Two-complete stable motivic stems over finite fields

0.34 MB·

by  Glen M. Wilson

TWO-COMPLETE STABLE MOTIVIC STEMS
OVER FINITE FIELDS
6
1
GLENM.WILSONANDPAULARNEØSTVÆR
0
2
Abstract. Let ℓbe aprime,andq=pn wherep isaprimedifferent fromℓ
n
and 2. We show that the ℓ-completion of the nth stable homotopy group of
a
J spheresisasummandoftheℓ-completionofthe(n,0)stablehomotopygroup
4 of motivic spheres over the finite field with q elements Fq. With this, and
assistedbycomputer calculations, weareableto compute explicitlythetwo-
2 complete stable motivic stems πn,0(Fq)∧2 for 0 ≤ n ≤ 20 when q ≡ 1mod4
andfor0≤n≤18whenq≡3mod4.
]
T
A
.
h 1. Introduction
t
a
The homotopy groups of spheres belong to the most important and puzzling
m
invariants in topology. See [Koc90] and the more recent works [Isa14b], [WX16]
[ for amazing computer assisted ways of computing these invariants based on the

1 Adamsspectralsequence;thisisamuchstudiedapproachtocomputestablehomo-
v topy group of spheres [Ada95], [Rav86]. With two-primary coefficients, the second
8 page of the Adams spectral sequence — converging to the two-completed stable
9
homotopygroupsofspheres—iscomprisedofExtgroupsoverthemod2Steenrod
3
algebra,
6
0 E2s,t =ExtsA,t∗(F2,F2).
. Extensive computer calculations of these Ext groups are carried out in [Bru93],
1
0 [Bru16]. However, even if one knew completely the answer for the Ext groups in
6 the Adams spectral sequence, one is still not finished with computing the stable
1 homotopygroupsofspheres. Oneneedstoknowinadditionthedifferentialsandall
:
v the groupextensionshiddeninthe associatedgradedofthe filtration. Onlypartial
i results have been obtained in spite of an enormous effort.
X
Given any field k, the stable motivic homotopy category SH(k) over k has the
r
a structure of a triangulated category, which encodes both algebro-geometric and
topological information. An application of this setting is Voevodsky’s proof of
Milnor’s conjecture on Galois cohomology[Voe03a]. Just as for the classicalstable
homotopycategorySH,itisaninterestinganddeepproblemtocomputethestable
motivic homotopy groups of spheres π (k) over k, i.e., SH(k)(Σp,q , ), where
p,q
1 1 1
denotesthemotivicspherespectrumoverk. Whenkhasfinitemod2cohomological
dimension,themotivicAdamsspectralsequence,henceforthabbreviatedtoMASS,
converges to the two-completed stable motivic stems
Ef,(s,w) =Extf,(s+f,w)(H∗∗,H∗∗)=⇒(π )∧.
2 A∗∗ s,w1 2
Date:January26,2016.
2010 Mathematics Subject Classification. 14F42,16-04,18G15,55T15.
Key words and phrases. Motivic Adams spectral sequence, stable motivic stems over finite
fields,computer assistedmotivicExtgroupcalculations.
1
2 G.M.WILSONANDP.A.ØSTVÆR
This is a tri-graded spectral sequence, where A∗∗ is the bi-graded mod 2 motivic
Steenrod algebra, see e.g., [HKØ13] and [Voe03a], and H∗∗ is the bi-graded mod
2 motivic cohomology ring of k. (We refer to Section 5 for the construction of
MASS.) The calculational challenges consists basically of: (1) identify the motivic
Extgroups, (2) determine the differentials, and (3) reconstructthe abutment from
the filtration quotients.
Based on the MASS, Dugger and Isaksen have carried out calculations of the
2-complete stable motivic homotopy groups of spheres up to the 34-stem over the
complex numbers [DI10]. Isaksen has extended this work largely up to the 70-
stem [Isa14a,Isa14b]. We are led to wonder, how do the stable motivic homotopy
groupsvary for different base fields? Morelhas givena complete descriptionof the
0-line π (k) in terms of Milnor-Witt K-theory [Mor12]. The 1-line π (k) is
p,p p+1,p
determined by Hermitian and Milnor K-theory groups [RSØ16], generalizing the
partial results obtained in [OØ14]. Ormsby has investigated the case of related
invariants over p-adic fields [Orm11] and the rationals [OØ13], and Dugger and
Isaksen have analyzed the case over the real numbers [DI15]. It is now possible to
performsimilarcalculationsoverfieldsofpositivecharacteristic,thankstoworkon
the motivic Steenrod algebra in positive characteristic [HKØ13]. In this paper we
use computer assisted motivic Ext group calculations in tandem with theoretical
arguments to determine stable motivic stems π in weight zero over finite fields.
