Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Annala’s derived algebraic cobordism over a general base Talk I July 7, 2020 Talk I Annala’s derived algebraic cobordism over a general base
Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Bibliography [A] Chern classes in precobordism theories, T. Annala (2019) [AY] Bivariant algebraic cobordism with bundles, T. Annala, S. Yokura (2019) Talk I Annala’s derived algebraic cobordism over a general base
Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Contents 1 Introduction 2 Bivariant derived algebraic cobordism with bundles Construction Comparison theorem 3 Precobordism theories Definition and the universal precobordism Weak projective bundle formula and Chern classes Talk I Annala’s derived algebraic cobordism over a general base
Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Algebraic cobordism (Voevodsky) The bigraded theory MGL ❻ , ❻ on Sm k was defined as the theory represented by the algebraic Thom spectrum MGL ❃ ❙❍ ❼ k ➁ in the stable motivic homotopy category and was used in order to prove the Milnor conjecture. Talk I Annala’s derived algebraic cobordism over a general base
Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Algebraic cobordism over a field k of characteristic 0 (Levine-Morel) Ω ❻ ✂ Sm op k � GrRing is the universal oriented cohomology theory. Ω ❻ ✂ Sch ➐ k � GrAb is the universal oriented Borel-Moore homology theory. A theorem of Levine establishes MGL 2 ❻ , ❻ ❼ X ➁ ☞ Ω ❻ ❼ X ➁ . The graded group Ω n ❼ X ➁ can be described in terms of cobordism f cycles of the form � Y � X , L 1 ,..., L r ✆ with Y irreducible, f Ð projective and L 1 ,..., L r line bundles on Y such that n � dim ❼ Y ➁ ✏ r . Talk I Annala’s derived algebraic cobordism over a general base
Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Double point cobordism over a field k of characteristic 0 (Levine-Pandharipande) For X ❃ Sch k , the group ω n ❼ X ➁ is the free abelian group on f cobordism cycles of the form � Y � X ✆ with Y ❃ Sm k irreducible of Ð dimension n and f projective modulo the double-point relations. A theorem of Levine-Pandharipande establishes ω ❻ ❼ X ➁ ☞ Ω ❻ ❼ X ➁ . Talk I Annala’s derived algebraic cobordism over a general base
Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Algebraic cobordism of bundles over a field k of characteristic 0 (Lee-Pandharipande) For X ❃ Sch k , the group ω n , r ❼ X ➁ is the free abelian group on f cobordism cycles of the form � Y � X , E ✆ with Y ❃ Sm k irreducible Ð of dimension n , E a vector bundle of rank r on Y and f projective modulo analogously defined double-point relations. In particular, one has ω ❻ , 0 ❼ X ➁ ☞ ω ❻ ❼ X ➁ . Talk I Annala’s derived algebraic cobordism over a general base
Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Derived algebraic cobordism over a field k of characteristic 0 (Lowrey-Sch¨ urg) Lowrey and Sch¨ urg define algebraic cobordims groups d Ω ❻ ❼ X ➁ for quasi-projective derived schemes. For a quasi-projective derived scheme X , one has an isomorphism d Ω ❻ ❼ X ➁ ☞ Ω ❻ ❼ tX ➁ , where tX denotes the truncation of X . Talk I Annala’s derived algebraic cobordism over a general base
Introduction Bivariant derived algebraic cobordism with bundles Precobordism theories Bivariant derived algebraic cobordism over a field k of characteristic 0 (Annala-Yokura) Derived algebraic cobordism as a bivariant theory over the homotopy category of quasi-projective derived k -schemes with f groups Ω ❻ ❼ X � Y ➁ for any morphism f between quasi-projective Ð derived schemes over k . id Associated cohomology theory: Ω ❻ ❼ X ➁ ✂ � Ω ❻ ❼ X � X ➁ . Ð Associated homology theory: Ω ❻ ❼ X ➁ ✂ � Ω ✏❻ ❼ X � pt ➁ . The associated homology theory coincides with the derived algebraic cobordism as defined by Lowrey-Sch¨ urg. Next step: Bivariant derived algebraic cobordism of bundles over a field of characteristic 0! Talk I Annala’s derived algebraic cobordism over a general base
