Semi-topologization in motivic homotopy theorykiem/Yeosu/ParkJH.pdf · 2013-02-21 · Basics on motivic homotopy theory Descent theorems Semi-topologization Semi-topologization in

Basics on motivic homotopy theoryDescent theorems

Semi-topologization

Semi-topologization inmotivic homotopy theory

: a report of recent works, joint with A. Krishna (Tata)

Jinhyun Park

Department of Mathematical SciencesKAIST, Daejeon, Korea

February 21, 2013 at Yeosu Symposium

Jinhyun Park Semi-topologization in motivic homotopy theory


Semi-topologization

Contents

I. Basics on motivic homotopy theoryII. Descent theoremsIII. Semi-topologization



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Preliminary picturesModel structuresMotivic spacesMotivic homotopy categories

I. Basics on motivic homotopy theory

1. Preliminary pictures2. Model structures3. Motivic spaces4. Motivic homotopy categories



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motive

algebraicgeometry

algebraictopology

number theory



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Algebraic Topology vs. Algebraic Geometry

algebraic topology // algebraic geometry

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tt...

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Motivic homotopy theory tries to do homotopy theory onvarieties. How? Use “model structures” for instance.

Although not said, model structure is implicitly used by variousalgebraic geometers.

Definition (Quillen’s homotopical algebra)A model structure on a category C allows one to do homotopytheory.This has three classes of morphisms:

W: weak-equivalencesF : fibrationsC: cofibrations

subject to a list of axioms. Its homotopy category is,Ho(C) = C[W−1].



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Examples of model categories:

Example (1)Top: topological spaces

W: weak homotopy equivalences: π∗(X )'→ π∗(Y )

F : (Serre) fibrationsC: monomorphisms

⇒: Ho(Top) = H = the homotopy category (of spaces)



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Example (2)

A: commutative ring with 1, Kom(A): complexes of A-modules

W: quasi-isomorphisms : H∗(X )'→ H∗(Y )

F : “surjective” morphismsC: “injective” morphisms

⇒: Ho(Kom(A)) = D(A): the derived category of A.



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Example (2)







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Benefits of using model categories:

Allows a systematic study of Ho(C) using objects of C.Unifies various (seemingly unrelated) topics.Given a homotopy category H, we can possibly considerseveral different modelsM such that Ho(M) = H.(Some models are better suited for certain jobs than othermodels.)



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∆ = category with objects [n] = 0,1, · · · ,n, n ≥ 0,morphisms= nondecreasing functions.

DefinitionA simplicial set is a functor ∆op → (Set). Let SSet be thecategory of simplicial sets.

This is equivalent to having a collection of sets K = Knn≥0with “faces” and “degeneracies” subject to some relations.

Example (3)SSet withW: weak-equivalences of simplicial sets (definedusing “Kan complexes”)



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TheoremHo(SSet) ' Ho(Top) given by | · |: geometric realization andSing∗: singular simplicial set.

: up to weak-equivalence, all topological spaces are obtained ina combinatorial way.

A simplicial set is an algebraist’s way of constructing atopological space.



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Q) Can we do homotopy theory for (smooth) algebraicvarieties? For instance, we want to contract X × A1 to X , etc.

Naive attempt: take C = Smk . This is inadequate. For instance,if X ∈ Smk contains Y ' A1, then Y is contractible, butX/Y 6∈ Smk . In Top, this makes sense with the quotienttopology.

Correct idea: use Yoneda lemma:

C → Funct(Cop,Set) → Funct(Cop,SSet)

Funct(Cop,Set) = Presh(C): presheavesFunct(Cop,SSet) = SPresh(C): simplicial presheaves



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DefinitionA motivic space over k is a simplicial presheaf on Smk . LetSpc(k) be the category of motivic spaces.

: this category of closed under all small limits and colimits. Inparticular, products, coproducts, quotients, etc. are all possible.Similarly, for Spt : the category of spectra (a sequence ofpointed simplicial sets (E0,E1, · · · ) with bonding mapsS1 ∧ En → En+1), we consider presheaves with values in Spt :

DefinitionA motivic spectrum over k is a presheaf on Smk with values inSpt . Let Spt(k) be the category of motivic spectra.



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For a Grothendieck topology τ on Smk , a point is a functorx∗ : Sh(Smk )τ → (SSet) that commutes with all finite limits andsmall colimits. We say a morphism f : F → G in Spc(k) (resp.Spt(k)) is a τ -local w.e. (weak-equivalence) if each Fx → Gx isa weak-equivalence of simplicial sets (spectra), where stalksare taken after τ -sheafification.

Theorem ((1) Jardine, Morel, Voevodsky)

Then, Spc(k) (resp. Spt(k)) has a model structure withW: τ -local weak-equivalencesC: monomorphismsF : morphisms satisfying the “right lifting property" withrespect to all “trivial cofibrations”.