n,0
Let πs denote the nth topological stable stem. Over the complex numbers,
n
Levine showed there is an isomorphism πns ∼=πn,0(C) [Lev14, Cor.2]. We obtain a
similar result over the algebraic closure of a finite field, after ℓ-completion. For a
prime ℓ and an Abelian group G, we write the ℓ-completion of G by G∧.
ℓ
Theorem A. Letpbeaprime,andletF denotethealgebraicclosureofthefinite
p
field with p elements. For any prime ℓ6=p there is an isomorphism
πn,0(Fp)∧ℓ ∼=(πns)∧ℓ.
Using theorem A, we are able to perform calculations in the MASS over finite
fields at the prime 2 up to the 18-stem.
Theorem B. Let F be the finite field with q elements, where q = pi with p an
q
odd prime. For any natural number 0≤n≤18 there is an isomorphism
πn,0(Fq)∧2 ∼=(πns)∧2 ⊕(πns+1)∧2.
In particular, the groups π (F ) and π (F ) are trivial.
4,0 q 12,0 q
InthecaseofafinitefieldF withq ≡3mod4,werequirecomputercalculations
q
toidentifythestructureoftheE pageoftheMASS.Thedetailsofthesecomputer
2
calculations, and a careful analysis of the differentials can be found in [KW15].
It is interesting to note that the pattern πn,0(Fq)∧2 ∼= (πns)∧2 ⊕(πns+1)∧2 obtained
in Theorem B does not hold in general. We show that if q ≡5mod8, then
π19,0(Fq)∧2 ∼=(Z/8⊕Z/2)⊕Z/4 and π20,0(Fq)∧2 ∼=Z/8⊕Z/2.
We shall leave open for further investigations the question of whether or not an
isomorphismπn,0(Fq)∧2 ∼=(πns)∧2 ⊕(πns+1)∧2 holds when q ≡3mod4 and n=19,20.
STABLE MOTIVIC STEMS OVER Fq 3
2. The stable motivic homotopy category
We firstsketcha constructionofthe stable motivic homotopycategorythatwill
beconvenientforourpurposes,andinthe process,setournotation. Treatmentsof
stable motivic homotopy theory can be found in [Ayo07,DRØ03,DLØ+07,Jar00,
Hu03].
2.1. Base schemes. A base scheme B is a Noetherian separated scheme of finite
Krull dimension. We write Sm/B for the category of smooth schemes of finite
type over B. Let Pshv(Sm/B) denote the categoryof presheaves of sets on Sm/B.
A space over B is a simplicial presheaf on Sm/B. The collection of spaces over
B forms the category Spc(B) = ∆opPshv(Sm/B), where morphisms are natural
transformations of functors. We write Spc (B) for the category of pointed spaces.
∗
Note that Spc(B) is naturally equivalent to the category of presheaves on Sm/B
with values in the category of simplicial sets sSet. We will occasionally switch
between the two perspectives.
2.2. The projective model structure. The first model category structure we
endow Spc(B) with is the projective model structure, see, e.g., [Bla01, Thm.1.4],
[DRØ03, Thm.2.7], [Hir03, Thm.11.6.1].
Definition 2.1. A map f : X → Y in Spc(B) is a (global) weak equivalence if
for any U ∈ Sm/B the map f(U) : X(U) → Y(U) of simplicial sets is a weak
equivalence. The projective fibrations are those maps f : X →Y for which f(U):
X(U) → Y(U) is a Kan fibration for any U ∈ Sm/B. The projective cofibrations
arethosemapsinSpc(B)whichsatisfytheleftliftingpropertyfortrivialprojective
fibrations. This defines the projective model structure on Spc(B).
The category Spc(B) equipped with the projective structure is cellular, proper,
and simplicial [Bla01, Thm.1.4]. Furthermore, Spc(B) has the structure of a sim-
plicialmonoidalmodel category,with product × and internalhom Hom. We write
Map(X,Y) for the simplicial mapping space for spaces X and Y.
