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Some stable splittings in motivic homotopy theory
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Some stable splittings in motivic homotopy theory

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Content Some Stable Spliings in Motivic Homotopy
Theory
Viktor Kleen
A dissertation presented to the faculty of The USC Graduate School at the
UniversityofSouthernCalifornia
in partial fulllment of the requirements for the degree of
DoctorofPhilosophy inMathematics
in August 2019.
Copyright©2019
All Rights Reserved
Contents
1 Introduction 2
2 Motivic Homotopy Theory 5
2.1 Motivic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Motivic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Extension to Smooth Ind-Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Traces and Transfers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Spliing BGL
n
24
3.1 Transfers of Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 The Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Bibliography 30
1
1 Introduction
The theme of this thesis is the use of a concrete geometrical construction to prove an abstract structure
result on the motivic classifying space of the general linear group. The theoretical framework for
this is motivic homotopy theory as rst developed by Fabien Morel and Vladimir Voevodsky in
[MV99]. The goal of this theory is to bring to bear on problems in algebraic geometry the formalism
of homotopy theory.
The main result of this thesis is inspired by a classical result in homotopy theory due to Snaith. In
Part 1 of his monograph [Sna79] Snaith proves among other things that the ltration BU
1
BU
2

: : : BU
n
of the classifying space BU
n
of the unitary group U
n
and an analogous ltration of BO
n
split after passing to suspension spectra in stable homotopy theory. That is to say, Snaith constructs
weak equivalences
BU
n;+
'
n
Ü
i=1
BU
i
BU
i1
and
BO
2k;+
'
k
Ü
i=1
BO
2i
BO
2i2
in the stable homotopy category. The unitary group U
n
is the maximal compact subgroup of the
complex general linear group GL
n
¹Cº. It is a classical fact that their classifying spaces are homotopy
equivalent. This leads to a natural extension of Snaith’s result to algebraic geometry, GL
n
being an
algebraic group.
Theorem. OveranyschemeS,thereisaP
1
S
–stablesplitting BGL
n;+
'
Ô
n
i=1
BGL
i
BGL
i1
ofthenatural
ltration BGL
0
: : : BGL
n1
BGL
n
.
Using topological realization, forS = Spec¹Cº, this theorem recovers Snaith’s result on BU
n
. For
S = Spec¹Rº, it recovers a renement of Snaith’s splitting result for BO
n
, namely an equivalence BO
n
'
Ô
n
i=1
BO
i
BO
i1
. However, real realization can be rened to take values in stableZ2–equivariant
homotopy theory. This way we even obtain aZ2–equivariant splitting theorem, essentially for free.
Snaith’s proof is based on transfers in the stable homotopy category. Transfer maps for ber
bundles in the stable homotopy category were rst constructed by Becker and Gottlieb in [BG75] in
order to prove the Adams conjecture. Given a ber bundleF E B over a nite CW complex
B with structure group a compact Lie groupG such thatF is a closed smooth manifold with a smooth
action ofG, Becker and Gottlieb construct a transfer mapB
+
E
+
. They later generalized their
2
construction to Hurewicz brations in [BG76]. In [DP80] Dold and Puppe show that the transfer is in
fact an instance of a very general phenomenon in symmetric monoidal categories. This perspective
is also what allows one to construct the required transfers in motivic homotopy theory. In [Lev18]
Levine uses duality results by Riou, [Rio05], and Ayoub, [Ayo07b; Ayo07a], to construct such motivic
Becker–Gottlieb transfers.
Snaith’s proof uses the Becker–Gottlieb transfer associated with the inclusion of a maximal torus in
U
n
. Instead of using an analogous motivic construction, here we follow an approach due to Mitchell,
Priddy and Richter, [MP89]. They observe that the block–diagonal subgroup GL
i
¹Cº GL
ni
¹Cº
GL
n
¹Cº induces a map on classifying spaces. Its homotopy ber can be shown to be homotopy
equivalent to the Grassmannian ofi–dimensional linear subspaces ofn–dimensional space. Since
this space is a compact smooth manifold with a smooth GL
n
¹Cº action this situation ts into the
framework of [BG76] and there is an associated transfer map. This transfer map can then be used to
construct the splitting.
We show that an analogous approach works in motivic homotopy theory. The block–diagonal
subgroup GL
i
GL
ni
of GL
n
is of course algebraic and also induces a map on classifying spaces. To
show that it admits a transfer, we compute its homotopy ber again as the Grassmanian ofi–planes
inn–space. For this it is essential that ane space bundles are homotopy equivalences in motivic
homotopy theory. Furthermore, to be able to dene the transfer it is essential to invert the projective
lineP
1
with respect to the smash product. This ensures that a motivic version of Atiyah duality
holds for smooth proper schemes, the Grassmannian being an example of the latter. Putting all this
together we obtain the splitting theorem for BGL
n
in stable motivic homotopy theory.
Overview In Chapter 2 we summarize the foundations of motivic homotopy theory in an1–
categorical language. This language is very convenient for formulating the functoriality of the
motivic homotopy category and for constructing traces and transfers. We rst show how to construct
the1–topos of Nisnevich local spaces over a schemeS and the1–category of motivic spaces over
S. We then recall the construction of inverse and direct image functorialities for these. Then, we
describe the1–category of motivic spectra with its associated functorialities. Our key technical
contribution is the construction of a satisfactory stable motivic homotopy category over a smooth
ind-scheme. We also extend the relevant functorialities to these more general bases. Finally, we recall
the theory of dualizability and the associated traces and transfers.
In Chapter 3 we come to the proof of our main theorem. We rst recall that the classifying space
BGL
n
of interest admits an explicit geometric model in motivic homotopy theory as a smooth ind-
scheme, namely the innite Grassmannian Gr
n
ofn–planes. We then show analogously to [MP89] how
to use the geometry of nite Grassmannian to partially compute the transfer Gr
n;+
Gr
r;+
^Gr
nr;+
.
This partial computation is enough to construct the splittings maps for BGL
n
.
Acknowledgements I would like to thank Aravind Asok for his steady advice and patience. I
would also like to thank Marc Hoyois for countless fruitful conversations about higher category
3
theory and motivic homotopy theory. Finally, I would like to thank my family and friends for their
support during my time at USC. Without you this work would not have been possible.
4
2 Motivic Homotopy Theory
The foundations of motivic homotopy theory were laid in [MV99] in order to import methods from
homotopy theory into algebraic geometry. Since then, this approach has proven immensely useful.
For instance, the Voevodsky–Rost proof of the Bloch–Kato conjecture [Voe11], now the norm residue
isomorphism theorem, makes essential use of motivic homotopy theory.
According to Dugger [Dug01] and Robalo [Rob15] the motivic homotopy categoryH¹Sº of a scheme
S can be though of as the universal Nisnevich–local homotopy theory built on smooth schemes over
S in which the ane lineA
1
S
is contractible. As such it contains the homotopy theory of spacesSpc,
at least as long asS is nonempty. Additionally, it contains objects of geometric origin such as the
Tate circleG
m;S
=A
1
S
rf0g. Both the topological circleS
1
and the Tate circleG
m;S
are inverted with
respect to the smash product when passing to stable motivic homotopy theory. This ensures that
motivic Atiyah duality, Theorem 2.4.1, holds.
Notation We denote by Δ the category of nonempty nite ordinals with order preserving maps
between them. The1–category of spaces isSpc. We work mostly model independently with1–
categories, but if an explicit choice of foundations is required we use the theory of quasicategories as
introduced by [Joy02] and developed in [Lur09]. The terms "limit" and "homotopy limit" as well as
"colimit" and "homotopy colimit" will be used interchangably. When necessary,Cat
1
denotes the
1–category of1–categories. We will suppress any further mention of universes.
Overview We rst show how to construct the1–categories of motivic spaces and motivic spectra
following [CD09; Kha16; Lev18]. Both of these depend suitably functorially on schemes over a xed
baseS. Briey, for any morphism f : S
0
S of schemes there are1–categoriesH¹Sº andH¹S
0
º
and stable symmetric monoidal1–categoriesSH¹Sº andSH¹S
0
º with adjunctions
f

:H¹Sº
`
H¹S
0
º : f

and
f

:SH¹Sº
`
SH¹S
0
º : f

:
Both functors f

are symmetric monoidal with respect to the cartesian structure onH¹Sº andH¹S
0
º
and the smash product onSH¹Sº andSH¹S
0
º. If in addition f is smooth, then there are additional
adjunctions
f
#
:H¹S
0
º
`
H¹Sº : f