:τ -local injective model structure



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A Nisnevich cover is an étale cover together with residue-fieldpreserving point-wse sections. Let H be the homotopycategory with Nisnevich local model structure. An object F isA1-local if HomH(E ,F ) ' HomH(E × A1,F ). A morphismf : E ′ → E is a motivic weak-equivalence if it induces a bijectionHomH(E ,F ) ' HomH(E ′,F ) for all A1-local F .

Theorem ((2) Morel, Voevodsky)

Spc(k) (resp. Spt(k)) has a model structure withW: motivic weak-equivalencesC: monomorphismsF : morphisms satisfying the ”right lifting property” withrespec to all ”trivial cofibrations”.

The homotopy categories are: H(k) and SHS1(k).Jinhyun Park Semi-topologization in motivic homotopy theory


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Remarks:

One great property of Nisnevich topology: when X is asmooth variety of pure dimension n and p ∈ X , we havethe following “Thom isomorphism” in motivic homotopycategories:

X/(X − p) ' An/(An − p).

Here, we also have

P1 ' S1 ∧Gm ' A1/(A1 − 0).

There is also the stable motivic homotopy category SH(k)obtained from various models, e.g. from T -spectra,(s, t)-bispectra, (s,p)-bispectra, etc.



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Remarks:


X/(X − p) ' An/(An − p).

Here, we also have

P1 ' S1 ∧Gm ' A1/(A1 − 0).




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Remarks:


X/(X − p) ' An/(An − p).

Here, we also have

P1 ' S1 ∧Gm ' A1/(A1 − 0).




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Remarks:


X/(X − p) ' An/(An − p).

Here, we also have

P1 ' S1 ∧Gm ' A1/(A1 − 0).




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Remarks:


X/(X − p) ' An/(An − p).

Here, we also have

P1 ' S1 ∧Gm ' A1/(A1 − 0).




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(co)fibrant replacementsObject-wise weak-equivalences and quasi-fibrancyNisnevich descent theoremsMotivic descent theorems

II. Descent theorems

1. (co)fibrant replacments2. object-wise weak-equivalences and quasi-fibrancy3. Nisnevich descent theorems4. motivic descent theorems



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Homological algebra replaces an object by its injective /projective resolutions. Model categories have fibrant / cofibrantreplacements. For a model category, let ∗= initial and terminalobject. An object A is called fibrant if A→ ∗ is a fibration andcofibrant if ∗ → A is a cofibration.LemmaLet A be an object of a model category. Then, there areweak-equivalences

A→ B, with B: fibrant (fibrant replacement)C → A, with A: cofibrant (cofibrant replacement)

Example

In Kom(A), an injective resolution = fibrant replacement, aprojective resolution = cofibrant replacement.



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DefinitionLet f : E → F be a morphism of motivic spaces. We say f is anobject-wise weak-equivalence if f (U) : E(U)→ F (U) is aweak-equivalence for each U ∈ Smk .

Local weak-equivalences and motivic weak-equivalences aredifficult to handle, but we have:

object-wise w.e.⇒1 Nisnevich local w.e.⇒2 motivic w.e.

But, the reverse arrows are NOT true.Q) When is it true? Furthermore, when is a Nisnevich / motivicfibrant replacement an object-wise w.e.?



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Nisnevich descent theorem shows the converse for⇒1:

Theorem (Nisnevich, Morel-Voevodsky)Let E ,F be motivic spaces (spectra).

E is B.G. if and only if any Nisnevich fibrant replacement isan object-wise w.e.Let E ,F be B.G. Then, f : E → F is a Nisnevich local w.e.if and only if f is an object-wise w.e.

B.G. stands for Brown-Gersten, but it means Nisnevichdescent: E maps any elementary Nisnevich square to ahomotopy Cartesian square. (homotopical analogue ofMayer-Vietoris for Nisnevich topology)



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For motivic weak-equivalences, we have:

TheoremLet E ,F be motivic spaces (spectra).

E is A1-B.G. if and only if any motivic fibrant replacementis an object-wise w.e.Let E ,F be A1-B.G. Then, f : E → F is a motivic w.e. ifand only if it is an object-wise w.e.

Here A1-B.G. means: (1) B.G. + (2) E(X )→ E(X ×A1) is a w.e.

We also have a stable version for “(s,p)-bispectra”, that we willnot say much in detail. We just say that “A1-B.G. motivicΩ-bispectra” play the corresponding roles.



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Simply put, our motivic descent theorems show:

motivic homotopy ∼ homological algebra of (pre)shavesmotivic fibrant objects ∼ injective objects

A1-B.G. objects ∼ flasque objects

On the other hand, in topology, fibrant objects are infinite loopspaces.In stable motivic homotopy category, flasque objectscorrespond to “A1-B.G. motivic Ω-bispectra.”



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Brief descriptionHomotopy semi-topologizationRepresentabilitySemi-topological cobordismSemi-topological Hopkins-Morel spectral sequence

III. Semi-topologization

1. Brief description2. Homotopy semi-topologization3. Representability4. Semi-topological cobordism5. Semi-topological Hopkins-Morel spectral sequence



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Friedlander-Walker’s (singular) semi-topologization. What is itfor?