Definition 2.2. ForasmoothschemeX overB,wewriterX fortherepresentable
presheaf of simplicial sets. For U ∈ Sm/B, the simplicial set rX(U) is given by
rX(U) =Sm/B(U,X) for all n∈∆ where the face and degeneracy maps are the
n
identity map. We will occasionally abuse notation and write X for rX.
The constantpresheaffunctor c:sSet→Spc(B) associates to a simplicialset A
the presheaf cA defined by cA(U)=A for any U ∈Sm/B.
The functor c is a left Quillen functor when Spc(B) is equipped with the pro-
jective model structure. Its right adjoint Ev :Spc(B)→sSet satisfies Ev (X)=
B B
X(B). One can show that representable presheaves and constant presheaves in
Spc(B) are cofibrant with the projective model structure.
2.3. The Nisnevich local model structure. Althoughrepresentablepresheaves
embed into Spc(B), colimits which exist in Sm/B are not necessarily preserved in
Spc(B). Thatis,ifX =colimX inSm/B,itneednotbetruethatrX =colimrX ,
i i
e.g., colim(rA1 ← rG → rA1) 6= rP1 (as one can check by applying the Picard
m
group functor). To fix this, one introduces the Nisnevich topology on Sm/B.
Definition2.3(Nisnevichtopology). LetBbeabasescheme. ForanyX ∈Sm/B,
letU ={U →X}beafinitefamilyof´etalemapsinSm/B. WesayU isaNisnevich
i
4 G.M.WILSONANDP.A.ØSTVÆR
covering of X if for any x∈X there exists a map U →X ∈U and a point u∈U
i i
for which the induced map of residue fields k(x) → k(u) is an isomorphism. The
Nisnevich covers determine a Grothendieck topology on Sm/B.
Definition 2.4 (Elementary distinguished squares). An elementary distinguished
square is a diagram in Sm/B
V′ // X′
f
(cid:15)(cid:15) j (cid:15)(cid:15)
V //X
for which f is an´etale map, j is anopen embedding, and f−1(X−V)→X−V is
an isomorphism, where these subschemes are given the reduced structure.
Proposition 2.5. The Nisnevich topology is the smallest topology containing the
covers {V →X,X′ →X} which come from an elementary distinguished square.
Definition 2.6. For a pointed space X, and n≥0, let π X denote the Nisnevich
n
sheafification of the presheaf U 7→π (X(U)).
n
Let W denote the class of maps f : X →Y for which f : π X →π Y is an
Nis ∗ n n
isomorphismofNisnevichsheavesforalln≥0. TheNisnevichlocalmodelstructure
on Spc (B) is the left Bousfield localizationof the projective model structure with
∗
respect to W .
Nis
Definition 2.7. Let WA1 be the class of maps πX : (X ×A1)+ → X+ for X ∈
Sm/B. The motivic model structure on Spc (B) is the left Bousfield localization
∗
A1
of the projective model structure with respect to WNis∪WA1. We write Spc∗ (B)
forthe categoryofpointedspaces equippedwith the motivic model structure. The
pointedmotivichomotopycategoryHA1(B)isthehomotopycategoryofSpcA1(B).
∗ ∗
Comparewith [HKØ13, p.5]. For pointed spaces X and Y, we write [X,Y] for the
set of maps HA1(B)(X,Y).
∗
There are two circles in the category of pointed spaces: the constant simplicial
presheafS1 pointed at its 0-simplex and the representable presheafG =A1\{0}
m
pointedat1. Thesedetermineabi-gradedfamilyofspheresSp,q =(S1)∧p−q∧G∧q.
m
Definition 2.8. For a pointed space X over B and natural numbers i and j, let
π X denote the set of maps [Si,j,X].
i,j
The category of pointed spaces Spc (B) equipped with the induced motivic
∗
model category structure has many good properties which make it amenable to
Bousfieldlocalization. Inparticular,Spc (B)isclosedsymmetricmonoidal,pointed
∗
simplicial, left proper, and cellular.
2.4. The stable Nisnevich local model structure. With the unstable motivic
modelcategoryinhand,we nowconstructthe stablemotivic model categoryusing
the general framework laid out in [Hov01].
