ = H ∗ , ∗ ( X ) ⊗ M R H ∗ , ∗ ( Y )as the relevant Kunneth spectral sequence collapses. Further, if H ∗ , ∗ ( Y ) is free as aleft M R -module, then so is H ∗ , ∗ ( X ∧ Y ). The A R -module structure of H ∗ , ∗ ( X ∧ Y )can then be computed using the Cartan formula. The comultiplication map of A R is left M R -linear, coassociative and cocommutative [V, Lemma 11.9], which isalso reflected in the fact that its M R -linear dual is a commutative and associativealgebra. Thus, when H ∗ , ∗ ( X ) is a free left M R -module, the elements of F [Σ n ] actson H ∗ , ∗ ( X ∧ n ) ∼ = H ∗ , ∗ ( X ) ⊗ M R · · · ⊗ M R H ∗ , ∗ ( X ) N R -MOTIVIC v –SELF-MAP OF PERIODICITY 1 13 via permutation and commutes with the action of A R . This also implies that F [Σ n ]also acts onH ∗ , ∗ ( X ∧ n ) / ( ρ, τ ) ∼ = (H ∗ , ∗ ( X ) / ( ρ, τ )) ⊗ · · · ⊗ H ∗ , ∗ ( X ) / ( ρ, τ )and commutes with the action of A R // M R . From the above discussion we mayconclude that(3.13) H ∗ , ∗ ( A R ) ∼ = Σ − e (H ∗ , ∗ ( Q R ) ⊗ )is an isomorphism of A R -module.We will also rely upon the following important property of the action of the motivicSteenrod algebra on the cohomology of a motivic space (as opposed to a motivicspectrum): Remark 3.14 (Instability condition for R -motivic cohomology) . If X is an R -motivic space then H ∗ , ∗ ( X ) admits a ring structure, and, for any u ∈ H n,i ( X ), the R -motivic squaring operations obey the ruleSq i ( u ) = (cid:26) n < iu if n = 2 i .This is often referred to as the instability condition .To understand the A R -module structure of H ∗ , ∗ ( Q R ), we first make the followingobservation regarding H ∗ , ∗ (C R ( h )) (as C R ( h ) is a sub-complex of Q R ) using anargument very similar to [DI1, Lemma 7.4]. Proposition 3.15.
There are two extensions of A R (0) to an A R -module, and these A R -modules are realized as the cohomology of C R ( h ) and C R (2) . y , y , y , y , H ∗ , ∗ R (C R ( h )) H ∗ , ∗ R (C R (2)) Figure 3.16.
The x -axis represents the negative of topologicaldimension, y -axis represents the negative of motivic weight, verti-cal lines of length (0 ,
1) represent τ -multiplication, diagonal linesof length (1 ,
1) represent ρ -multiplication, blue lines represent Sq -action and red lines represent Sq -action. Proof.
For degree reasons, the only choice in extending A R (0) to an A R -module isthe action of Sq on the generator in bidegree (0 , y , for the generatorin degree (0, 0) and y , for Sq ( y , ) in (cohomological) bidegree (1 , • Sq ( y , ) = 0 and • Sq ( y , ) = ρ · y , .We can realize the degree 2 map as an unstable map S , −→ S , , and we will writeC R (2) u for the cofiber. We deduce information about the A R -module structure ofH ∗ , ∗ (C R (2)) by analyzing the cohomology ring of S , ∧ C R (2) u using the instabilitycondition of Remark 3.14. First, note that inH ∗ , ∗ (S , ) ∼ = M R · ι , we have the relation ι , = ρ · ι , [V, Lemma 6.8]. Also note thatH ∗ , ∗ ((C R (2) u ) + ) ∼ = M R [ x ] / ( x )where x is in cohomological degrees (1 , ∗ , ∗ (S , ∧ C R (2) u ) = M R · ι , ⊗ M R M R { x, x } the instability condition impliesSq ( ι , ⊗ x ) = ι , ⊗ x = ρ · ι , ⊗ x . Here the space-level cohomology class x corresponds to the spectrum-level class y , . Therefore, Sq ( y , ) = ρ · y , in H ∗ , ∗ (C R (2)). This is also reflected in the factthat multiplication by 2 is detected by h + ρh in the R -motivic Adams spectralsequence [DI1, § h is the ‘zeroth R -motivic Hopf-map’ detected by the element h in the motivic Adams spectral sequence. It follows that Sq ( y , ) = 0. (cid:3) In order to express the A R -module structure on H ∗ , ∗ ( X ) for a finite spectrum X ,it is enough to specify the action of A R on its left M R -generators as the action of τ and ρ multiples are determined by the Cartan formula. Example 3.17.