5
and
f
#
:SH¹S
0
º
`
SH¹Sº : f

:
All these adjunctions are functorial in f .
Our main technical contribution is then to use these adjunctions to extend the possible bases for
motivic homotopy theory. We construct a stable symmetric monoidal1–category of motivic spectra
over any smooth ind-scheme over a xed baseS. These1–categories are again connected by the
expected adjunctions. This is the key ingredient for Chapter 3.
Finally, we briey describe the formalism of traces and transfers in stable symmetric monoidal
1–categories. This is used to construct motivic Becker–Gottlieb transfers in [Hoy14] and [Lev18].
Hoyois [Hoy14] proves a Grothendieck–Lefschetz–Verdier trace formula in motivic homotopy theory
while Levine [Lev18] constructs an Euler characteristic with values inπ
0
End¹1
S
º over any scheme
S, rening the classicalZ–valued Euler characteristic. In Chapter 3 we will use transfers to prove a
splitting result following ideas of Mitchell and Priddy [MP89] in classical homotopy theory.
2.1 Motivic Spaces
Let S be a scheme. We will denote by Sm
S
the category of schemes X equipped with a smooth
morphismX S. The maps in Sm
S
are commuting triangles
X Y
S
f
with the horizontal map f an arbitrary morphism of schemes. A space over S is a presheaf of spaces
on Sm
S
. These assemble into an1–categorySpc¹Sº =Fun¹Sm
op
S
;Spcº.
A Nisnevich square is a cartesian square
V Y
U X
f
i
in Sm
S
withi an open immersion and f an étale map, such that the induced map on the complement
f
1
¹XrUº
red
¹XrUº
red
is an isomorphism. Here,¹_º
red
denotes the underlying reduced scheme.
Definition 2.1.1 (cf. [AHW17, Theorem 3.2.5]). A spaceF over S satises Nisnevich excision if
6
F¹;º' and for every Nisnevich square as above, the square
F¹Vº F¹Yº
F¹Uº F¹Xº
is a homotopy pullback of spaces. The full subcategory ofSpc¹Sº on spaces satisfying Nisnevich
excision is denotedSpc
Nis
¹Sº.
It turns out that Nisnevich excision characterizes Nisnevich descent, see [AHW17]. The Nisnevich
topology on Sm
S
is the coarsest Grothendieck topology on Sm
S
such that the empty sieve is a covering
sieve and for each Nisnevich square
V Y
U X
f
i
the sieve onX generated byi and f is a covering sieve. A spaceFoverS satises Nisnevich descent
if for every Nisnevich covering familyff
i
: U
α
Ug
α
the natural map
F¹Uº holim
»n¼2Δ
Ö
α
0
;:::;α
n
F¹U
α
0

U

U
U
α
n
º
is an equivalence.
Theorem2.1.2 (cf. [Voe10, Corollary 5.10] and [AHW17]). AspaceFoverS satisesNisnevichexcision
as in Denition 2.1.1 if and only if it satises Nisnevich descent.
Definition2.1.3. A spaceFoverS isA
1
–homotopy invariant if for every smooth schemeX overS
the imageF¹Xº F¹XA
1
º of the projection is an equivalence. The full subcategory ofSpc¹Sº
onA
1
–homotopy invariant spaces overS is denotedSpc
A
1¹Sº.
Theorem 2.1.4 (cf. [Kha16, section 2.4], [MV99; Rob15; Hoy14]). The1–categoriesSpc
Nis
¹Sº and
Spc
A
1¹Sº are accessible localizations of Spc¹Sº. Write
L
Nis
: Spc¹Sº Spc
Nis
¹Sº
and
L
A
1: Spc¹Sº Spc
A
1¹Sº
for the localization functors. In particular, bothSpc
Nis
¹Sº andSpc
A
1¹Sº are presentable1–categories.
(i) The functors L
Nis
and L
A
1 preserve colimits and nite products.
(ii) The functor L
Nis
preserves nite limits.
7
(iii) Colimits inSpc
Nis
¹Sº andSpc
A
1¹Sº are universal.
Proof. The Nisnevich topology is a Grothendieck topology on Sm
S
. Therefore,Spc
Nis
¹Sº is a topolog-
ical localization ofSpc¹Sº by [Lur09, Propoisition 6.2.2.7]. This implies thatSpc
Nis
¹Sº is an1–topos
and provides us with anexact localization functor L
Nis
. Furthermore, colimits in any1–topos are
universal.
The case ofSpc
A
1¹Sº is more delicate. The key ingredient is the construction of L
A
1. Forn 0
denote byΔ
n
the algebraicn–simplex
Δ
n
= SpecZ»x
0
; : : :;x
n
¼¹x
0
++x
n
1º
and byΔ
n
S
=Δ
n
S its basechange toS. The standard face and degeneracy maps provideΔ

with the
structure of a cosimplicial scheme andΔ
n
A
n
for alln. Morel and Voevodsky [MV99, section 2.3]
construct theA
1
–localization of a spaceFoverS by
L
A
1¹FºB hocolim
»n¼2Δ
op
Map
Spc¹Sº
¹Δ
n
S
;Fº:
They prove that this denes a left adjoint functor L
A
1 to the inclusionSpc
A
1¹SºSpc¹Sº. As such it
of course preserves colimits.
To see that L
A
1 preserves nite products, note that products inSpc
A
1¹Sº are also products inSpc¹Sº
and products inSpc¹Sº are computed pointwise. Since L
A
1 is dened by asifted colimit and sifted
colimits commute with nite products this nishes the proof of (i).
The family of generatingA
1
–local equivalencesfXA
1
Xg
X2Sm
S
is stable under basechange.
By [GK12, Proposition 1.3] this shows thatSpc
A
1¹Sº is a locally cartesian localization ofSpc¹Sº. In
particular, colimits inSpc
A
1¹Sº are universal.
Definition2.1.5 (cf. [MV99; Hoy14; Kha16]). A space overS ismotivic if it isA
1
–homotopy invariant
and satises Nisnevich descent. The full subcategory ofSpc¹Sº on motivic spaces is the motivic
homotopy categoryH¹Sº ofS.
Theorem 2.1.6 (cf. [MV99; Hoy14; Kha16]). The1–categoryH¹Sº is an accessible localization of
Spc¹Sº. Write
L
mot
: Spc¹Sº H¹Sº
for the localization functor. In particular,H¹Sº is a presentable1–category.
(i) The functor L
mot
preserves colimits and nite products.
(ii) Colimits inH¹Sº are universal.
(iii) The1–categoryH¹Sº is cartesian closed.
Proof. The localization functor L
mot
can be constructed as the transnite composite
L
mot
¹Fº = hocolim
n0
¹L
A
1 L
Nis
º
n
¹Fº
8
following [Kha16, section 2.4], essentially because Nisnevich locality andA
1
–invariance are stable
under ltered colimits. Since both L
A
1 and L
Nis
commute with nite products, this description of
L
mot
also shows that it commutes with nite products. Since both L
A
1 and L
Nis
are locally cartesian
localizations we also have thatH¹Sº is a locally cartesian localization. In particular, colimits inH¹Sº
are universal andH¹Sº is cartesian closed.
Composing the Yoneda embedding Sm
S
Spc¹Sº with the motivic localization functor L
mot
denes a functor Sm
S
H¹Sº which we will elide from our notation. That is, for a smooth scheme
X overS we still writeX for the motivic localization of the space Map
S
¹_;Xº overS.
Suppose f : S
0
S is an arbitrary morphism of schemes. Then pullback denes a functor
f

: Sm
S
Sm
S
0 which extends by Kan extension to an adjunction
f

:Spc¹Sº
`
Spc¹S
0
º : f

which is characterized by the property that f

¹Xº'X
S
S
0
forX2 Sm
S
. Here, we have implicitly
identiedX with its image under the Yoneda embedding. Additionally, f

is given by the equation
f

¹Fº¹Yº'F¹Y
S
S
0
º forF2Spc¹S
0
º andY2 Sm
S
.
Since products inSpc¹Sº preserve colimits in both entries and f

clearly preserves products of
representable presheaves, the functor f

is symmetric monoidal with respect to the cartesian monoidal
structures onSpc¹Sº andSpc¹S
0
º. Similarly, f

is symmetric monoidal since it commutes with limits,
being a right adjoint.
If f : S
0
S is a smooth morphism, then postcomposition with f denes a left adjoint functor
f
#
: Sm
S
0 Sm
S
to the pullback functor f

: Sm
S
Sm
S
0. This implies that, on representable
presheavesF , we have f

¹Fº¹Xº' F¹f
#
Xº for every X 2 Sm
S
0. Since both sides commute with
colimits inF —colimits in presheaves are computed pointwise—we conclude that f

coincides with
the functor given by precomposing with f
#
. That is, f
#
Kan extends to an adjunction
f
#
:Spc¹S
0
º
`
Spc¹Sº : f

:
These adjunctions restrict toSpc
Nis
¹Sº andH¹Sº: Basechange fromS toS
0
preserves Nisnevich
covering families. Hence, from the description of f