Let T be a cofibrantreplacementof A1/A1−{0}. One can show that T is weak
equivalent to S2,1 in SpcA1(B). The functor T ∧− on SpcA1(B) is a left Quillen
∗ ∗
functor, which we may invert by creating a category of T-spectra.
A1
Definition 2.9. A T-spectrum is a sequence of spaces X ∈ Spc (B) equipped
n ∗
with structure maps σ : T ∧X → X . A map of T-spectra f : X → Y is a
n n n+1
STABLE MOTIVIC STEMS OVER Fq 5
collection of maps f :X →Y which is compatible with the structure maps. We
n n n
write Spt (B) for the category of T-spectra of spaces.
T
The levelmodelstructureonSpt (B)is givenby declaringamapf :X →Y to
T
be a cofibration (resp. weak equivalence, fibration) if for every map f : X →Y
n n n
is a cofibration (resp. weak equivalence, fibration) in the motivic model structure
on Spc (B).
∗
Definition 2.10. Let X be a T-spectrum. For integers i and j, the (i,j) stable
homotopysheafofX isthesheafπ X =colim π X . Amapf :X →Y is
i,j n i+2n,j+n n
a stable weakequivalence if for allintegers i and j, the induced maps f :π X →
∗ i,j
π Y are isomorphisms.
i,j
Definition 2.11. The stable model structure on Spt (B) is the model category
T
where the weak equivalences are the stable weak equivalences and the cofibrations
are the cofibrations in the level model structure. The fibrations are those maps
withtherightlifting propertywithrespecttotrivialcofibrations. We writeSH(B)
for the homotopy category of Spt (B) equipped with the stable model structure.
T
The stable model structure on Spt (B) can be realizedas a left Bousfieldlocal-
T
ization of the level-wise model structure [Hov01, Def.3.3].
Just as for the category Spt of simplicial S1-spectra, there isn’t a symmetric
S1
monoidalcategorystructureonSpt (B)whichliftsthesmashproduct∧inSH(B).
T
OneremedyistouseacategoryofsymmetricT-spectraSptΣ(B). Theconstruction
T
ofthiscategoryisgivenin[Hov01,Def.7.7],[Jar00]. By[Hov01,Thm.9.1],thereis
azig-zagofQuillenequivalencesfromSptΣ(B)toSpt (B),soSH(B)isequivalent
T T
to the homotopy category of SptΣ(B) as well. The category SH(B) therefore is
T
a symmetric monoidal, triangulated category, where the shift functor is given by
[1]=S1,0∧−.
In addition to the categoryof T-spectra,we will find it convenientto work with
the category of (G ,S1) bi-spectra, see [Jar00,DLØ+07].
m
Definition 2.12. An S1-spectrum over the base scheme B is a sequence of spaces
X ∈Spc (B) equipped with structure maps σ :S1∧X →X . A map of S1-
n ∗ n n n+1
spectraf :X →Y isasequenceofmapsf :X →Y whichiscompatiblewiththe
n n n
structure maps. The collection of S1-spectra with compatible maps between them
formsa categorySpt (B). Similarto the constructionindefinition 2.9, the stable
S1
model structure on Spt (B) is the left Bousfield localization of the level model
S1
structure on Spt (B) with respect to the motivic model structure on Spc (B) at
S1 ∗
the stable equivalences, cf. [Hov01, Def.3.3].
Definition 2.13. In the projective model structure on Spc (B), the space G
∗ m
pointed at 1 is not cofibrant. We abuse notation, and write G for a cofibrant
m
replacementofG . A(G ,S1)bi-spectrumoverBisaG -spectrumofS1-spectra.
m m m
More concretely, a (G ,S1) bi-spectrum E is a bi-graded family of spaces E
m m,n
with bonding maps σ : S1∧E →E and γ : G ∧E →E
m,n m,n m,n+1 m,n m m,n m+1,n
which are compatible, meaning that the following diagram commutes.
G ∧S1∧E τ∧Em,n // S1∧G ∧E
m m,n m m,n
Gm∧σ S1∧γ
G ∧E(cid:15)(cid:15) γ //E oo σ S1∧E(cid:15)(cid:15)
m m,n+1 m+1,n+1 m+1,n
6 G.M.WILSONANDP.A.ØSTVÆR
Thinking of Spt as the category of G -spectra of S1-spectra, we first equip
Gm,S1 m
SptGm,S1 with the level model category structure with respect to the stable model
category structure on Spt (B). The stable model category structure on Spt
S1 Gm,S1
is the left Bousfield localization at the class of stable equivalences.