Let { y , , y , } ⊂ H ∗ , ∗ (C R ( h )) denote a left M R -basis of H ∗ , ∗ (C R ( h )).The data that • Sq ( y , ) = y , • Sq ( y , ) = 0completely determines the A R -module structure of H ∗ , ∗ (C R ( h )). Proposition 3.18. H ∗ , ∗ ( Q R ) is a free M R -module generated by a, b and c in coho-mological bidegrees (0 , , (1 , and (3 , , and the relations(1) Sq ( a ) = b ,(2) Sq ( b ) = c , N R -MOTIVIC v –SELF-MAP OF PERIODICITY 1 15 (3) Sq ( a ) = 0 .completely determine the A R -module structure of H ∗ , ∗ ( Q R ) .Proof. H ∗ , ∗ ( Q R ) is a free M R -module because the attaching maps of Q R inducetrivial maps in H ∗ , ∗ ( − ). The first two relations can be deduced from the obviousmaps(1) C R ( h ) → Q R (2) Q R → Σ , C R ( η , )which are respectively surjective and injective in cohomology.Let h u : S , → S , and η u , : S , → S , denote the unstable maps that stabilizeto h and η , , respectively. The unstable R -motivic space Q u R (which stabilizes to Q R ) can be constructed using the fact that the composite of the unstable mapsS , S , S , , η u , h u is null. Thus H ∗ , ∗ ( Q u R ) consists of three generators a u , b u and c u in bidegrees (3 , ,
2) and (6 , ( a u ) = 0. (cid:3) Proof of Theorem 3.5.
From Remark 3.9 and Lemma 3.11, we deduce that A R isa type (2 ,
1) complex. To show that the bi-graded R -motivic cohomology of A R isfree as an A R (1), we make use of Corollary 2.7.Since H ∗ , ∗ ( A R ) is a summand of a free M R -module, it is projective as an M R -module. In fact, H ∗ , ∗ ( A R ) is free, as projective modules over (graded) local ringsare free. Also note that the elementsP , P , P ∈ A R (1) / ( ρ, τ ) ∼ = Λ(P , P , P )are primitive. Hence we have a Kunneth isomorphism in the respective Margolishomologies, in particular we have, M (H ∗ , ∗ ( A R ) / ( ρ, τ ) , P st ) = e ( M (H ∗ , ∗ ( Q R ) / ( ρ, τ ) , P st ) ⊗ )for ( s, t ) ∈ { (0 , , (1 , , (0 , } . Since dim F M (H ∗ , ∗ ( Q R ) / ( ρ, τ ) , P st ) = 1, byLemma 3.1 M ( A R / ( ρ, τ ) , P st ) = 0for ( s, t ) ∈ { (0 , , (1 , , (0 , } . Thus, by Corollary 2.7 we conclude that H ∗ , ∗ ( A R )is a free A R (1)-module. A direct computation shows thatdim F H ∗ , ∗ ( A R ) / ( ρ, τ ) = 8 , hence H ∗ , ∗ ( A R ) is A R (1)-free of rank one. (cid:3) The A R -module structure. Using the description (3.13) and Cartan for-mula we make a complete calculation of the A R -module structure of H ∗ , ∗ ( A R ). Let a, b, c ∈ H ∗ , ∗ ( Q R ) as in Proposition 3.18. In Figure 3.20 we provide a pictorial repre-sentation with the names of the generators that are in the image of the idempotent e . For convenience we relabel the generators in Figure 3.20, where the indexing on a new label records the cohomological bidegrees of the corresponding generator. Thefollowing result is straightforward, and we leave it to the reader to verify. Lemma 3.19. In H ∗ , ∗ ( A R ) , the underlying A R (1) -module structure, along with therelations(1) Sq ( v , ) = τ · w , ,(2) Sq ( v , ) = w , ,(3) Sq ( v , ) = 0 ,(4) Sq ( v , ) = 0 = Sq ( w , ) ,(5) Sq ( v , ) = 0 ,completely determine the A R -module structure. baa + aba = v , bab + abb = v , v , = caa + acav , = cab + cba + bca + acb cbc + bcc = w , w , = cac + acc cbb + bcb = w , cab + bac + acb + abc = w , Figure 3.20.