: Spc¹S
0
º Spc¹Sº given above, the functor
f

sends spaces satisfying Nisnevich excision overS
0
to spaces satisfying Nisnevich excision over
S, that is, it restricts to a functor f

: Spc
Nis
¹S
0
º Spc
Nis
¹Sº. Then L
Nis
f

is left adjoint to the
restricted functor f

: we have natural equivalences
Map
Spc
Nis
¹S
0
º
¹L
Nis
¹f

Fº;Gº' Map
Spc¹S
0
º
¹f

F;Gº
' Map
Spc¹Sº
¹F; f

Gº
' Map
Spc
Nis
¹Sº
¹F; f

Gº
forF2 Spc
Nis
¹Sº and G2 Spc
Nis
¹S
0
º. By abuse of notation we continue to write f

instead of
L
Nis
f

if there is no danger of confusion.
9
Similarly, the basechange of anA
1
–projection is again anA
1
–projection. Combining this with the
previous argument shows that f

sends motivic spaces overS
0
to motivic spaces overS and admits
the left adjoint L
mot
f

. Again, if there is no danger of confusion, we will simply write f

instead of
L
mot
f

.
If f : S
0
S happens to be smooth, then f
#
: Sm
S
0 Sm
S
preserves Nisnevich covering
families as well asA
1
–projections. Consequently, by the same argument as above, precomposition
with it sends spaces over S satisfying Nisnevich excision to spaces over S
0
satisfying Nisnevich
excision and motivic spaces over S to motivic spaces over S
0
. We have already seen that in this
case the functor f

coincides with precomposition with f
#
. In other words, if f is smooth, then f

restricts to a functor f

: Spc
Nis
¹Sº Spc
Nis
¹S
0
º and further to a functor f

: H¹Sº H¹S
0
º. By
[Lur09, Proposition 5.2.7.4] the restrictions of the localization functors L
Nis
and L
mot
toSpc
Nis
¹Sº and
H¹Sº respectively are naturally equivalent to the identity functors. Consequently, L
Nis
f

' f

and
L
mot
f

' f

. Then, the same arguments as above show that we have an adjunction
L
Nis
f
#
:Spc
Nis
¹S
0
º
`
Spc
Nis
¹Sº : L
Nis
f

and similarly for motivic spaces. Again, by abuse of notation we write f
#
a f

for both adjunctions if
there is no danger of confusion. We summmarize:
Theorem2.1.7 (cf. [MV99; Ayo07b; Kha16]). A morphism f : S
0
S induces adjunctions
f

:Spc
Nis
¹Sº
`
Spc
Nis
¹S
0
º : f

and
f

:H¹Sº
`
H¹S
0
º : f

:
The functor f

is characterized by f

¹L
Nis
Xº' L
Nis
¹X
S
S
0
º for representable spaces X 2 Spc¹Sº
and f

¹Fº¹Yº'F¹Y
S
S
0
º for Y2 Sm
S
. Similar statements hold for motivic spaces.
If f is smooth, then there are additional adjunctions
f
#
:Spc
Nis
¹S
0
º
`
Spc
Nis
¹Sº : f

and
f
#
:H¹S
0
º
`
H¹Sº : f

:
Thefunctor f
#
ischaracterizedby f
#
¹L
Nis
Xº = L
Nis
X for X2 Sm
S
0 whereontherighthandsideX is
considered as an object of Sm
S
by composing with f. A similar statement holds for motivic spaces.
For our purposes, the most convenient formulation of the functoriality properties for motivic
homotopy theory is found using the theory of presentable1–categories. The latter is developed in
[Lur09, section 5.5.3]. Very briey, there is an1–categoryPr
L
of presentable1–categories with
left adjoint functors as morphism and an1–categoryPr
R
of presentable1–categories with right
adjoint functors as morphisms. There is an equivalencePr
L
'¹Pr
R
º
op
of1–categories which is
10
the identity on objects and sends a left adjoint functor to its right adjoint. BothPr
L
andPr
R
are
complete and cocomplete and the homotopy limits in bothPr
L
andPr
R
coincide with homotopy
limits in the1–category of1–categories.
Following [Kha16, section 5.1] the adjunction f

a f

in Theorem 2.1.7 can be extended to two
functors
H

: Sch
S
Pr
R
and H

: Sch
op
S
Pr
L
:
The functor H

is obtained by starting with the cartesian bration Sch

Sch sending a
morphism between schemes to its target. Restricting this bration to the full subcategory of Sch

on smooth morphisms produces another cartesian bration. By the straightening construction, [Lur09,
Theorem 3.2.0.1], this latter cartesian bration corresponds to a functor Sch
op
Cat
1
sending a
schemeS to the discrete1–category Sm
S
of smooth schemes overS and a morphism f : S
0
S
to the pullback functor f

: Sm
S
0 Sm
S
. From here, we compose with the functor sending an
1–category to its category of presheaves of spaces and functors to their Kan extension. This way we
obtainH

and we can compose with the antiequivalence¹Pr
L
º
op
'Pr
R
and getH

.
RestrictingH

to the wide subcategory Sch
S;sm
of Sch
S
on the smooth morphisms, we have
shown that it gives a functor
H

: Sch
op
S;sm
Pr
R
and postcomposing with the antiequivalence¹Pr
L
º
op
'Pr
R
yields a functor
H
#
: Sch
S;sm
Pr
L
:
This is also an outline of how we will construct the functorialities for stable motivic homotopy theory
in Section 2.2.
Proposition2.1.8 (Weak Nisnevich Descent, cf. [Kha16, Proposition 6.1.6]). SupposeX isaschemewith
a Nisnevich covering familyfp
α
: X
α
Xg
α
. Then the familyfp

α
: Spc
Nis
¹Xº Spc
Nis
¹X
α
ºg
α
of pullback functors is jointly conservative. That is, a morphism f :F Gof Nisnevich local spaces
over X is an equivalence if and only if p

α
¹fº is an equivalence inSpc
Nis
¹X
α
º for allα.
Proof. Suppose f :F Gis such thatp

α
¹fº is an equivalence for allα. BecauseFandGsatisfy
Nisnevich descent, it will be enough to show that
F¹E
X
X
α
º
f
α ;E
G¹E
X
X
α
º
is an equivalence for every E2 Sm
X
. But E
X
X
α
' p
α#
p

α
¹Eº when considered as an object of
Spc
Nis
¹Xº. By adjunction we have a commutative square
Map¹p
α#
p

α
E;Fº Map¹p
α#
p

α
;Gº
Map¹p

α
E;p

α
Fº Map¹p

α
E;p

α
Gº:
f

' '
f

'
That is, under the identication given by the Yoneda lemma, f
α;E
= f

is an equivalence.
11
2.2 Motivic Spectra
We denote byH¹Sº

the undercategoryH¹Sº

of pointed motivic spaces overS. Coproduct with the
contractible space overS denes a functor¹_º
+
: H¹Sº H¹Sº

.
Proposition 2.2.1 ([Kha16, Proposition 3.2.5], cf. [Rob15]). There is a unique closed symmetric
monoidal structure onH¹Sº

such that the functor¹_º
+
is symmetric monoidal with respect to the
cartesianmonoidalstructureonH¹Sº. WedenotethissymmetricmonoidalstructureonH¹Sº

by _^_.
Definition2.2.2. SinceH¹Sº

is closed symmetric monoidal, the functorP
1
S
^_, whereP
1
S
is considered
pointed at1, is left adjoint to the functorΩ
P
1 =Map¹P
1
S
; _º ofP
1
–loops. We dene the1–category
of motivic spectra overS to be the homotopy colimit
SH¹SºB hocolim

H¹Sº

P
1
S
^_
H¹Sº

P
1
S
^_


inPr
L
. Since colimits along left adjoints inPr
L
correspond to limits along their right adjoints inPr
R
and those limits coincide with limits in the1–category of1–categories, we can equivalently dene
SH¹SºB holim


Ω
P
1
H¹Sº

Ω
P
1
H¹Sº


in the1–category of1–categories. The colimit description ofSH¹Sº immediately supplies us with a
functor
Σ
1
P
1
: H¹Sº

SH¹Sº
with a right adjointΩ
1
P
1
. We often suppressΣ
1
P
1
from the notation if there is no danger of confusion.
An objectX in a symmetric monoidal1–category is calledn–symmetric, for somen 1, if the
cyclic permutation onX

n
is homotopic to the identity. The main result of [Rob15], Corollary 4.24
and Corollary 4.25, shows thatP
1
S
is 3–symmetric and that this implies thatSH¹Sº admits a unique
closed symmetric monoidal structure such that the functorΣ
1
P
1
: H¹Sº