There are evidently left Quillen functors Σ∞S1 :Spc∗(B) →SptS1(B) and Σ∞Gm :
SptS1(B) → SptGm,S1(B). Additionally, the category SptGm,S1(B) equipped with
the stable model structure is Quillen equivalent to Spt (B).
T
Definition 2.14. To any spectrum of simplicial sets E ∈SptS1 we may associate
the constant S1-spectrum cE over B with value E. That is, cE is the sequence of
spaces cE with the evident bonding maps. For a simplicial spectrum E, we also
n
write cE for the (G ,S1) bi-spectrum Σ∞ cE. This defines a left Quillen functor
m G
m
c : SptS1 → SptGm,S1(B) with right adjoint given by evaluation at B. Compare
with [Lev14, Lemma 6.5].
2.5. Base change of stable model categories.
Definition 2.15. Let f : C → B be a map of base schemes. Pull-back along
f determines a functor f−1 : Sm/B → Sm/C, which induces Quillen adjunctions
(f∗,f ) : SpcA1(B) → SpcA1(C) and (f∗,f ) : Spt (B) → Spt (C). See e.g.,
∗ ∗ ∗ ∗ T T
[Mor05, §5] for more details.
Let Q (resp. R) denote the cofibrant (resp. fibrant) replacement functor in
Spt (B). The derived functors Lf∗ and Rf are givenby the formulasLf∗ =f∗Q
T ∗
and Rf =f R.
∗ ∗
3. Comparison to the classical stable homotopy category
We start this section by recalling [Lev14, Thm.1].
Theorem 3.1. If B =Spec(C), the map Lc:SH→SH(B) is fully faithful.
Proposition3.2. Letf :C →Bbeamapofbaseschemes. Thefollowingdiagram
of stable homotopy categories commutes.
SH
{{✈✈L✈c✈✈✈✈✈✈Lf∗❍❍❍❍❍❍L❍c❍❍##
SH(B) // SH(C)
Proof. The result follows by considering the following diagram.
f∗
Spt (B) // Spt (C)
S1 ff▼▼▼▼▼▼c▼▼▼▼ qqqqqcqqqqq88 S1
Σ∞Gm (cid:15)(cid:15) xxqqqqcqqqqqqSpft∗S1▼▼▼▼▼▼c▼▼▼▼&& (cid:15)(cid:15) Σ∞Gm
SptGm,S1(B) // SptGm,S1(C)
The left and right triangles are evidently commutative. The upper triangle com-
mutes by the computation
(f∗c E)(U)= colim c E(V)=E,
B B
U→f−1V
STABLE MOTIVIC STEMS OVER Fq 7
wherethe colimitis overthe comma category(U ↓f−1). Hence the entirediagram
is commutative, and the result follows. (cid:3)
Proposition 3.3. Let B be a base scheme equipped with a map Spec(C) → B.
Then Lc:SH→SH(B) is faithful.
Proof. ForsymmetricspectraX andY,themapLc:SH(X,Y)→SH(C)(cX,cY)
factors through SH(B)(cX,cY) by proposition 3.2. Hence Lc : SH(X,Y) →
SH(B)(cX,cY) is injective by theorem 3.1. (cid:3)
Corollary 3.4. Write W(F ) for the ring of Witt vectors of F and K for the
p p
fraction field of W(F ) [Ser79, II §6]. Then Lc:πs →π (W(F )) is an injection,
p n n,0 p
as we have maps W(F )→K →C.
p
4. Motivic cohomology
SpitzweckhasconstructedaspectrumHZinSptΣ(B)whichrepresentsLevine’s
T
motivic cohomology Hi (X,n) — see [Lev01] — when B is the Zariski spectrum
mot
ofaDedekinddomain. SpitzweckestablishesenoughnicepropertiesofHZforusto
constructthe MASSovergeneralbaseschemes,andestablishcomparisonsbetween
the MASS over a Hensel local ring and its residue field.
