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Cohomology theories in algebraic geometry Cohomology theories in motivic stable homotopy theory Andreas Holmstrom Universitetet i Oslo 12 Sep 2012 Andreas Holmstrom Cohomology theories in algebraic geometry

Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

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Page 1: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometry

Cohomology theories in motivic stable homotopytheory

Andreas Holmstrom

Universitetet i Oslo

12 Sep 2012

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 2: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometry

Outline

1 Introduction and motivation

2 The motivic stable homotopy category

3 Representability theorems and axiom systems

4 Properties of cohomology theories

5 Comparison theorems

6 Final remarks on arithmetic geometry

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 3: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic topology

Question:What is a cohomology theory?

Answer:A cohomology theory is a functor from spaces into graded abeliangroups, which is representable by an object in the stable homotopycategory SH.

Spaces Σ∞−→ SH Hom(−,ΣnE)−→ Ab∗

Equivalently, it is a sequence of functors satisfying the(generalized) Eilenberg-Steenrod axioms.

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 4: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic topology

Examples:Singular cohomology, various versions of K-theory, complexcobordism, BP-theory, Morava K-theories and E-theories, tmf, . . .

Topologists have studied the structure of SH for a long time, andhave developed many tools for working with cohomology theories.

Properties of a cohomology theory are a reflection of properties ofthe representing spectrum. Basic examples:

Multiplicative cohomology theories correspond to ring spectraOriented cohomology theories correspond to MU-algebras

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 5: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic geometry

Question:What is a cohomology theory?

Answer:?????

There is no framework in which all cohomology theories fit.There are no textbooks or surveys titled ”Cohomology inalgebraic geometry” - the subject is to big!Problem, not for the experts in algebraic geometry, but forstudents.Lack of framework makes it harder to make progress oncohomology in settings more general than algebraic varieties.

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 6: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic geometry

Many examples of cohomology1-motives Absolute cohomologyAbsolute cohomology of stacks Absolute Hodge cohomologyAdditive Chow groups Additive higher Chow groupsAdditive K-theory Algebraic cobordismAlgebraic cycle groups Algebraic de Rham cohomologyAlgebraic elliptic cohomology Algebraic G-theoryAlgebraic K-homology Algebraic K-theoryAlgebraic K-theory with compact supports Algebraic L-theoryAlgebraic Morava K-theory Amitsur cohomologyAnalytic cyclic cohomology Arakelov Chow groupsArakelov motivic cohomology Archimedean cohomologyArithmetic Chow groups Arithmetic cohomologyArithmetic homology Arithmetic K-theoryArtin motives Betti cohomologyBi-relative algebraic K-theory Bi-relative cyclic homologyBi-relative topological cyclic homology Big de Rham-Witt cohomologyBirelative homology Bivariant cycle cohomology

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 7: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic geometry

Many examples of cohomologyBivariant cycle homology Bivariant K-theoryBloch-Ogus cohomology Borel-Moore homologyBorel-Moore motivic homology Boundary cohomologyBounded K-theory Bousfield’s united K-theoryBredon style cohomology of stacks cdh-cohomologyCech cohomology Chow groupsChow groups of stacks Chow groups with coefficientsChow motives Chow-Witt groupsCisinski-DŐglise motives Classical motivesCM motives Cohomology of coherent sheavesCohomology with compact supports Compact K-theoryCompleted cohomology Continuous cohomologyContinuous etale cohomology Continuous hypercohomologyCristalline Deligne cohomology Crystalline cohomologyCyclic homology de Rham cohomologyde Rham-Witt cohomology Deligne cohomologyDeligne homology Deligne-Beilinson cohomology

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 8: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic geometry

Many examples of cohomologyDeninger cohomology Derivator K-theoryDwork cohomology Dynamical cohomologyEichler cohomology Eisenstein cohomologyElliptic Bloch groups Equivariant algebraic K-theoryEquivariant Cech cohomology Equivariant etale cohomologyEquivariant higher Chow groups Equivariant operational Chow groupsEquivariant pretheory Equivariant smooth Deligne cohomologyEtale BP2 Etale cobordismEtale cohomology Etale cohomology of rigid analytic spacesEtale cohomology with compact support Etale homologyEtale K-theory Etale K-theory of ring spectraEtale Morava K-theory Etale motivic cohomologyf-cohomology Faltings cohomologyFinite polynomial cohomology Finite-dimensional motivesFlat cohomology Flat homologyFontaine-Messing cohomology Formal cohomology