We depict the A R -module structure of H ∗ , ∗ ( A ).The black, blue, and red lines represent the action of motivic Sq ,Sq , and Sq , respectively. Black dots represent M R -generators,and a dotted line represents that the action hits the τ -multiple ofthe given M R -generator. Remark 3.21.
In upcoming work, we show that A R (1) admits 128 different A R -module structures. Whether all of the 128 A R -module structures can be realizedby R -motivic spectra, or not, is currently under investigation.4. An R -motivic v –self-map With the construction of A R , we hope that any one of Y R ( i,j ) fits into an exacttriangle(4.1) Σ , Y R ( i,j ) Y R ( i,j ) A R Σ , Y R ( i,j ) . . . v Σ v in Ho( Sp R , fin ). The motivic weights prohibit A R from being the cofiber of a self-map on Y triv or Y ( h , . We will also see that the spectrum Y R (2 , cannot be a part of N R -MOTIVIC v –SELF-MAP OF PERIODICITY 1 17 (4.1) because of its A R -module structure (see Lemma 4.6). If Y ( i,j ) = Y R ( h , in (4.1),then the map v will necessarily be a v , nil –self-map because Y R ( h , is of type (1 , A R is of type (2 , Y R ( h , does fit into an exact triangle very similarto (4.1) Σ , Y R ( i,j ) Y R ( i,j ) C R ( v ) Σ , Y R ( i,j ) . . . v Σ v where C R ( v ) is of type (2 ,
1) and H ∗ , ∗ (C R ( v )) ∼ = H ∗ , ∗ ( A R ) as A R -modules. Remark 4.2.
The fact that H ∗ , ∗ (C R ( v )) is isomorphic to H ∗ , ∗ ( A R ) as A R -modulesdoes not imply that C R ( v ) and A R are equivalent as R -motivic spectra. There area plethora of examples of Steenrod modules that are realized by spectra of differenthomotopy types.We begin by discussing the A R -module structures of H ∗ , ∗ ( Y R ( h , ). Using Ademrelations, one can show that the element Q := Sq Sq + Sq Sq ∈ A R (1)squares to zero. Let Λ( Q ) denote the exterior subalgebra M R [ Q ] / ( Q ) of A R (1).Let B R (1) denote the A R (1)-module B R (1) := A R (1) ⊗ Λ( Q ) M R . Both Y R (2 , and Y R ( h , are realizations of B R (1). In other words: Proposition 4.3.
There is an isomorphism of A R (1) -modules H ∗ , ∗ ( Y R ( i,j ) ) ∼ = B R (1) for ( i, j ) ∈ { (2 , , ( h , } .Proof. Note that H ∗ , ∗ ( Y R ( i,j ) ) is cyclic as an A R (1)-module for ( i, j ) ∈ { (2 , , ( h , } .Thus we have an A R (1)-module map(4.4) f i : A R (1) → H ∗ , ∗ ( Y R ( i,j ) ) . The result follows from the fact that Q acts trivially on H ∗ , ∗ ( Y R ( i,j ) ) and a dimensioncounting argument. (cid:3) Remark 4.5.
Let { y , , y , } be the M R -basis of H ∗ , ∗ (C R ( h )) or H ∗ , ∗ (C R (2)), sothat Sq ( y , ) = y , , and let { x , , x , } a basis of C R ( η , ), so that Sq ( x , ) = x , . If we consider the M R -basis { v , , v , , v , , v , , w , , w , , w , , w , , w , } of A R (1) from Subsection 3.3, then the maps f i of (4.4) are given as in Table 1. Lemma 4.6.
The A R -module structures on H ∗ , ∗ ( Y R (2 , ) and H ∗ , ∗ ( Y R ( h , ) are givenas in Figure 4.7. Table 1.
The maps f and f h x f ( x ) f h ( x ) v , y , x , y , x , v , y , x , y , x , v , y , x , + ρ · y , x , y , x , v , y , x , y , x , w , y , x , y , x , w , w , w , ∗ , ∗ ( Y R ( h , ) ρ H ∗ , ∗ ( Y R (2 , ) Figure 4.7.
Black, blue, and red lines represent the action of Sq ,Sq , and Sq , respectively. Black dots represent M R -generators,and in the case of Y R (2 , , Sq on the bottom cell is ρ times the topcell. Proof.