SH¹Sº is symmetric
monoidal. We continue to denote the symmetric monoidal structure onSH¹Sº by _^ _. Note that
_^ _ preserves colimits in either entry, since _^A admits a right adjoint for allA2SH¹Sº. This is
just a restatement of the smash product being a closed symmetric monoidal structure.
Recall that a suspension functorΣ¹Xº =q
X
can be dened in any nitely cocomplete1–
category with a terminal object. Sending a pointed spaceT to the constant presheaf and then applying
motivic localization yields a pointed motivic space over S which we continue to denote byT . In
particular, we have the pointed circleS
1
2H¹Sº

. Because smash product preserves colimits in either
entry andS
1
'Σ¹S
0
º it follows thatΣ'S
1
^ _.
Consider the schemeP
1
S
overS pointed by 1. We have an elementary Nisnevich square
G
m;S
A
1
S
A
1
S
P
1
S
t
t
1
12
becauseP
1
S
is covered by two ane lines in the Zariski topology. Consequently, we obtain a homotopy
pushoutP
1
S
'A
1
S
q
G
m;S
A
1
S
inSpc
Nis
¹Sº. Passing toH¹Sº, we haveA
1
S
' by construction and we
nd thatP
1
S
'Σ¹G
m;S
º inH¹Sº

and therefore also inSH¹Sº. In summary, we have the fundamental
relationP
1
S
'S
1
^G
m;S
inH¹Sº

and also inSH¹Sº.
Proposition2.2.3. The1–categorySH¹Sº is stable in the sense of [Lur12].
Proof. Recall that a nitely complete and cocomplete1–category is stable if it admits a zero object
and the suspension functor Σ is an equivalence. But we have seen that Σ' S
1
^ _, so this last
condition is equivalent toS
1
being invertible with respect to the smash product. By the denition
ofSH¹Sº the objectP
1
S
is invertible with respect to the smash product. But we have constructed an
equivalenceP
1
S
'S
1
^G
m;S
, soS
1
is invertible as well. Of course,SH¹Sº contains a zero object by
construction.
The six functor formalism forSHwas established in [Ayo07b; Ayo07a] for noetherian schemes and
extended to arbitrary schemes in [Hoy14, Appendix C]. A comprehensive treatment of the standard
functorialities on the level of triangulated categories can be found in [CD09]. [Kha16] develops a
detailed account of the1–categorical six functor formalism, including exchange transformations. In
particular, they show that the functorialities from Theorem 2.1.7 extend naturally toSH.
Here, we will content ourselves with describing just a fragment of the full formalism. First, for
every morphism f : Y X between smooth schemes overS we have an adjunction
f

:SH¹Xº
`
SH¹Yº : f

between the stable presentable1–categoriesSH¹Xº andSH¹Yº. These adjunctions extend the
ones constructed forH in Theorem 2.1.7 in the following sense. Since f

: H¹Xº

H¹Yº

is a symmetric monoidal functor between symmetric monoidal, stable, presentable1–categories
admitting a right adjoint and f

¹P
1
X
º'P
1
Y
, passing to the colimit along the rows in the diagram
H¹Xº

H¹Xº

: : :
H¹Yº

H¹Yº

: : :
P
1
X
^_
f

P
1
X
^_
f

P
1
Y
^_ P
1
Y
^_
yields the functor f

: SH¹Xº SH¹Yº as a morphism in the subcategory ofPr
L
consisting of
symmetric monoidal functors between symmetric monoidal, stable, presentable1–categories. The
functor f

is then its right adjoint.
In other words, these adjunctions assemble into functorsSH

: Sm
op
S
Pr
L
with respect to
pullback andSH

: Sm
S
Pr
R
with respect to push forward. These are naturally equivalent after
composing with the equivalencePr
L
'¹Pr
R
º
op
. Additionally, the functorSH

factors through the
subcategory ofPr
L
consisting of symmetric monoidal functors between symmetric monoidal, stable,
presentable1–categories.
13
Second, if f : Y X is smooth, then there is an additional adjunction
f
#
:SH¹Yº
`
SH¹Xº : f

:
Again, these adjunctions extend the ones from Theorem 2.1.7 and they naturally assemble into a
functorSH
#
: Sm
S;sm
Pr
L
from the wide subcategory of Sm
S
consisting of smooth morphisms
between smooth schemes overS. Given a cartesian square


g
q p
f
in Sm
S
with f and henceg smooth, there is a natural exchange equivalence
Ex

#
:g
#
q


p

f
#
:
This can be checked by noting that both functors commute with colimits and therefore reducing
to the case of suspension spectra of smooth schemes overS. In this case, the statement is obvious;
see also [Kha16, Proposition 6.2.6]. More details on other exchange transformations can be found in
[CD09; Kha16].
Finally, we will need a dierent description of the suspension spectrum Σ
1
P
1
X
+
2 SH¹Sº for a
smooth schemeX overS. This is provided by a lemma of Ayoub’s.
Lemma2.2.4 ([Ayo14, Lemma C.2]). Let π: X S be a smooth scheme over S. There is a canonical
equivalenceΣ
1
P
1
X
+
'π
#
¹1
X
º inSH¹Sº.
Proof. The monoidal unit 1
X
2SH¹Xº is the 0–sphereS
0
2SH¹Xº. It is the image of the point
under the composite functorΣ
1
P
1
¹_º
+
: H¹Xº SH¹Xº, that is, the schemeX considered as a
smooth scheme overX. By our denition ofπ
#
: H¹Xº H¹Sº, we haveπ
#
¹º = X2H¹Sº. But
we have a commutative diagram
H¹Xº H¹Xº

SH¹Xº
H¹Sº H¹Sº

SH¹Sº
π
#
¹_º
+
π
#
Σ
1
P
1
π
#
¹_º
+
Σ
1
P
1
by construction. Chasing the image of the point around the outsides of this diagram yields the
lemma.
Definition2.2.5 (cf. [Lev18]). Given a smooth schemeX overS with structure mapπ: X S we
writeXS B π
#
¹1
X
º'Σ
1
P
1
X
+
2SH¹Sº.
14
2.3 Extension to Smooth Ind-Schemes
Our approach to the stable splitting of BGL
n
requires a stable motivic homotopy theory over smooth
ind-schemes. The most expedient way to obtain such a theory is to formally extend the functorSH

,
SH

andSH
#
of the last section to smooth ind-schemes. This section describes how to do this.
Definition2.3.1. Asmoothind-scheme overS is an object of Ind¹Sm
S
º, the1–category of ind-objects
in the category of smooth schemes overS with arbitrary morphisms between them. A morphism of
ind-schemes is smooth if it can be presented as a colimit of smooth morphisms in Sm
S
.
BecausePr
R
is cocomplete, the functorSH

naturally extends to a functor
SH

: Ind¹Sm
S
º Pr
R
:
Composing with the equivalencePr
L
'¹Pr
R
º
op
, we then obtain the functor
SH

: Ind¹Sm
S
º
op
Pr
L
:
More explicitly, if¹X
i
º
i2I
is a ltered diagram of smooth schemes overS andX = colim
i
X
i
as an
ind-scheme overS, then
SH

¹Xº = holim
i
SH

¹X
i
º and SH

¹Xº = hocolim
i
SH

¹X
i
º:
Note thatSH

¹Xº andSH

¹Xº are equivalent1–categories since homotopy limits along left adjoints
inPr
L
correspond to homotopy colimits along their right adjoints inPr
R
, see [Lur09, section 5.5.3].
Because of this fact, if we don’t need to consider the functoriality, we will just writeSH¹Xº for this
1–category. This description ofSH¹Xº also shows that it inherits the structure of a closed symmetric
monoidal, stable, presentable1–category, see [Lur12, section 3.4.3, Proposition 4.8.2.18].
The adjunction f

a f

for a morphism f : X Y of ind-schemes is obtained by presenting f as
a colimit of maps f
i
: X
i
Y
i
between schemes overS and then taking f

to be the functor induced
on the homotopy limits inPr
L
and f

the functor induced on the homotopy colimits inPr
R
. Since
each f

i
is symmetric monoidal, [Lur12, Section 3.4.3] shows that the limit f

is again symmetric
monoidal.
It remains to construct the extra left-adjoint f
#
for a smooth map f between ind-schemes overS.
First, a morphism f : X Y between ind-schemes is smooth if and only if it is a ltered colimit of
smooth maps f
i
: X
i
Y
i
. Each f

i
admits a left adjoint f
i#
and sincePr
R
is stable under limits, the
functor f

: SH

¹Yº SH

¹Xº admits a left adjoint as well. That is,SH

: Ind¹Sm
S
º
op
Pr
L
restricts to a functorSH

: Ind¹Sm
S
º
op
sm
Pr
R
from the wide subcategory of Ind¹Sm
S
º consisting of
smooth maps between smooth ind-schemes overS. Composing with the equivalencePr
L
'¹Pr
R
º
op
then yields the functor
SH
#
: Ind¹Sm
S
º
sm
Pr
L
:
In summary, we have the following proposition.
15
Proposition2.3.2. For every smooth ind-schemeX over S, there is a closed symmetric monoidal, stable,
presentable1–categorySH¹Xº. For every morphism f : X Y between smooth ind-schemes there is
an associated adjunction
f