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 9: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic geometry

Many examples of cohomologyFriedlander-Suslin cohomology Frobenius cohomologyG-theory G-theory of algebraic stacksGeneralized de Rham cohomology Generalized etale cohomologyGeneralized sheaf cohomology Generic cohomologyGeometric cohomology Gillet cohomologyGorenstein cohomology Grothendieck motivesGrothendieck-Witt groups Hermitian K-theoryHermitian-holomorphic Deligne cohomology Higher Arakelov Chow groupsHigher arithmetic Chow groups Higher arithmetic K-theoryHigher Chow groups Hochschild homologyHodge-Witt cohomology Holomorphic K-theoryHomology of schemes Homotopy K-theoryHyodo-Kato cohomology HypercohomologyInfinitesimal cohomology Infinitesimal K-theoryIntersection Chow groups Intersection cohomologyK’-theory K-theory of stacks

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 10: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic geometry

Many examples of cohomologyK-theory with coefficients K-theory with supportsKaroubi L-theory Karoubi-Villamayor K-theoryKato homology l-adic algebraic K-theoryl-adic cohomology l-adic parabolic cohomologyLaumon 1-motives Lawson homologyLocal cohomology Locally analytic cohomologyLog Betti cohomology Log convergent cohomologyLog crystalline cohomology Log de Rham cohomologyLog etale cohomology Log Hodge groupsLogarithmic cohomology Logarithmic Hodge-Witt cohomologyMilnor K-theory Mixed motivesMixed Tate motives Mixed Weil cohomologyModified syntomic cohomology Monsky-Washnitzer cohomologyMorphic cohomology MotivesMotives for absolute Hodge cycles Motivic cobordismMotivic cohomology Motivic cohomology of stacks

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 11: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic geometry

Many examples of cohomologyMotivic cohomology with compact supports Motivic homologyNegative cyclic homology Negative K-theoryNisnevich cohomology Nisnevich K-theoryNon-standard Őtale cohomology Nonabelian cohomologyNoncommutative motives Operational Chow groupsOrientable cohomology Oriented Borel-Moore homologyOriented Chow groups Oriented cohomologyOriented homology p-adic Chow motivesp-adic cobordism p-adic cohomologyp-adic etale cohomology p-adic K-theoryParabolic cohomology Periodic cyclic homologyPositive cohomology PretheoryPrimitive cohomology Pure motivesQuillen K-theory Relative algebraic K-theoryRelative Bloch groups Relative K-theoryRelative rigid cohomology Rigid cohomologyRigid syntomic cohomology Rost’s cycle modules

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 12: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic geometry

Many examples of cohomologySchmidt homology Semi-topological K-homologySemi-topological K-theory Sharp cohomologySharp motives Sheaf cohomologySmooth cohomology Smooth Deligne cohomologyStable cohomology Stack cohomologySuslin homology Syntomic cohomologyt-motives Thomason-Trobaugh K-theoryTopological cycle cohomology Topological cyclic homologyTopological Hochschild homology Twisted cohomologyTwisted duality theory Unramified cohomologyVoevodsky motives Volodin K-theoryWaldhausen K-theory Weak Hodge cohomologyWeil cohomology Weil-etale cohomologyWeil-etale motivic cohomology Witt cohomologyWitt cohomology with supports Witt groupsWitt vector cohomology Zariski cohomology

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 13: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic geometry

Properties of cohomology theoriesPick your favourite cohomology theory. It probably satisfies a longlist of formal properties which makes it useful for applications.

Contravariant functoriality Chern classesCovariant functoriality Cycle classesPoincare duality Product structureMayer-Vietoris Descent for various topologiesProjective bundle formula Blow-up formulaKunneth formula Finiteness propertiesVanishing properties Versions of rigidityVersions of purity Localization sequencesBase change theorems + + + + +

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 14: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic geometry

Questions:

How can we understand which formal properties to expectfrom a randomly picked theory?Can we characterize the theories satisfying such-and-suchproperties?For example, which of the above theories satisfy Poincareduality?Which of the theories have pushforwards? (i.e. covariantfunctoriality for proper morphisms)

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 15: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic geometry

Axiom systems

Weil cohomology theoryBloch-Ogus theoryOriented theoryGeometric theoryPretheory. . .

Question:How can we understand these axiom systems in a unified way?

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 16: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryIntroduction and motivation

Algebraic geometry

Question:What is a cohomology theory?

Answer:?????

Many possible answers, none of them completely satisfactory.