The result is an easy consequence of a calculation using the Cartan formulaSq ( xy ) = Sq ( x ) y + τ Sq ( x )Sq ( y ) + Sq ( x )Sq ( y ) + τ Sq ( x )Sq ( y ) + x Sq ( y )and the fact that Sq ( y , ) = ρy , in H ∗ , ∗ (C R (2)), whereas Sq ( y , ) = 0 inH ∗ , ∗ (C R ( h )) (see Proposition 3.15). (cid:3) Remark 4.8.
Comparing Lemma 4.6 and Lemma 3.19, we see that the A R (1)-module map f , as in Remark 4.5, cannot be extended to a map of A R -modules. Corollary 4.9.
There is an exact sequence of A R -modules (4.10) 0 H ∗ , ∗ (Σ , Y R ( h , ) H ∗ , ∗ ( A R ) H ∗ , ∗ ( Y R ( h , ) 0 . π ∗ ι ∗ Proof.
From the description of the map f h in Remark 4.5, along with Lemma 3.19and Lemma 4.6, it is easy to check that f h extends to an A R -module map and thatker f h ∼ = H ∗ , ∗ ( Y R ( h , )as A R -modules. (cid:3) N R -MOTIVIC v –SELF-MAP OF PERIODICITY 1 19 The exact sequence (4.10) corresponds to a nonzero element in the E -page of the R -motivic Adams spectral sequence (also see Remark 4.13 and Remark 4.15 )(4.11) v ∈ Ext , , A R (H ∗ , ∗ ( Y R ( h , ∧ D Y R ( h , ) , M R ) ⇒ [ Y R ( h , , Y R ( h , ] , , where D Y R ( h , := F( Y R ( h , , S R ) is the Spanier-Whitehead dual of Y R ( h , . If Notation 4.12.
Note that we follow [DI1, BI] in grading Ext A R as Ext s,f,w A R , where s is the stem, f is the Adams filtration, and w is the weight. We will also follow[GI1] in referring to the difference s − w as the coweight . Remark 4.13.
Since H ∗ , ∗ ( Y R ( h , ) is M R -free, an appropriate universal-coefficientspectral sequence collapses and we get H ∗ , ∗ ( D Y R ( h , ) ∼ = hom M R (H ∗ , ∗ ( Y R ( h , ) , M R ) . Further, the Kunneth isomorphism of (3.12) gives usH ∗ , ∗ ( Y R ( h , ∧ D Y R ( h , ) ∼ = H ∗ , ∗ ( Y R ( h , ) ⊗ M R H ∗ , ∗ ( D Y R ( h , ) , and therefore,Ext ∗ , ∗ , ∗A R ( M R , H ∗ , ∗ ( Y R ( h , ∧ D Y R ( h , )) ∼ = Ext ∗ , ∗ , ∗A R (H ∗ , ∗ ( Y R ( h , ) , H ∗ , ∗ ( Y R ( h , )) . Theorem 1.11 follows immediately if we show that the element v is a nonzeropermanent cycle. The following lemma implies that a d r -differential (for r ≥ v has no potential nonzero target. Proposition 4.14.
For f ≥ , Ext ,f, A R (H ∗ , ∗ ( Y R ( h , ) , H ∗ , ∗ ( Y R ( h , )) = 0 . In order to calculate Ext ∗ , ∗ , ∗A R (H ∗ , ∗ ( Y R ( h , ) , H ∗ , ∗ ( Y R ( h , )), we filter the spectrum Y R ( h , via the evident maps Y Y Y Y . S R C R ( h ) C R ( h ) ∪ S R C R ( η , ) Y R ( h , Note that H ∗ , ∗ ( Y j ) are free M R -modules. The above filtration results in cofibersequences Y Y Σ , S R ,Y Y Σ , S R , and Y Y Σ , S R , which induce short exact sequences of A R -modules as the connecting mapC R ( Y j → Y j +1 ) −→ Σ Y j induces the zero map in H ∗ , ∗ ( − ). Thus, applying the functor Ext ∗ , ∗ , ∗A R (H ∗ , ∗ ( Y ( h , ) , − )to these short-exact sequences, we get long exact sequences, which can be spliced together to obtain an Atiyah-Hirzebruch like spectral sequenceE ∗ , ∗ , ∗ , ∗ = Ext ∗ , ∗ , ∗A R (H ∗ , ∗ ( Y ( h , ) , M R ) { g , , g , , g , , g , } Ext ∗ , ∗ , ∗A R (H ∗ , ∗ ( Y R ( h , ) , H ∗ , ∗ ( Y R ( h , )) . An element x · g i,j in the E -page contributes to the degree | x | − ( i, , j ) of theabutment. Thus, Proposition 4.14 is a straightforward consequence of the followingProposition 4.16. Remark 4.15.