4.1. Integral motivic cohomology.
Definition 4.1. Overthe baseschemeSpec(Z), thespectrumHZSpec(Z) isdefined
in[Spi13,Def.4.27]. ForageneralbaseschemeB,wedefineHZB tobef∗HZSpec(Z)
where f :B →Spec(Z) is the unique map.
LetB =Spec(D)forDaDedekinddomain. ForX ∈Sm/B,thereisacanonical
isomorphismSH(B)(Σ∞X ,Σi,nHZ)∼=Hi (X,n),whereHi denotesLevine’s
+ mot mot
motivic cohomology defined using Bloch’s cycle complex [Spi13, Cor.7.19]. The
isomorphismisfunctorialwithrespecttomapsinSm/B. Additionally,ifi:{x}→
B is the inclusion of a closed point with residue field k(x), there is a commutative
diagram.
SH(B)(Σ∞X ,Σi,nHZ) ∼= //Hi (X,n)
+ mot
(cid:15)(cid:15) (cid:15)(cid:15)
SH(k(x))(Li∗Σ∞X ,Σi,nHZ) ∼= // Hi (Li∗X,n)
+ mot
If the residue field k(x) has positive characteristic, there is a canonical isomor-
phism of ring spectra Li∗HZB ∼=HZk(x) [Spi13, Thm.9.16].
Proposition4.2. Iff :C →B isasmoothmapofbaseschemes,thenLf∗HZB ∼=
HZ .
C
Proof. Since f is smooth, Lf∗ =f∗, see [Mor05, p.44]. It is straightforwardto see
that f∗HZB ∼=HZC, so the result follows. (cid:3)
8 G.M.WILSONANDP.A.ØSTVÆR
4.2. Motivic cohomology with coefficients Z/ℓ. For a prime ℓ, write HZ/ℓ
for the cofiber of the map ℓ· : HZ → HZ in SH(B). The spectrum HZ/ℓ
represents motivic cohomology with Z/ℓ coefficients. For a smooth scheme X
over B, we write H∗,∗(X;Z/ℓ) for the motivic cohomology of X with Z/ℓ coef-
ficients. The mod ℓ Steenrod algebra over B is the bi-graded algebra A∗∗(B) =
SH(B)(HZ/ℓ,HZ/ℓ). The dual mod ℓ Steenrod algebra over B is the Hopf al-
gebroid A (B) = π (HZ/ℓ∧HZ/ℓ). When B is the spectrum of a ring R, we
∗∗ ∗∗
write A∗∗(R) for the Steenrod algebra instead of A∗∗(Spec(R)). We use the same
convention for the dual Steenrod algebra and motivic cohomology.
Let B =Spec(D) be the spectrumofa Dedekind domainD in whichℓ is invert-
ible. Spitzweck shows in [Spi13, Thm.11.24] that the dual Steenrod algebra with
Z/ℓ coefficients A (D) has the expected structure as a Hopf algebroid, as given
∗∗
in [HKØ13, Thm.5.5].
Definition4.3. LetDbeaDedekinddomain,andletC denotethesetofsequences
(ǫ ,r ,ǫ ,r ,...) with ǫ ∈ {0,1}, r ≥ 0, and only finitely many non-zero terms.
0 1 1 2 i i
The elements τi ∈ A2ℓi−1,ℓi−1(D) and ξi ∈ A2ℓr−2,ℓr−1(D) are constructed in
[Spi13, Cor.11.23]. For any sequence I =(ǫ ,r ,ǫ ,r ,...) in C, write ω(I) for the
0 1 1 2
element τǫ0ξr1··· and (p(I),q(I)) for the bi-degree of the operation ω(I).
0 1
WerecordSpitzweck’scalculationofthedualSteenrodalgebra[Spi13,Thm.11.24]
and the Steenrod algebra [Spi13, Remark 11.25] in the following propositions.
Proposition 4.4. Let D be a Dedekind domain. As an HZ/ℓ-module, there is a
weakequivalence Σp(I),q(I)HZ/ℓ→HZ/ℓ∧HZ/ℓ. The mapisgivenbyω(I)
I∈B
W
on the factor Σp(I),q(I)HZ/ℓ.