Today: Will present the best answer given so far, namely motivicstable homotopy theory(Morel, Voevodsky, Ayoub, Cisinski, Deglise, . . . )

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 17: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryThe motivic stable homotopy category

The category SH(S)

For any finite-dimensional scheme S, there is an associated stablehomotopy category SH(S).

Sketch of construction:Take the category of smooth S-schemes, with the NisnevichtopologyLet

Spaces(S) = ∆opSh•(Sm/S)

be the category of pointed simplicial sheaves on this categoryLet T ∈ Spaces(S) be a ”motivic sphere” (e.g. A1/Gm)Let Spectra(S) be the category of symmetric T -spectraSH(S) is the homotopy category of Spectra(S)

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 18: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryThe motivic stable homotopy category

Six functors formalism

Key difference between algebraic geometry and topology!

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 19: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryThe motivic stable homotopy category

Six functors formalism

For any scheme X , the triangulated category SH(X ) is closedsymmetric monoidal (has tensor product and internal Hom).For any morphism f : Y → X , there is a pair of adjointfunctors

f ∗ : SH(X ) � SH(Y ) : f∗and f ∗ is a monoidal functor.For any separated morphism of finite type f : Y → X , there isa pair of adjoint functors

f! : SH(Y ) � SH(X ) : f !

In addition to the shift functor [n], there is another invertibleoperator (m), called ”twist”. (Here m, n ∈ Z)These “six functors” satisfy a long list of properties.

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 20: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryThe motivic stable homotopy category

Cohomology theory represented by a spectrum

For E in SH(S), and X a finite type S-scheme, we defineE -cohomology of X by:

En(X , p) = HomSH(S)(1, f∗f ∗E (p)[n])

We can also define other theories:Homology: En(X , p) = Hom(1, f!f !E (−p)[−n])

Cohomology with compact support:

Enc (X , p) = Hom(1, f!f ∗E (p)[n])

Borel-Moore homology:

EBMn (X , p) = Hom(1, f∗f !E (−p)[−n])

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 21: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryThe motivic stable homotopy category

Algebraic geometry

Main points of this talk:1 Cohomology theories for S-schemes should be representable

by objects in SH(S).2 Formal properties of a cohomology theory should be governed

byThe six functors formalismThe ”geography” of SH

Questions:To what extent is this true?To what extent is it helpful?

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 22: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryRepresentability theorems and axiom systems

Representability theorems

General ”theorems”:Any cohomology theory which is A1-invariant and satisfiesNisnevich descent should be representable (Morel?)Nisnevich descent should be automatically satisfied for anytheory defined as sheaf cohomologyAny mixed Weil cohomology theory is representable (Cisinskiand Deglise)Any Bloch-Ogus theory should be representable (Levine?)Any oriented theory should be representable

Making all of these precise is not completely trivial.

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 23: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryRepresentability theorems and axiom systems

Example: Representability of Bloch-Ogus theories

Classical axioms:

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 24: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryRepresentability theorems and axiom systems

Example: Representability of Bloch-Ogus theories

New tentative definition:

A Bloch-Ogus theory is an HZ-algebra.

Much simpler!!

Similar simplifications should be possible for other axioms systems.

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 25: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryRepresentability theorems and axiom systems

More simplified axiom systems

An oriented theory is an MGL-algebra. (Vezzosi?)A Beilinson theory is an HQ-module(Tentative) A Weil theory is an HQ-algebra E such thatE (1) ' E .(Tentative) An ordinary theory is a theory whose cohomologygroups are functorial with respect to correspondences.(For an oriented theory, this should be equivalent to theformal group law being the additive one.)

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 26: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryRepresentability theorems and axiom systems

More representability theorems

Specific examples:Algebraic K-theory (KGL)Voevodsky’s motivic cobordism (MGL)Motivic cohomology (HZ)`-adic cohomology, algebraic de Rham cohomology, rigidcohomology (Cisinski-Deglise)Hermitian K-theory (Hornbostel)Deligne cohomology (Holmstrom-Scholbach)

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 27: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryRepresentability theorems and axiom systems

The geography of SH

With the above tentative definitions we can attempt to draw a”map of all cohomology theories in algebraic geometry”.

Many formal properties of cohomology theories should be governedby where they are located on such a map.