:SH¹Yº
`
SH¹Xº : f

with f

a monoidal functor. If f is smooth, then there is an additional adjunction
f
#
:SH¹Xº
`
SH¹Yº : f

:
If X happenstobeasmoothschemeover S,thenthisversionof SH¹Xºisnaturallyequivalenttothe
usual construction.
Following [Lev18] and as in Section 2.2, for a smooth morphism f : X Y of smooth ind-schemes
overS we deneXY = f
#
¹1
X
º2SH¹Yº where 1
X
denotes the monoidal unit inSH¹Xº. In particular,
ifY = S, we see that any smooth ind-scheme X over S determines an object XS2SH¹Sº. Note
however that at this point we do not yet have a functor _S. To obtain it we will need the following.
Proposition 2.3.3. Suppose an ind-scheme X is presented as a colimit X = colim
i
X
i
in Ind¹Sm
S
º.
Then there is a natural equivalenceXS' hocolim
i
X
i
S inSH¹Sº.
Proof. Write π: X S and π
i
: X
i
S for the structure morphisms. Suppose Y 2 SH¹Sº is
arbitrary. Then we have natural equivalences
Map
SH¹Sº
¹π
#
1
X
;Yº' Map
SH¹Xº
¹1
X
;π

Yº
' holim
i
Map
SH¹X
i
º
¹1
X
i
;π

i
Yº
' holim
i
Map
SH¹Sº
¹π
i#
1
X
i
;Yº
' Map
SH¹Sº
¹hocolim
i
X
i
S;Yº
of mapping spaces. The Yoneda lemma implies thatXS =π
#
1
X
' hocolim
i
X
i
S inSH¹Sº.
Now, Proposition 2.3.3 allows us to extend the denition of the functor _S: Sm
S
SH¹Sº in
[Lev18] to smooth ind-schemes. The functor _S: Sm
S
SH¹Sº extends uniquely up to natural
equivalence to a functor _S: Ind¹Sm
S
º SH¹Sº becauseSH¹Sº is cocomplete. By Proposition 2.3.3
this coincides on objects with the previous construction XS = π
#
¹1
X
º for a smooth ind-scheme
π: X S.
2.4 Traces and Transfers
Becker and Gottlieb introduced their eponymous transfer maps in [BG75] as a tool for giving a simple
proof of the Adams conjecture. They considered a compact Lie groupG and a ber bundleE B
over a nite CW complex with structure groupG and whose ber F is a closed smooth manifold
16
with a smooth action byG. There is a smoothG–equivariant embedding F V of F into a nite
dimensional representationV ofG. Then there is an associated Pontryagin–Thom collapse map
S
V
F
ν
whereν is the normal bundle of F inV and F
ν
is its Thom space. Denoting byτ the
tangent bundle ofF one obtains a morphism
S
V
F
ν
F
τν
' F
+
^S
V
inG–equivariant homotopy theory. Assuming thatE B is associated with a principalG–bundle
e
E B one gets a map
e
ES
V
e
E¹F
+
^S
V
º
and passing to homotopy orbits with respect to the diagonal G–actions yields the transfer map
B
+
E
+
in the stable homotopy category. This construction of the transfer was generalized in
[DP80, Theorem 6.1]: The mapS
V
F
+
^S
V
arises from a duality datum in parameterized stable
homotopy theory over the base spaceB.
To be precise, rst supposeC is an ordinary symmetric monoidal category with an objectX2 C.
A duality datum for X in C is an object X
_
2 C together with maps e: X
_

X 1
C
and
c: 1
C
X
X
_
such that we have the equations
¹X
id
Xº =¹X' 1
C

X
c
id
X
X
_

X
id
e
X
1
C
'Xº
and
¹X
_ id
X
_
º =¹X
_
'X
_

1
C
id
c
X
_

X
X
_ e
id
1
C

X
_
'X
_
º
inC.
Now, ifC is a symmetric monoidal1–category andX2 C is an object, then a duality datum for
X is an objectX
_
2 C with a morphisme: X
_

X 1
C
such that¹X
_
;eº can be extended to a
duality datum in the homotopy category hC. Lurie proves in [Lur12, Lemma 4.6.1.10] that the space
of duality data for an objectX is either empty or contractible. In this sense, there is no ambiguity in
speaking of the dual of an object if it exists. Ife: X
_

X 1
C
andc: 1
C
X
X
_
are part of
a duality datum forX inC, we will calle theevaluationmap and write ev for it; similarly we will call
c the coevaluation map and write coev. Furthermore, ifX admits a duality datum at all, we callX
dualizable.
In the case of a closed symmetric monoidal1–category there is always a canonical candidate
for the dual of an objectX. For instance, this is the case for the stable motivic homotopy category
SH¹Sº. SinceSH¹Sº is closed symmetric monoidal, we have the mapping objectMap¹X; 1
S
º and the
evaluation map
ev: Map¹X; 1
S
º^X 1
S
:
The latter is the image of the identity map under the adjunction equivalence
Map¹Map¹X; 1
S
º;Map¹X; 1
S
ºº' Map¹Map¹X; 1
S
º^X; 1
S
º:
17
The easiest way to see thatMap¹X; 1
S
º must be equivalent toX
_
ifX is dualizable is to observe that
the denition of a duality datum exhibits the functorX
_
^ _ as left and right adjoint to the functor
X^ _. But in a closed symmetric monoidal1–category the right adjoint ofX^ _ is by denition the
internal mapping space functorMap¹X; _º. By the uniqueness of adjoints this implies that there is a
natural equivalenceMap¹X; _º'X
_
^ _ and in particular an equivalence
Map¹X; 1
S
º'X
_
^ 1
S
'X
_
:
Furthermore, we always have a natural transformation
α
X
: Map¹X; 1
S
º^ _
unit
Map¹X;X^Map¹X; 1
S
º^ _º
ev

Map¹X; _º:
If this transformation is invertible, then this shows thatMap¹X; 1
S
º^ _ is right adjoint toX^ _ and
implies thatMap¹X; 1
S
º is the dual ofX , cf. [Lur12, Lemma 4.6.1.6].
The fundamental result about dualizability inSH¹Sº is the following motivic analogue of Atiyah
duality. This was originally announced by Voevodsky but the rst complete proof was published by
Ayoub. A detailed account of the argument is given in [CD09]. Here, we will only state the result.
Theorem2.4.1 (cf. [Lev18, Proposition 1.2], [Voe01; Rio05; Ayo07b; CD09]). If π: X S isasmooth
andpropermorphismofschemes,thenπ
#
¹1
X
º =XS2SH¹Sºisdualizablewithdualπ

¹1
X
º2SH¹Sº.

In general, dualizability is a strong niteness property and nontrivial to check. Most arguments
are based on the homotopy invariance of dualizability coupled with motivic Atiyah duality. Another
powerful tool is the following locality property.
Theorem2.4.2 (cf. [Lev18, Proposition 1.2]). DualizabilityislocalwithrespecttoNisnevichcoversinthe
followingsense. If A2SH¹Xºandfj
i
: U
i
Xg
i
isaNisnevichcoverof X suchthat j

i
¹Aº2SH¹U
i
º
is dualizable for every i, thenAis dualizable inSH¹Xº.
Proof. Consider the natural transformationα
A
: Map¹A; 1
X
º^ _ Map¹A; _º. Since each j

i
is
monoidal, the image ofα
A
under j

i
is
j

i
α
A
'α
j

i
A
: Map¹j

i
A; 1
U
i
º^ _ Map¹j

i
A; _º:
Because j

i
A is assumed to be dualizable, j

i
α
A
is invertible. Consequently,α
A
is invertible by Proposi-
tion 2.1.8. Hence,A is dualizable.
A useful consequence of this locality property is a criterion for the dualizability of Nisnevich locally
trivial bundles. Suppose f : E B is such a Nisnevich locally trivial bundle over a smooth scheme
B overS. This means that there is a Nisnevich covering familyfj
i
: U
i
Bg
i
and isomorphisms
j

i
E U
i
F withF the ber of f . Furthermore, the pullbackj

i
f of f must under these isomorphisms
coincide with the projection ontoU
i
. Assume thatF is itself a smooth scheme overS. In this situation,
18
Theorem 2.4.2 applies and shows thatEB will be dualizable ifU
i
FU
i
is dualizable for everyi. But
U
i
FU
i
'π