Remark 4.5. Let D be a Dedekind domain in which ℓ is invertible, and consider
the map f : Z[1/ℓ] → D. A key observation in the proof of [Spi13, Thm.11.24] is
that the map Lf∗ : A (Z[1/ℓ])→A (D) satisfies Lf∗τ = τ and Lf∗ξ = ξ for
∗∗ ∗∗ i i i i
all i. For a map j :D →D of Dedekind domains in which ℓ is invertible, it follows
that Lj∗τi =τi and Lj∗ξie=ξi for all i.
Proposition 4.6. Let D be a Dedekind domain in which ℓ is invertible. The
Steenrod algebra over D is isomorphic to the dual of A∗∗(D), i.e., A∗∗(D) ∼=
Hom(A (D),HZ/ℓ (D)).
∗∗ ∗∗
5. Motivic Adams spectral sequence
The motivic Adams spectral sequence over a base scheme B may be defined
using the appropriate notion of an Adams resolution, just as it is done classically
[Ada95], [Rav86]. We recount the definition and establish some basic properties
of the MASS under base change. We follow [DI10, §3] for the definition of the
(cohomological)motivic Adams spectral sequence. See also [HKO11, §6].
Let p6=2 andℓ be fixedprimes, p6=ℓ. We will be interested in the specific case
oftheMASSoverafieldandoveradiscretevaluationring(DVR)withresiduefield
of characteristic p. We write H for the spectrum HZ/ℓ over the base scheme B
and H∗∗(B) for the motivic cohomology of B with Z/ℓ coefficients. The spectrum
H is a ring spectrum and is cellular in the sense of [DI05] by [Spi13, Cor.11.4].
5.1. Construction of the mod ℓ MASS.
STABLE MOTIVIC STEMS OVER Fq 9
Definition 5.1. Consider X ∈ SptΣ(B), and let H denote the spectrum in the
T
cofibrationsequence H → →H →ΣH. The standard H-Adams resolution of X
1 ∧f
is the tower of cofibration sequences X → X → W given by X = H ∧X
f+1 f f f
and W =E∧X . Compare with [Ada95, §15].
f f
X =X oo i1 H ∧X oo i2 H ∧H ∧X oo ···
0 ❑❑❑❑j0❑❑❑❑❑❑%% tttt•t∂t0tttt:: ▼▼▼▼▼j1▼▼▼▼▼&& ♦♦♦♦•♦∂♦1♦♦♦♦♦♦77
H ∧X H ∧H ∧X
Definition 5.2. Acollectionofindices{(p ,q )∈Z2|α∈A}issaidtobemotivi-
α α
cally finite if for any α ∈ A there are only finitely many β ∈ A for which p ≥ p
α β
and 2q −p ≥2q −p .
α α β β
Definition 5.3 (H∗∗-MASS). Let X ∈SptΣ(B). An H∗∗-Adams resolution of X
T
is a tower of cofibration sequences X → X → W in SH(B) of the following
f+1 f f
form. Each spectrum Wf has a description Wf ∼= ∨αΣpα,qαH where the set of
indices {(p ,q )} is motivically finite, and the induced map H∗∗W → H∗∗X is
α α f f
a surjection.
X =X oo i1 X oo i2 X oo ···
0 ■■■j■0■■■■■$$ ④④④•④④∂④0④④== 1❈❈j❈1❈❈❈❈❈!! ④④④•④④∂④1④④== 2
W W
0 1
Proposition5.4. LetX ∈SptΣ(B)becellular,andsupposeH∗∗(X)isafreeH∗∗-
T
module which is motivically finitely generated. The standard H-Adams resolution
of X is an H∗∗-Adams resolution.
Proof. The conditions on X ensure that H ∧X is a motivically finite wedge of
suspensions of H. Furthermore,the map X →H∧X =W induces a surjection
f f f
H∗∗W →H∗∗X . Ifx:X →H isamapinH∗∗X ,themapH∧x:H∧X →H
f f f f f
in H∗∗W maps to x under j∗. (cid:3)
f f
Definition 5.5. Let X ∈ SptΣ(B) and let {X ,W } be the standard H-Adams
T f f
resolution of X. The H-motivic Adams spectral sequence for X is obtained from
the following exact couple.
⊕π X i∗ //⊕π X
∗∗ f ∗∗ f
ee▲▲▲∂▲∗▲▲▲▲▲▲ yyrrrrrrjr∗rrr
⊕π W
∗∗ f
The E term of the MASS is Ef,(s,w) =π W . The index f is called the Adams
1 1 s,w f
filtration, s is the stem, and w is the motivic weight.