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 28: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

A map of cohomology theories (CTs) for schemes

Omitted are also CTs for stacks, for log schemes, and other more general algebraic geometry objects

All CTs

Non- A 1 -invariant CTs CTs not satisfying Nisnevich descent CTs related to noncommutative motives

E.g. arithmetic Chow gps , E.g. truncated Deligne coh., Many versions of cyclic and Hochschild homology,

crystalline coh., Weil-etale coh., various presheaf coh. gps. algebraic K-theory, bivariant CTs, +++++

p-adic etale coh., +++++

Representable CTs (arrows are inclusions) Omitted: Chromatic theory

Motivic Morava K-th

Representable Q -CTs -----> Representable ordinary CTs Motivic BP, +++

Λ All Q-sheaf coh? Λ All sheaf coh?

ǀ ǀ

ǀ ǀ

ǀ ǀ

ǀ ǀ

Mixed Weil CTs ---> Beilinson CTs ------> Bloch-Ogus CTs -----------> Orientable CTs

Etale Z/l-coh. Rational motivic Motivic coh. Algebraic K-theory

Alg. de Rham Deligne R-coh. Morphic coh. Motivic cobordism

Rigid coh. Arakelov motivic Deligne Z-coh. +++

Betti coh. +++ +++

+++

Page 29: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryProperties of cohomology theories

Properties of CTs

Properties we would like to understand:Contravariant functoriality Chern classesCovariant functoriality Cycle classesPoincare duality Product structureMayer-Vietoris Descent for various topologiesProjective bundle formula Blow-up formulaKunneth formula Finiteness propertiesVanishing properties Versions of rigidityVersions of purity Localization sequencesBase change theorems + + + + +

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 30: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

A map of cohomology theories (CTs) for schemes

Omitted are also CTs for stacks, for log schemes, and other more general algebraic geometry objects

All CTs

Non- A 1 -invariant CTs CTs not satisfying Nisnevich descent CTs related to noncommutative motives

E.g. arithmetic Chow gps , E.g. truncated Deligne coh., Many versions of cyclic and Hochschild homology,

crystalline coh., Weil-etale coh., various presheaf coh. gps. algebraic K-theory, bivariant CTs, +++++

p-adic etale coh., +++++

Representable CTs (arrows are inclusions) Omitted: Chromatic theory

Motivic Morava K-th

Representable Q -CTs -----> Representable ordinary CTs Motivic BP, +++

Λ All Q-sheaf coh? Λ All sheaf coh?

ǀ ǀ

ǀ ǀ

ǀ ǀ

ǀ ǀ

Mixed Weil CTs ---> Beilinson CTs ------> Bloch-Ogus CTs -----------> Orientable CTs

Etale Z/l-coh. Rational motivic Motivic coh. Algebraic K-theory

Alg. de Rham Deligne R-coh. Morphic coh. Motivic cobordism

Rigid coh. Arakelov motivic Deligne Z-coh. +++

Betti coh. +++ +++

+++

Page 31: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryProperties of cohomology theories

Properties of CTs

Example 1: Pushforwards

Proposition: Let E be a Beilinson cohomology theory. Letf : X → Y be a smooth projective morphism of relative dimensiond . Then there are functorial pushforward maps

f∗ : En(X , p)→ En−2d (Y , p − d).

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 32: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryProperties of cohomology theories

Six functors formalism: further properties

For any morphism f separated of finite type, there is a naturaltransformation f! → f∗ which is an isomorphism if f is proper.If f is an open immersion, we have f ! = f ∗.For an HQ-module E , we have more generally thatf ∗ = f !(−d)[−2d ]

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 33: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryProperties of cohomology theories

Properties of CTs

Example 1: Pushforwards Proposition: Let E be a Beilinsoncohomology theory. Let f : X → Y be a smooth projectivemorphism of relative dimension d . Then there are functorialpushforward maps

f∗ : En(X , p)→ En−2d (Y , p − d).

Proof: Let pX and pY be the structural morphisms of X and Yrespectively. It is enough to construct a mappX∗p∗

X = pY ∗f∗f ∗p∗Y → pY ∗p∗

Y (d)[2d ]. But f∗ = f! since f isproper, and we have f ∗ = f !(−d)[−2d ] for any Beilinson modulewhenever f is smooth and quasi-projective. So for f smooth andprojective, the counit f!f ! → id gives the desired map.