i
¹FSº whereπ
i
: U
i
S denotes the (smooth) structure map ofU
i
. In summary, we
have the following corollary.
Corollary2.4.3 (cf. [Lev18, Theorem 1.10]). Suppose f : E B is a Nisnevich locally trivial bundle
overasmoothscheme B over S. Let F betheberof f andassumethatitisasmoothschemeover S.
If FS2SH¹Sº is dualizable, then so is EB2SH¹Bº.
Finally, we can also characterize dualizability inSH¹Xº for ind-schemesX.
Lemma2.4.4. SupposeX is a smooth ind-scheme over S and E2SH¹Xº. If X is presented as a ltered
colimit X = colim
i
X
i
ofsmoothschemesin Ind¹Sm
S
º,let f
i
: X
i
X bethecanonicalmapforeachi.
ThenE2SH¹Xº is dualizable if and only if f

i
E2SH¹X
i
º is dualizable for every i.
Proof. This follows from [Lur12, Proposition 4.6.1.11] since we haveSH¹Xº' holim
i
SH¹X
i
º along
the monoidal pullback functors f

i
.
SupposeX is a dualizable object in a symmetric monoidal1–categoryC. Then we can dene the
Euler characteristic ofX to be the composition
1
C
coev
X
X
_ switch
X
_

X
ev
1
C
in Map
C
¹1
C
; 1
C
º. In [DP80, Theorem 6.1] Dold and Puppe show that for a ber bundle E B
with ber a compact smooth manifold, there is a duality datum in the homotopy category of B–
parameterized spectra. It exhibits the berwise Thom spectrum of the berwise stable normal bundle
of E as a dual of the suspension spectrum of E. For the case B = this proves the dualizability of
suspension spectra of smooth closed manifolds. Dold and Puppe also show that in this case the
general abstract denition of the Euler characteristic coincides with the usual one, as long as one
identies Map
SH
¹S
0
;S
0
º withZ via the mapping degree. They then also show that the transfer in
[BG75] is an instance of the following general construction.
Definition 2.4.5. In a symmetric monoidal1–category C, suppose that a dualizable objectX is
equipped with a mapΔ: X X
C for some other objectC. The transfer of X with respect toΔ is
dened as the composition
tr
X ;Δ
: 1
C
coev
X
X
_ switch
X
_

X
id
Δ
X
_

X
C
ev
id
1
C

C'C:
We sometimes write tr
X
= tr
X ;Δ
if the mapΔ is clear from the context.
In the setting of stable motivic homotopy theory over ind-schemes, it turns out to be advantageous
to consider further variants of this denition. For ordinary schemes this was done in [Lev18].
19
Definition 2.4.6. For a smooth map f : E B between smooth ind-schemes over S such that
EB2SH¹Bº is dualizable we dene the relative transfer Tr¹fBº: 1
B
EB as follows. Applying
f
#
to the diagonal E E
B
E gives a morphismΔ: EB EB^ EB inSH¹Bº and we set
Tr¹EBº = Tr¹fBº = tr
EB;Δ
.
Additionally, sinceπ: B S is a smooth ind-scheme overS, we can dene the absolute transfer
of f as
Tr¹fSº =π
#
¹Tr¹fBºº: ES BS:
We refer to the relative transfer and the absolute transfer collectively as themotivicBecker–Gottlieb
transfer.
We summarize the properties of the motivic Becker–Gottlieb transfer that will be needed in our
splitting of BGL
n;+
. The proof will occupy the rest of this section.
Theorem2.4.7. The motivic Becker–Gottlieb transfer enjoys the following properties.
(i) Thetransferisadditiveinhomotopypushouts: SupposeX,Y,U andV aresmoothind-schemes
over a smooth ind-scheme B over S. Further suppose that there is a homotopy cocartesian square
XB UB
VB YB
inSH¹Bº. Assume that YB, UB and VB are dualizable. Then Tr¹YBº is a sum of the
compositions
1
B
Tr¹UBº
UB YB
1
B
Tr¹VBº
VB YB
and
1
B
Tr¹XBº
XB YB
inSH¹Bº.
(ii) The relative transfer is compatible with pullback: If p: B
0
B and f : E B are maps
of smooth ind-schemes over S and EB is dualizable inSH¹Bº, then the pullback p

¹EBº'
¹E
B
B
0
ºB
0
is dualizable inSH¹B
0
º and Tr¹p

fB
0
º'p

Tr¹fBº.
(iii) The absolute transfer is natural in cartesian squares: If
E
0
E
B
0
B
f
0
f
20
is a cartesian square of smooth ind-schemes over S and the vertical maps are smooth, then the
square
E
0
S ES
B
0
S BS
Tr¹f
0
Sº Tr¹fSº
commutes inSH¹Sº.
To prove part (i) of Theorem 2.4.7 we appeal to a general additivity result of May’s. In the
context of symmetric monoidal triangulates categories, [May01] proves that the transfer is additive in
distinguished triangles. However, since duality in symmetric monoidal1–categories is characterized
at the level of homotopy categories, May’s theorem admits the following reformulation.
Theorem 2.4.8 ([May01, Theorem 1.9]). Let Cbe a symmetric monoidal stable1–cateogry and
let X Y Z be a cober sequence in C. AssumeC2 Cis such that _
C preserves cober
sequences. Suppose that Y is equipped with a mapΔ
Y
: Y Y
C and that X and Y are dualizable.
ThenZ is dualizable and there are mapsΔ
X
andΔ
Z
such that
X Y Z
X
C Y
C Z
C
Δ
X
Δ
Y
Δ
Z
commutes. Furthermore, we have tr
Y ;Δ
Y
= tr
X ;Δ
X
+tr
Z ;Δ
Z
inπ
0
Map
C
¹1
C
;Cº.
Proof of Theorem 2.4.7, (i). The homotopy cocartesian square induces a cober sequence
UB_VB YB S
1
^XB
inSH¹Bº. Shifting this sequence and introducing diagonal maps gives a diagram
XB UB_VB YB
XB^XB ¹UB^UBº_¹VB^VBº
XB^YB ¹UB_VBº^YB YB^YB
Δ
X
Δ
U
_Δ
V
Δ
Y
¹1º ¹2º
in which the outer two rows are cober sequences and the maps¹1º and¹2º are induced from the
maps XB YB,UB YB andVB YB respectively. Then we can conclude using
Theorem 2.4.8.
21
Part (ii) of Theorem 2.4.7 is proven in [Lev18, Lemma 1.6]. For the convenience of the reader we
reproduce the proof here.
Proof of Theorem 2.4.7, (ii). First note thatp

is a monoidal functor. This immediately implies that
p

¹EBº is dualizable and p

¹tr
EB;Δ
º = tr
p

¹EBº;p

¹Δº
. Here, Δ: EB EB^ EB denotes the
standard diagonal map. For the cartesian square
E
B
B
0
E
B
0
B
g
p

f=q f
p
we have an exchange equivalence
p

¹EBº =p

f
#
¹1
E
º

q
#
g

¹1
E
º'
¹1º
q
#
¹1
E
B
B
0º = E
B
B
0
B
0
;
where¹1º follows fromg

being a monoidal functor. This also implies thatp

¹Δº coincides with the
standard diagonal map forE
B
B
0
B
0
. Consequently, we nd for the relative transfer that
Tr¹p

fB
0
º = tr
E
B
B
0
B
0
;Δ
= tr
p

¹EBº;p

¹Δº
=p

¹tr
EB;Δ
º =p

Tr¹fBº:
We formulate the proof of part (iii) of Theorem 2.4.7 as a lemma.
Lemma2.4.9. Let S be a scheme and B and B
0
smooth ind-schemes over S. Suppose that f : E B is
smooth withEB2SH¹Bº dualizable and
E
0
E
B
0
B
i
0
f
0
f
i
is cartesian. Then the square
E
0
S ES
B
0
S BS
i
0
S
Tr¹f
0
Sº
iS
Tr¹fSº
is commutative.
Proof. Writep: B
0
S andq: B S for the structure morphsims. There is a natural transforma-
tionp
#
i

q
#
dened as the composition
p
#
i
unit
p
#
i

q

q
#
'p
#
p

q
#
counit
q
#
:
22
For a smooth schemeX overB, this natural transformation is the composition
X
B
B
0
S X
S
B
0
S
proj
XS;
that is, it is just the projection onto the rst factor in a ber product. Consequently, we obtain a
commutative diagram
B
0
S =p
#
p