Because, H ∗ , ∗ ( Y R ( h , ) is M R -free and finite, we haveH ∗ , ∗ ( Y R ( h , ) ∼ = hom M R (H ∗ , ∗ ( Y ( h , ) , M R ) , and therefore, Ext s,f,w A R (H ∗ , ∗ ( Y R ( h , ) , M R ) ∼ = Ext s,f,w A R ∗ ( M R , H ∗ , ∗ ( Y R ( h , )) . Proposition 4.16.
For f ≥ and ( i, j ) ∈ { (0 , , (1 , , (2 , , (3 , } , we have that Ext i,f, j A R ∗ ( M R , H ∗ , ∗ ( Y R ( h , )) = 0 . Proof.
Our desired vanishing concerns only the groups Ext A R ∗ ( M R , H ∗ , ∗ ( Y R ( h , )) incoweights 0, 1 and 2. These groups can be easily calculated starting from the com-putations of Ext ∗ , ∗ , ∗A R ∗ ( M R , M R ) in [DI1] and [BI] and using the short exact sequencesin Ext A R ∗ arising from the cofiber sequencesΣ , S R η , −→ S R −→ C R ( η , ) andC R ( η , ) h −→ C R ( η , ) −→ C R ( h ) ∧ C R ( η , ) = Y R ( h , . We display Ext A R ∗ ( M R , H ∗ , ∗ (C R ( η , ))) in coweights 0, 1 and 2 in the charts below. Ext A R ∗ (cid:0) M , H ∗ , ∗ (C R ( η , ) (cid:1) in coweight zero − − ρ h Ext A R ∗ (cid:0) M , H ∗ , ∗ (C R ( η , ) (cid:1) in coweight one − − τh h h [2] Ext A R ∗ (cid:0) M , H ∗ , ∗ (C R ( η , ) (cid:1) in coweight two − − τ h ( τh ) h ρτh [2] h [2] We find that Ext A R ∗ ( M R , H ∗ , ∗ ( Y R ( h , )) is, in coweights zero, one, and two, also given N R -MOTIVIC v –SELF-MAP OF PERIODICITY 1 21 by the charts below. Ext A R ∗ (cid:0) M , H ∗ , ∗ ( Y R ( h , ) (cid:1) in coweight zero − − ρ Ext A R ∗ (cid:0) M , H ∗ , ∗ ( Y R ( h , ) (cid:1) in coweight one − − τh h h [2] ρ [1] Ext A R ∗ (cid:0) M , H ∗ , ∗ ( Y R ( h , ) (cid:1) in coweight two − − τ h ( τh ) h ρτh [2] h [2] τh [1] h h [1] ρh [1] The result follows from the above charts. (cid:3)
Remark 4.17.
One can also resolve Proposition 4.16 directly using the ρ -Bocksteinspectral sequence(4.18) E := Ext A C ∗ ( F [ τ ] , H ∗ , ∗ ( Y C ( h , )) ⊗ F [ ρ ]Ext A R ∗ ( M R , H ∗ , ∗ ( Y R ( h , ))and identifying a vanishing region for Ext s,f,w A C ∗ ( F [ τ ] , H ∗ , ∗ ( Y C ( h , )). Even a roughestimate of the vanishing region using the E -page of the C -motivic May spectralsequence leads to Proposition 4.16. Such an approach would avoid explicit calcu-lations of Ext A R as in [DI1] and [BI]. Proof of Theorem 1.11.
Since Proposition 4.16 = ⇒ Proposition 4.14, every map v : Σ , Y R ( h , Y R ( h , detected by v of (4.11) is a nonzero permanent cycle. In order to finish the proofof Theorem 1.11 we must show that v is necessarily v (1 , nil) –self-map of periodicity1. It is easy to see that the underlying mapΦ e ( β ( v )) : Σ Y Y is a v –self-map of periodicity 1 asC(Φ e ( β ( v ))) (cid:39) Φ e ( β (C R ( v ))) (cid:39) A [10]is of type 1 (see Remark 3.9). On the other hand,Φ C ( β ( v )) : Σ (ΣM (1) ∨ M (1)) ΣM (1) ∨ M (1) is necessarily a nilpotent map because of [HS, Theorem 3(ii)] and the fact that a v –self-map of M (1) has periodicity at least 4 (see [DM] for details) which lives in[M (1) , M (1)] k for k ≥ (cid:3) Proof of Theorem 1.14.