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 34: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

A map of cohomology theories (CTs) for schemes

Omitted are also CTs for stacks, for log schemes, and other more general algebraic geometry objects

All CTs

Non- A 1 -invariant CTs CTs not satisfying Nisnevich descent CTs related to noncommutative motives

E.g. arithmetic Chow gps , E.g. truncated Deligne coh., Many versions of cyclic and Hochschild homology,

crystalline coh., Weil-etale coh., various presheaf coh. gps. algebraic K-theory, bivariant CTs, +++++

p-adic etale coh., +++++

Representable CTs (arrows are inclusions) Omitted: Chromatic theory

Motivic Morava K-th

Representable Q -CTs -----> Representable ordinary CTs Motivic BP, +++

Λ All Q-sheaf coh? Λ All sheaf coh?

ǀ ǀ

ǀ ǀ

ǀ ǀ

ǀ ǀ

Mixed Weil CTs ---> Beilinson CTs ------> Bloch-Ogus CTs -----------> Orientable CTs

Etale Z/l-coh. Rational motivic Motivic coh. Algebraic K-theory

Alg. de Rham Deligne R-coh. Morphic coh. Motivic cobordism

Rigid coh. Arakelov motivic Deligne Z-coh. +++

Betti coh. +++ +++

+++

Page 35: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryProperties of cohomology theories

Properties of CTs

Example 2: Localization sequences

For a closed immersion Z → X with complementary openimmersion j : U → X , we would like to relate the cohomologygroups of Z , X and U in some way.

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 36: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryProperties of cohomology theories

Six functors formalism: Localization

Let Z → X be a closed immersion with complementary openimmersion j : U → X . Then the following holds:

The functor j! is left adjoint to j∗.The functor i∗ is left adjoint to i !.The functors j! and i∗ are fully faithful.We have i∗j! = 0For any object M of SH, there are natural distinguishedtriangles

j!j !M → M → i∗i∗M → j!j !M[1]

andi!i !M → M → j∗j∗M → i!i !M[1]

where the maps are given by units and counits of the relevantadjoint pairs of functors. These triangles are referred to aslocalization triangles.

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 37: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

A map of cohomology theories (CTs) for schemes

Omitted are also CTs for stacks, for log schemes, and other more general algebraic geometry objects

All CTs

Non- A 1 -invariant CTs CTs not satisfying Nisnevich descent CTs related to noncommutative motives

E.g. arithmetic Chow gps , E.g. truncated Deligne coh., Many versions of cyclic and Hochschild homology,

crystalline coh., Weil-etale coh., various presheaf coh. gps. algebraic K-theory, bivariant CTs, +++++

p-adic etale coh., +++++

Representable CTs (arrows are inclusions) Omitted: Chromatic theory

Motivic Morava K-th

Representable Q -CTs -----> Representable ordinary CTs Motivic BP, +++

Λ All Q-sheaf coh? Λ All sheaf coh?

ǀ ǀ

ǀ ǀ

ǀ ǀ

ǀ ǀ

Mixed Weil CTs ---> Beilinson CTs ------> Bloch-Ogus CTs -----------> Orientable CTs

Etale Z/l-coh. Rational motivic Motivic coh. Algebraic K-theory

Alg. de Rham Deligne R-coh. Morphic coh. Motivic cobordism

Rigid coh. Arakelov motivic Deligne Z-coh. +++

Betti coh. +++ +++

+++

Page 38: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryProperties of cohomology theories

Properties of CTs

Example 3: Descent propertiesAny representable theory satisfies Nisnevich descent, so inparticular has a long exact Mayer-Vietoris sequence:

. . . En(X , p)→ En(U, p)⊕En(U, p)→ En(U∩V , p)→ En+1(X , p)→ . . .

Any Beilinson theory satisfies h-descent, so in particular Galoisdescent and faithfully flat descent.

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 39: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

A map of cohomology theories (CTs) for schemes

Omitted are also CTs for stacks, for log schemes, and other more general algebraic geometry objects

All CTs

Non- A 1 -invariant CTs CTs not satisfying Nisnevich descent CTs related to noncommutative motives

E.g. arithmetic Chow gps , E.g. truncated Deligne coh., Many versions of cyclic and Hochschild homology,

crystalline coh., Weil-etale coh., various presheaf coh. gps. algebraic K-theory, bivariant CTs, +++++

p-adic etale coh., +++++

Representable CTs (arrows are inclusions) Omitted: Chromatic theory

Motivic Morava K-th

Representable Q -CTs -----> Representable ordinary CTs Motivic BP, +++

Λ All Q-sheaf coh? Λ All sheaf coh?