1
S
p
#
f
0
#
i
0
f

q

1
S
= E
0
S
p
#
i

q

1
S
p
#
i

f
#
f

q

1
S
BS =q
#
q

1
S
q
#
f
#
f

q

1
S
= ES:
p
#
Tr¹f
0
B
0
º
Ex

#
p
#
i

Tr¹fBº
q
#
Tr¹fBº
Chasing through the denition ofp
#
i

q
#
shows that the leftmost composite vertical map isiS
and that the rightmost vertical map isi
0
S. More explicitly, the leftmost vertical map is the composite
B
0
S' B
B
B
0
S B
S
B
0
S
proj
BS:
while the rightmost vertical map is the composite
E
0
S' E
B
B
0
S E
S
B
0
S
proj
ES:
23
3 Spliing BGL
n
This chapter is concerned with establishing a motivic analogue of a theorem due to Snaith, [Sna79;
Sna78]. He described splittings of the natural ltrations on BGL
n
¹Cº and BGL
n
¹Hº in the classical
stable homotopy category, using the Becker–Gottlieb transfer associated with the normalizer of a
maximal torus in GL
n
. We will prove the following theorem.
Theorem3.0.1. Over any schemeS, there is aP
1
S
–stable splitting BGL
n;+
'
Ô
n
i=1
BGL
i
BGL
i1
of the
natural ltration BGL
0
: : : BGL
n1
BGL
n
.
Our approach follows the technique of Mitchell and Priddy [MP89] for obtaining Snaith’s results.
Briey, the idea is to use the transfer associated with the block-diagonal subgroup GL
i
GL
ni
GL
n
instead of the one associated with the normalizer of a maximal torus. The key ingredient is then
proving that these transfers are suciently natural in the ltration of BGL
n
and assemble into a
splitting map BGL
n;+
Ô
n
i=1
BGL
i
BGL
i1
. This step makes essential use of the additivity of
Becker–Gottlieb transfers inSH¹BGL
n
º as constructed in Section 2.3.
In classical homotopy theory, Mitchell and Priddy [MP89] accomplish this task and recover Snaith’s
results on BGL
n
¹Cº and BGL
n
¹Hº while improving the ones on BGL
n
¹Rº and BGL
n
¹F
q
º. In fact, the
applicability of this approach to the case of BGL
n
¹Cº was rst noted by Richter.
Overview To obtain a Becker–Gottlieb transfer for the inclusion BGL
i
BGL
ni
BGL
n
we
use an explicit model of BGL
n
as the innite Grassmannian Gr
n
. We give a presentation of the
inclusion BGL
i
BGL
ni
BGL
n
as a smooth Zariski–locally trivial bundleU Gr
n
. The bers
of this bundle areA
1
–equivalent to Gr
i
¹nº, which is smooth and proper, and thereforeU denes a
dualizable object inSH¹Gr
n
º. The image of the monoidal trace ofU inSH¹Sº can be used to dene a
map BGL
n;+
BGL
i
BGL
i1
. Then varyingi and taking the wedge sum of these maps gives the
splitting.
3.1 Transfers of Grassmannians
LetU
r
¹Nº denote the scheme of monomorphismsO
r
O
N
. Along the inclusionO
N
0O
N+1
,
there are closed embeddingsU
r
¹Nº U
r
¹N + 1º and [MV99, Proposition 4.2.3] shows that the
colimitU
r
¹1º = colim
N
U
r
¹Nº along these embeddings is contractible inH¹Sº: theU
r
¹Nº are part
of an "admissible gadget" by [MV99, Example 4.2.2]. The quotientU
r
¹NºGL
r
is the Grassmannian
24
Gr
r
¹Nº ofr–planes in N –space andU
r
¹1ºGL
r
Gr
r
is a model for BGL
r
inH¹Sº. It is a smooth
ind-scheme, namely Gr
r
colim
N
Gr
r
¹Nº. Let
j
r;n
: U
n
¹1º¹GL
r
GL
nr
º U
n
¹1ºGL
n
Gr
r
be the quotient map by GL
n
.
Lemma3.1.1. Themorphism j
r;n
isaZariski–locallytrivialbundleover Gr
n
. Itsberisthequotient
GL
n
¹GL
r
GL
nr
º.
Proof. By construction, the morphism j
r;n
: U
n
¹1º¹GL
r
GL
nr
º U
n
¹1ºGL
n
is isomorphic to
the colimit of the quotient mapsU
n
¹Nº¹GL
r
GL
nr
º U
n
¹NºGL
n
Gr
n
¹Nº. But these are all
Zariski–locally trivial with ber GL
n
¹GL
r
GL
nr
º.
We note that GL
n
¹GL
r
GL
nr
º is equivalent to Gr
r
¹nº inH¹Sº and this equivalence is compatible
with the respective GL
n
actions. This is shown for example in [AHW18, Lemma 3.1.5] and implies
that the image inSH¹Gr
n
º of the associated bundleU
n
¹1º
GL
n
Gr
r
¹nº Gr
n
is equivalent to that
of the quotientU
n
¹1º¹GL
r
GL
nr
º Gr
n
.
Lemma 3.1.2. The morphism j
r;n
: U
n
¹1º¹GL
r
GL
nr
º Gr
n
denes a dualizable object G
r;n
inSH¹Gr
n
º.
Proof. By Lemma 2.4.4 it will be enough to show that the pullbacki: E Gr
n
¹Nº ofj
r;n
along the in-
clusion Gr
n
¹Nº Gr
n
denes a dualizable object inSH¹Gr
n
¹Nºº for allN . But, by Lemma 3.1.1 the
morphismi is a Zariski–locally trivial ber bundle over Gr
n
¹Nº with berX = GL
n
¹GL
r
GL
nr
º.
Hence, to show thati denes a dualizable object inSH¹Gr
n
¹Nºº, by Theorem 2.4.2 it is enough to
show thatXS2SH¹Sº is strongly dualizable.
As mentioned above, [AHW18, Lemma 3.1.5] shows thatX' Gr
r
¹nº inH¹Sº and therefore also in
SH¹Sº. The scheme Gr
r
¹nº is smooth and proper overS, so motivic Atiyah duality, Theorem 2.4.1, im-
plies that Gr
r
¹nºS and therefore alsoXS is dualizable inSH¹Sº; see for example [Lev18, Proposition
1.2].
Definition3.1.3. Direct sum denes a morphismU
r
¹NºU
nr
¹Nº U
n
¹2Nº which is equivariant
with respect to the block diagonal inclusion GL
r
GL
nr
GL
n
. Passing to the colimit N 1
and taking quotients yields a morphism
i
r;n
: Gr
r
Gr
nr
Gr
r
:
The morphismi
r;n
is a model for the map BGL
r
BGL
nr
BGL
n
induced by the block diagonal
inclusion GL
r
GL
nr
GL
n
inH¹Sº. The following lemma shows thati
r;n
admits an absolute
transfer inSH¹Sº.
25
Lemma 3.1.4. InH¹Sº there is an equivalence Gr
r
Gr
nr
U
n
¹1º¹GL
r
GL
nr
º. Along this
equivalence,i
r;n
corresponds to the quotient map
j
r;n
: U
n
¹1º¹GL
r
GL
nr
º U
n
¹1ºGL
n
Gr
r
by GL
n
. Consequently,i
r;n
admits an absolute transfer Tr
n;r
: Gr
n;+
Gr
r;+
^ Gr
nr;+
inSH¹Sº.
Proof. Writeφ: U
r
¹NºU
nr
¹Nº U
n
¹2Nº for the map induced by taking direct sums. We obtain
a commutative diagram
U
r
¹NºU
nr
¹Nº U
n
¹2Nº
Gr
r
¹Nº Gr
nr
¹Nº U
n
¹2Nº¹GL
r
GL
nr
º
Gr
n
¹2Nº:
φ
i
r ;n
Passing to the colimitN 1, the horizontal maps become equivalences.
Lemma3.1.5. Supposer < n. Theopencomplementoftheclosedimmersion Gr
r
¹n1º Gr
r
¹nºis
the total space of an ane space bundle of ranknr over Gr
r1
¹n1º.
Dually,thecomplementoftheclosedimmersion Gr
r1
¹n1º Gr
r
¹nºisthetotalspaceofanane
space bundle of rankr over Gr
r
¹n1º.
Proof. Suppose Spec¹Aº is an ane scheme mapping toS. On Spec¹Aº–valued points, the inclusion
Gr
r
¹n1º Gr
r
¹nº is given by considering a projective submoduleP ofA
n1
as a submodule of
A
n
=A
n1
A. It follows that the complementU of Gr
r
¹n1º has Spec¹Aº–valued points
U¹Spec¹Aºº =fP A
n
: P is projective of rankr andP 1A
n1
0g:
GivevnP2U¹Spec¹Aºº, the moduleP\¹A
n1
0º will be locally free of rankr1. This gives a map
φ: U Gr
r1
¹n1º which is trivial over the standard Zariski–open cover of Gr
r1
¹n1º with
berA
nr