Introduction Construction Bivariant derived algebraic cobordism with bundles Comparison theorem Precobordism theories Setting k : field of characteristic 0 dSch k : homotopy category of quasi-projective derived k -schemes with - proper morphisms as confined morphisms, - all homotopy Cartesian squares as independent squares, - quasi-smooth morphisms as specialized morphisms, - smooth morphisms as specialized projections. Talk I Annala’s derived algebraic cobordism over a general base
Introduction Construction Bivariant derived algebraic cobordism with bundles Comparison theorem Precobordism theories Step I: Construction of ▼ ❻ , ❻ L f Let ▼ i , r L ❼ X � Y ➁ be the quotient of the free L -module on the set Ð h of cycles of the form � V � X , E ✆ , where Ð - h ✂ V � X is proper, - the composite f ❳ h is quasi-smooth of virtual relative dimension ✏ i , - E is a vector bundle on V of rank r , by the relation that taking the disjoint union on the sources corresponds to summation. Pushforward: For f ✂ X � X ➐ proper and g ✂ X ➐ � Y , one defines the pushforward map g ❳ f L ❼ X ➐ g f ❻ ✂ ▼ i , r � Y ➁ � ▼ i , r L ❼ X � Y ➁ Ð Ð Ð h f ❳ h � X ➐ , E ✆ . by f ❻ ❼� V � X , E ✆➁ � � V Ð Ð Ð Talk I Annala’s derived algebraic cobordism over a general base
� � � � Introduction Construction Bivariant derived algebraic cobordism with bundles Comparison theorem Precobordism theories Step I: Construction of ▼ ❻ , ❻ L Pullback: For f ✂ X � X ➐ , g ✂ Y ➐ � Y and X ➐ � Y ➐ ✕ R Y X , one defines the pullback map f L ❼ X ➐ f ➐ g ❻ ✂ ▼ i , r � Y ➁ � ▼ i , r � Y ➐ ➁ L ❼ X Ð Ð � X , E ✆➁ � � V ➐ h ➐ h � X ➐ , E ➐ ✆ , where V ➐ � Y ➐ ✕ R by g ❻ ❼� V Y V and Ð Ð E ➐ � ❼ g ➐➐ ➁ ❻ ❼ E ➁ is the pullback of E along the projection g ➐➐ ✂ V ➐ � V . h ➐ f ➐ � Y ➐ V ➐ X ➐ g g ➐➐ g ➐ h f � X � Y V Talk I Annala’s derived algebraic cobordism over a general base
� � � � Introduction Construction Bivariant derived algebraic cobordism with bundles Comparison theorem Precobordism theories Step I: Construction of ▼ ❻ , ❻ L Bivariant products: We define two bivariant products g g ❳ f f ❨ ❵ ✂ ▼ i , r � Y ➁ ✕ ▼ j , s � Z ➁ � ▼ i ✔ j , r ✔ s L ❼ X L ❼ Y ❼ X � Z ➁ Ð Ð Ð Ð L f g g ❳ f ❨ ❛ ✂ ▼ i , r � Y ➁ ✕ ▼ j , s � Z ➁ � ▼ i ✔ j , rs L ❼ X L ❼ Y ❼ X � Z ➁ Ð Ð Ð Ð L h k as follows: For cycles � V � X , E ✆ and � W � Y , F ✆ , we form the Ð Ð following homotopy Cartesian diagram h ➐ f ➐ � W V ➐ X ➐ k ➐➐ k ➐ k g h f � X � Y � Z V We let E ➐ � ❼ k ➐➐ ➁ ❻ ❼ E ➁ and F ➐ � ❼ f ➐ ❳ h ➐ ➁ ❻ ❼ F ➁ and define � Y , F ✆ � � V ➐ h ❳ k ➐➐ h k � X , E ➐ ❵ F ➐ ✆ , � V � X , E ✆ ❨ ❵ � W Ð Ð Ð Ð � Y , F ✆ � � V ➐ h ❳ k ➐➐ h k � X , E ➐ ❛ F ➐ ✆ . � V � X , E ✆ ❨ ❛ � W Ð Ð Ð Ð Talk I Annala’s derived algebraic cobordism over a general base
Introduction Construction Bivariant derived algebraic cobordism with bundles Comparison theorem Precobordism theories Step I: Construction of ▼ ❻ , ❻ L Proposition [AY, Proposition 5.3] ▼ ❻ , ❻ is a commutative bivariant theory with respect to both L products ❨ ❵ and ❨ ❛ . We may also choose natural orientations along quasi-smooth morphisms: If f ✂ X � Y is quasi-smooth of relative virtual dimension ✏ i , then we may define the two orientations id X f � X , 0 ✆ ❃ ▼ i , 0 θ ❵ ❼ f ➁ ✂ � � X L ❼ X � Y ➁ Ð Ð Ð id X f � X , ❖ X ✆ ❃ ▼ i , 1 θ ❛ ❼ f ➁ ✂ � � X L ❼ X � Y ➁ Ð Ð Ð for the two bivariant theories ❼ ▼ ❻ , ❻ L , ❨ ❵ ➁ and ❼ ▼ ❻ , ❻ L , ❨ ❛ ➁ respectively. Talk I Annala’s derived algebraic cobordism over a general base
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