Since v is a v (1 , nil) –self-map and Y R ( h , is of type (1 , R ( v ) is of type (2 , ∗ , ∗ (C R ( v )) ∼ = H ∗ , ∗ ( A R )as v is detected by v of (4.11) in the E -page of the Adams spectral sequence. Thus,H ∗ , ∗ (C R ( v )) is a free A R (1)-module on single generator. (cid:3) Remark 4.19.
It is likely that realizing a different A R -module structure on A R (1)as a spectrum (see also Remark 3.21) may lead to a 1-periodic v –self-map on Y R (2 , as well as on Y C (2 , . We explore such possibilities in upcoming work.5. Nonexistence of v , –self-map on C R ( h ) and Y R ( h , Let X be a finite R -motivic spectrum and let f : Σ i,j X → X be a map suchthat Φ C ( β ( f )) : Σ i − j Φ C ( β ( X )) Φ C ( β ( X ))is a v –self-map. Then it must be the case that i = j , as v –self-maps preservedimension. Note that both C R ( h ) and Y R ( h , are of type (1 , Proposition 5.1.
The v –self-maps of M (1) are not in the image of the underlyinghomomorphism Φ e ◦ β : [Σ k, k C R ( h ) , C R ( h )] R −→ [Σ k M (1) , M (1)] . Proof.
The minimal periodicity of a v –self-map of M (1) is 4. Let v : Σ k M (1) → M (1) be a 4 k -periodic v –self-map. It is well-known that the composite(5.2) Σ k S Σ k M (1) M (1) Σ S v is not null (and equals P k − (8 σ ) where P is a periodic operator given by the Todabracket (cid:104) σ, , −(cid:105) ).Suppose there exists f : Σ k, k C R ( h ) → C R ( h ) such that Φ e ◦ β ( f ) = v . Then (5.2)implies that the composition(5.3) Σ k, k S R Σ k, k C R ( h ) C R ( h ) Σ , S v is nonzero as the functor Φ e ◦ β is additive. The composite of the maps in (5.3) isa nonzero element of π ∗ , ∗ ( S R ) in negative coweight. This contradicts the fact that π ∗ , ∗ ( S R ) is trivial in negative coweights [DI1]. (cid:3) Proposition 5.4.
The v –self-maps of Y are not in the image of the underlyinghomomorphism Φ e ◦ β : [Σ k, k Y R ( h , , Y R ( h , ] R −→ [Σ k Y , Y ] . N R -MOTIVIC v –SELF-MAP OF PERIODICITY 1 23 Proof.
Let v : Σ k Y → Y denote a v –self-map of periodicity k . Notice that thecomposite(5.5) S k Σ k Y Y Y ≥ v where Y ≥ is the first coskeleton, must be nonzero. If not, then v factors throughthe bottom cell resulting in a map S k → Σ k Y → S which induces an isomorphismin K(1)-homology, contradicting the fact that S is of type 0.If f : Σ k, k Y R ( h , → Y R ( h , were a map such that Φ e ◦ β ( f ) = v , then (5.5) wouldforce one among the hypothetical composites ( A ), ( B ) or ( C ) in the diagramΣ k, k S R Σ k, k Y R ( h , Y R ( h , Σ , S R ( A )Fib( p ) Σ , S R ( B )Fib( p ) Σ , S R ( C ) p p p to exist as a nonzero map, thereby contradicting the fact that π ∗ , ∗ ( S R ) is trivial innegative coweights. (cid:3) Remark 5.6.
The above results do not preclude the existence of a v , –self-mapon C C ( h ) and Y C ( h , . Forthcoming work [GI2] of the second author and Isaksenshows that 8 σ is in the image of Φ e : π , ( S C ) −→ π ( S ) and suggests that C C ( h )supports a v , –self-map. References [AM] J. F. Adams and H. R. Margolis,
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Email address : [email protected] Department of Mathematics, The University of Kentucky, Lexington, KY 40506–0027
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