One way to see it: on algebraic cycles there are variousadequate equivalences

(rat. eq)⇒ (alg. eq)⇒ (homolog. eq)⇒† (numeric. eq)

-The standard conjecture + Voevodsky’s smash nilpotenceconjecture predict that⇒† is equality. So, there should betruly 3 essentially different ones.-(rat. eq) ∼ algebraic K -theory, motivic cohomology, etc.-(homolog. eq) ∼ topological K -theory, singularcohomology, etc.-(alg. eq) ∼ should be something semi-topological.



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Its definition is a bit involved: for a simplicial / spectralpresheaf E on SchC, we let

E(∆ntop × U) = colimV E(V × U)

where the colimit is taken over the category(∆n

top ↓ VarC)op.We let Esst = |E(∆•top × U)|: the geometric realization ofthe simplicial object.There is a natural functor E → Esst .At this moment, we don’t have a purely algebraic way toconstruct it.



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Example(1) Semi-topological K -theory of Friedlander-Walker.For the algebraic K -spectrum K, we consider

πn(Ksst (X )) =: K sstn (X ).

(2) Morphic cohomology LqHp(X ) of Friedlander-B. Lawson.Originally in terms of algebraic “cocycles”, but can also bedefined using semi-topologization of the “motivicEilenberg-Maclane spectrum”.



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Example(1) Semi-topological K -theory of Friedlander-Walker.For the algebraic K -spectrum K, we consider

πn(Ksst (X )) =: K sstn (X ).

(2) Morphic cohomology LqHp(X ) of Friedlander-B. Lawson.Originally in terms of algebraic “cocycles”, but can also bedefined using semi-topologization of the “motivicEilenberg-Maclane spectrum”.



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Q) Does the semi-topologization process respect motivicweak-equivalences?A) No, in general. So, sst does not define a functor on themotivic homotopy categories.

Q) Is there a derived functor of the semi-topologization onmotivic homotopy categories?The answer is yes, by combining the motivic descent theoremswith the “recognition principle” of Friedlander and Walker thatshows, semi-topologization preserves object-wise w.e.s



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Theorem (Krishna-P.)

The category of A1-B.G. Ω-bispectra on SmC is stableunder semi-topologization.Given a motivic w.e. f between A1-B.G. Ω-bispectra, f sst isalso a motivic w.e.sst : Spt(C)→ Spt(C) has its derived triangulatedidempotent endo-functor

host : SHS1(C)→ SHS1(C)

with a universal property.Similarly on the category of (s,p)-bispectra and the stablemotivic homotopy category SH(C).



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Homotopy semi-topologization is convenient because itrespects motivic weak-equivalences. Using it, we can prove:

Corollary (Krishna-P.)

(1) The semi-topological K -theory is representable in SH(C).(2) The morphic cohomology theory is representable in SH(C).



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The homotopy semi-topologization allows us to define thefollowing:

DefinitionLet MGL ∈ SH(C) be the motivic Thom spectrum (algebraiccobordism spectrum) of Voevodsky. Define MGLsst to be thehomotopy semi-topologization of MGL. The object MGLsst itsassociated bi-graded cohomology theory is called thesemi-topological cobordism.



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Since homotopy semi-topologization is a triangulated functor,we can apply it to distinguished triangles in SH(C). So, it is notso hard to deduce the following semi-topological analogue ofHopkins-Morel spectral sequence:

TheoremLet L be the “Lazard ring.” Then, for X ∈ SmC, there is aspectral sequence

Ep,q2 (X ) = Ln−qHp−q(X )⊗Z Lq ⇒ MGLp+q,n

sst (X ).



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sst (X ).



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sst (X ).



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Combining the above spectral sequence with (1) Hopkins-Morelspectral sequence and (2) equality of the motivic cohomologyand morphic cohomology with finite coefficients, we deduce:

TheoremVoevodsky algebraic cobordism and the semi-topologicalcobordism give identical cohomology theories with finitecoefficients.



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Combining the above spectral sequence with (1) Hopkins-Morelspectral sequence and (2) equality of the motivic cohomologyand morphic cohomology with finite coefficients, we deduce:

TheoremVoevodsky algebraic cobordism and the semi-topologicalcobordism give identical cohomology theories with finitecoefficients.



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Final remark: The speaker recently (to appear in J. K -theory)constructed and studied a theory of algebraic cobordism Ω∗algmodulo algebraic equivalence, modelled on Levine-Morel andLevine-Pandharipande.e.g.


For smooth projective X over C, Ω∗alg(X ) is a f.g. L-module ifand only if Griffiths group Griff ∗(X ) is a f.g. abelian group.

We probably have an isomorphism

Ω∗alg(X ) ' MGL2∗,∗sst (X )

for X ∈ SmC.



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for X ∈ SmC.



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for X ∈ SmC.



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The talk is based on a recent on arxiv, uploaded a few daysago.

Thank you!


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Semi-topologization in motivic homotopy theorykiem/Yeosu/ParkJH.pdf · 2013-02-21 · Basics on motivic homotopy theory Descent theorems Semi-topologization Semi-topologization in