ǀ ǀ

ǀ ǀ

ǀ ǀ

ǀ ǀ

Mixed Weil CTs ---> Beilinson CTs ------> Bloch-Ogus CTs -----------> Orientable CTs

Etale Z/l-coh. Rational motivic Motivic coh. Algebraic K-theory

Alg. de Rham Deligne R-coh. Morphic coh. Motivic cobordism

Rigid coh. Arakelov motivic Deligne Z-coh. +++

Betti coh. +++ +++

+++

Page 40: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryProperties of cohomology theories

Properties of CTs

Example 4: Cycle classesFor smooth varieties over a field, there is a comparisonisomorphism between Chow groups CHn(X ) and the motiviccohomology groups H2n

M (X , n).Therefore any HZ -algebra (Bloch-Ogus theory) admits cycleclasses.

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 41: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

A map of cohomology theories (CTs) for schemes

Omitted are also CTs for stacks, for log schemes, and other more general algebraic geometry objects

All CTs

Non- A 1 -invariant CTs CTs not satisfying Nisnevich descent CTs related to noncommutative motives

E.g. arithmetic Chow gps , E.g. truncated Deligne coh., Many versions of cyclic and Hochschild homology,

crystalline coh., Weil-etale coh., various presheaf coh. gps. algebraic K-theory, bivariant CTs, +++++

p-adic etale coh., +++++

Representable CTs (arrows are inclusions) Omitted: Chromatic theory

Motivic Morava K-th

Representable Q -CTs -----> Representable ordinary CTs Motivic BP, +++

Λ All Q-sheaf coh? Λ All sheaf coh?

ǀ ǀ

ǀ ǀ

ǀ ǀ

ǀ ǀ

Mixed Weil CTs ---> Beilinson CTs ------> Bloch-Ogus CTs -----------> Orientable CTs

Etale Z/l-coh. Rational motivic Motivic coh. Algebraic K-theory

Alg. de Rham Deligne R-coh. Morphic coh. Motivic cobordism

Rigid coh. Arakelov motivic Deligne Z-coh. +++

Betti coh. +++ +++

+++

Page 42: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryComparison theorems

What about the non-representable CTs?

Often there are comparison theorems between representablecohomology theories and non-representable theories. Thesecan be used to understand non-representable theories viaresults in motivic stable homotopy theory.Crystalline cohomology (non-representable) agrees with rigidcohomology (representable) for smooth projective varieties.Hence crystalline cohomology is a Weil cohomology for thesevarieties.Truncated Deligne cohomology groups Hn

D(X , p)(non-representable) agree with ordinary Deligne cohomologyfor n ≤ 2p. Hence truncated Deligne cohomology admits cyclemaps.

Andreas Holmstrom Cohomology theories in algebraic geometry

Page 43: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

A map of cohomology theories (CTs) for schemes

Omitted are also CTs for stacks, for log schemes, and other more general algebraic geometry objects

All CTs

Non- A 1 -invariant CTs CTs not satisfying Nisnevich descent CTs related to noncommutative motives

E.g. arithmetic Chow gps , E.g. truncated Deligne coh., Many versions of cyclic and Hochschild homology,

crystalline coh., Weil-etale coh., various presheaf coh. gps. algebraic K-theory, bivariant CTs, +++++

p-adic etale coh., +++++

Representable CTs (arrows are inclusions) Omitted: Chromatic theory

Motivic Morava K-th

Representable Q -CTs -----> Representable ordinary CTs Motivic BP, +++

Λ All Q-sheaf coh? Λ All sheaf coh?

ǀ ǀ

ǀ ǀ

ǀ ǀ

ǀ ǀ

Mixed Weil CTs ---> Beilinson CTs ------> Bloch-Ogus CTs -----------> Orientable CTs

Etale Z/l-coh. Rational motivic Motivic coh. Algebraic K-theory

Alg. de Rham Deligne R-coh. Morphic coh. Motivic cobordism

Rigid coh. Arakelov motivic Deligne Z-coh. +++

Betti coh. +++ +++

+++

Page 44: Cohomology theories in motivic stable homotopy theory · Cohomology theories in algebraic geometry Introduction and motivation Algebraic topology Examples: Singular cohomology, various

Cohomology theories in algebraic geometryFinal remarks on arithmetic geometry

Final remarks

An arithmetic scheme is a scheme of finite type over Spec(Z).Cohomology for such schemes is very poorly understood.Thanks to Ayoub, Deglise and Cisinski, motivic stablehomotopy theory can be useful for studying cohomology ofsuch schemes, but it is not good enough for all purposes.Question: What is the ”right” notion of stable homotopytheory for arithmetic schemes???

Andreas Holmstrom Cohomology theories in algebraic geometry