Definition 5.6. Let X ∈SptΣ(B) and consideran Adams resolution{X ,W } of
T f f
X. The Adams filtration of π X is given by F π X =im(π X →π X).
∗∗ i ∗∗ ∗∗ i ∗∗
Proposition 5.7. LetSdenote thecategoryofspectralsequencesinthecategory
of Abelian groups. The associated spectral sequence to the standard H-Adams
10 G.M.WILSONANDP.A.ØSTVÆR
resolution defines a functor MASS : SH(B) → S. Furthermore, the motivic
Adams spectral sequence is natural with respect to base change.
Proof. The construction of the standard H-Adams resolution is functorial. Given
X →X′ we get induced maps of H-Adams resolutions{X ,W }→{X′,W′}. As
f f f f
π (−) is a triangulated functor, we get an induced map of the associated exact
∗∗
couples, and hence of spectral sequences MASS(X)→MASS(X′).
Let f : C → B be a map of base schemes. The claim is that there is a natural
transformation between MASS : SH(B) → S and MASS ◦ Lf∗ : SH(B) →
SH(C) → S. Let X ∈ SH(B) and let {X ,W } be the standard H -Adams
f f B
resolutionofX inSH(B). WemayaswellassumeX iscofibrant,soQX =X where
Q is the cofibrant replacement functor. Let {X′,W′} denote the standard H -
f f C
resolutionof Lf∗X =f∗X. Observethat {f∗X ,f∗W }={X′,W′}, since f∗ =
and f∗H = H . We therefore have a mafp {Lf∗fX ,Lf∗fW }f → {X′,W1′}.
1Applying LBf∗ : SHC(B)( s,w,−) → SH(C)( s,w,Lf∗−) tfo {X ,Wf } givesfa mfap
f f
of exact couples, and the1refore a map Φ : M1 ASS (X)→MASS (Lf∗X). It is
X B C
straightforwardto verify that Φ determines a natural transformation. (cid:3)
Corollary 5.8. For a map of base schemes f : C → B, there is a map of mo-
tivic Adams spectral sequences Φ : MASS ( ) → MASS ( ). The map Φ is
B C
1 1
furthermore compatible with the induced map π ( )→π ( ).
∗∗ B ∗∗ C
1 1
Proposition 5.9. SupposeX ∈SH(B)satisfiesthehypothesesofproposition5.4.
The MASS for X has E -page given by
2
Ef,(s,w) =Extf,(s+f,w)(H∗∗X,H∗∗B)
2 A∗∗(B)
with differentials d :Ef,(s,w) →Ef+r,(s−1,w) for r ≥2.
r r r
Proof. Thesameargumentgiveninthetopologicalsituation[Ada58,Thm.2.1]goes
through with minor modifications. (cid:3)
Definition 5.10. ConsiderabaseschemeB =Spec(D)foraDedekinddomainD.
Let Ext(D) denote the tri-graded group ExtA∗∗(B)(H∗∗(B),H∗∗(B)).
Corollary 5.11. IfX andX′ areinSH(B)whichsatisfythehypothesesofpropo-
sition5.4,andX →X′ inducesanisomorphismH∗∗X′ →H∗∗X,thentheinduced
map MASS(X) → MASS(X′) is an isomorphism of spectral sequences from the
E -page onward.
2
Corollary 5.12. Let f : C → B be a map of base schemes. Suppose Lf∗ :
H∗∗(B) → H∗∗(C), Lf∗ : A∗∗(B) → A∗∗(C), and Lf∗ : H∗∗X → H∗∗(Lf∗X)
are all isomorphisms. Then MASS (X)→MASS (Lf∗X) is an isomorphism of
B C
spectral sequences from the E page onward.
2
5.2. Convergence of MASS.
Proposition5.13. LetB =Spec(k)betheZariskispectrumofafieldkwithchar-
acteristic p 6= ℓ. For X ∈ SH(B), the motivic Adams spectral sequence converges
Ef,(s,w) ⇒π X∧underthefollowingconditions: khasfinitemodℓcohomological
2 s,w ℓ
dimension, X is cellular and of finite type in the sense of [HKO11, §2], i.e., H∗∗X
has a set of generators which is motivically finite and H∗∗X is free over H∗∗(B).