.
The dual statement is proved similarly. In fact, the bundleV Gr
r
¹n 1º in question is the
tautologicalr–plane bundle on Gr
r
¹n1º.
The decomposition Gr
r
¹nº =U[V of the last lemma yields a homotopy cocartesian square
U r Gr
r1
¹n1º =U\V V' Gr
r
¹n1º
Gr
r1
¹n1º'U Gr
r
¹nº
26
in theA
1
–homotopy categoryH¹Sº. This decomposition is stable under the action of GL
n1
1 GL
n
.
Therefore, we can pass to the bundles over Gr
n1
associated to the universal GL
n1
–torsorU
n1
¹1º
over Gr
n1
. Then, the image of the resulting diagram is a homotopy cocartesian square
¹U
n1
¹1º
GL
n1
¹U\VººGr
n1
G
r;n1
G
r1;n1
¹U
n1
¹1º
GL
n1
Gr
r
¹nººGr
n1
inSH¹Gr
n1
º.
Proposition3.1.6. Supposer < n and consider the composition
φ: Gr
n1;+
incl
Gr
n;+
Tr
n;r
Gr
r;+
^ Gr
nr;+
where incl is given by the assignment P PA on Spec¹Aº–valued points. Then there is a map
ψ: Gr
n1;+
Gr
r1;+
^ Gr
nr;+
inSH¹Sº such thatφ is the sum of the compositions
Gr
n1;+
Tr
n1;r
Gr
r;+
^ Gr
n1r;+
id^ incl
Gr
r;+
^ Gr
nr;+
Gr
n1;+
Tr
n1;r1
Gr
r1;+
^ Gr
nr;+
incl^ id
Gr
r;+
^ Gr
nr;+
and
Gr
n1;+
ψ
Gr
r1;+
^ Gr
nr;+
incl^ id
Gr
r;+
^ Gr
nr;+
:
Proof. Consider the homotopy pullback
E =U
n1
¹1º
GL
n1
Gr
r
¹nº Gr
r
Gr
nr
Gr
n1
Gr
n
incl
inH¹Sº. By the discussion following Lemma 3.1.5 we obtain a cober sequence
XGr
n1
G
r;n1
_G
r1;n1
EGr
n1
inSH¹Gr
n1
º whereX =U
n1
¹1º
GL
n1
¹U\Vº. Theorem 2.4.7 then shows that
tr
EGr
n1
= tr
G
r ;n1
+tr
G
r1;n1
tr
XGr
n1
inSH¹Gr
n1
º. Passing to the absolute transfer and using Lemma 2.4.9 yields thatφ is the sum of the
compositions
Gr
n1;+
Tr
n1;r
Gr
r;+
^ Gr
n1r;+
id^ incl
Gr
r;+
^ Gr
nr;+
Gr
n1;+
Tr
n1;r1
Gr
r1;+
^ Gr
nr;+
incl^ id
Gr
r;+
^ Gr
nr;+
27
and
Gr
n1;+
X
+
Gr
r;+
^ Gr
nr;+
inSH¹Gr
n1
º. The mapX
+
Gr
r;+
^ Gr
nr;+
is obtained from the inclusionU\V Gr
r
¹nº by
passing to associated bundles. Now, this inclusion factors through the inclusion ofU into Gr
r
¹nº.
By Lemma 3.1.5 the inclusion Gr
r1
¹n 1º U is anA
1
–equivalence, being the zero section of
an ane space bundle. Therefore the morphismX
+
Gr
r;+
^ Gr
nr;+
factors through the map
incl^ id: Gr
r1;+
^ Gr
nr;+
Gr
r;+
^ Gr
nr;+
as
X
+
Gr
r;+
^ Gr
nr;+
Gr
r1;+
^ Gr
nr;+
ψ
incl^ id
This way we obtain the mapψ and the required decomposition of Tr
n;r
incl.
3.2 The Spliing
We have seen in the last section that the natural ltrarion BGL
0
: : : BGL
m1
BGL
m
can be
geometrically modeled by the inclusionsi
n
: Gr
n
Gr
n+1
between innite Grassmannians. We
will construct splitting maps BGL
n;+
BGL
r
BGL
r1
and prove that they are compatible when
changingn. This way we will obtain the the natural splitting of Theorem 3.0.1. The last section
constructs transfers Tr
n;r
for the inclusion mapsi
r;n
: Gr
r
Gr
nr
Gr
n
in the motivic stable
homotopy categorySH¹Sº. Write f
n;r
for the composition
Gr
n;+
Tr
n;r
Gr
r;+
^ Gr
nr;+
proj
Gr
r;+
andϕ
n;r
for the composition
Gr
n;+
f
n;r
Gr
r;+
Gr
r
Gr
r1
:
The mapsϕ
n;r
will turn out to assemble into our splitting maps.
Lemma3.2.1. With notation as above, for r < n the compositions
Gr
n1;+
i
n
Gr
n;+
f
n;r
Gr
r;+
Gr
r
Gr
r1
and
Gr
n1;+
f
n1;r
Gr
r;+
Gr
r
Gr
r1
coincide inSH¹Sº.
28
Proof. By Proposition 3.1.6 f
n;r
i
n
is the of two compositions
Gr
n1;+
Tr
n1;r
Gr
r;+
^ Gr
n1r;+
id^ incl
Gr
r;+
^ Gr
nr;+
proj
Gr
r;+
and
Gr
n1;+
Gr
r1;+
^ Gr
nr;+
incl^ id
Gr
r;+
^ Gr
nr;+
proj
Gr
r;+
inSH¹Sº. But the composition
Gr
r1;+
incl
Gr
r;+
Gr
r
Gr
r1
vanishes by denition. Therefore,ϕ
n;r
i
n
is homotopic toϕ
n1;r
inSH¹Sº as required.
Lemma 3.2.1 is enough to prove that the mapsϕ
n;r
provide the splitting of Theorem 3.0.1:
Proof of Theorem 3.0.1. Proceeding by induction onn, assume that
Φ =
n1
Ü
r=1
ϕ
n1;r
: Gr
n1;+
n1
Ü
r=1
Gr
r
Gr
r1
is an equivalence inSH¹Sº. Because of Lemma 3.2.1 we have a commutative diagram
Gr
n;+
Gr
n1;+
n1
Ü
r=1
Gr
r
Gr
r1
Φ
0
Φ
i
n
whereΦ
0
=
Ô
n1
r=1
ϕ
n;r
. It follows thatΦ
1
Φ
0
i
n
' id, that is,i
n
admits a left inverse. This means
that the cober sequence
Gr
n1;+
i
n
Gr
n;+
Gr
n
Gr
n1
splits and yields an equivalence
Gr
n;+
¹Φ
1
Φ
0
º_ϕ
n;n
Gr
n1;+
_ Gr
n
Gr
n1
sinceϕ
n;n
is dened to be the canonical projection. Post-composing withΦ_ id then shows that the
stable mapΦ
0
_ϕ
n;n
: Gr
n;+
Ô
n
r=1
Gr
r
Gr
r1
is an eqivalence inSH¹Sº as well.
29
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Asset Metadata
Creator Kleen, Viktor (author) 
Core Title Some stable splittings in motivic homotopy theory 
Contributor Electronically uploaded by the author (provenance) 
School College of Letters, Arts and Sciences 
Degree Doctor of Philosophy 
Degree Program Mathematics 
Publication Date 07/24/2019 
Defense Date 06/14/2019 
Publisher University of Southern California (original), University of Southern California. Libraries (digital) 
Tag algebraic geometry,motivic homotopy theory,OAI-PMH Harvest 
Format application/pdf (imt) 
Language English
Advisor Asok, Aravind (committee chair), Ganatra, Sheel (committee member), Pilch, Krzysztof (committee member) 
Creator Email kleen@usc.edu,vkleen-usc@17220103.de 
Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-186974 
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Legacy Identifier etd-KleenVikto-7594-0.pdf 
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Document Type Dissertation 
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Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law.  Electronic access is being provided by the USC Libraries in agreement with the a... 
Repository Name University of Southern California Digital Library
Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Abstract (if available)
Abstract The filtration BGL₀ ⊂ ... ⊂ BGLₙ₋₁ ⊂ BGLₙ is split by motivic Becker-Gottlieb transfers in the motivic stable homotopy category over any scheme. ❧ This recovers results by Snaith on the splitting of BGLₙ(ℂ) in classical stable homotopy theory by passing to complex realizations. ❧ On the way, we extend motivic homotopy theory to smooth ind-schemes as bases and show how to construct the necessary fragment of the six operations and duality for this extension. 
Tags
algebraic geometry
motivic homotopy theory
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University of Southern California Dissertations and Theses
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University of Southern California Dissertations and Theses 
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