Bivariant derived algebraic cobordism: Bivariant theories June - PowerPoint PPT Presentation

Bivariant derived algebraic cobordism Bivariant derived algebraic cobordism: Bivariant theories June 16,2020 1 / 39

Bivariant derived algebraic cobordism Table of contents Overview 1 Bivariant theories: Definition & Examples 2 Properties of bivariant theories 3 Universal bivariant theory 4 Bivariant derived algebraic cobordism (brief introduction) 5 Summary of results 6 2 / 39

o O Bivariant derived algebraic cobordism Overview Algebraic cobordism [V] Voevodsky’s MGL ⇤ , ⇤ ( X ) (Solution to Milnor conjecture) [LM] Levine-Morel construct algebraic cobordism Ω ⇤ ( X ) geometrically [Lecture 1] [LP] Levine-Pandharipande’s double point cobordism ω ⇤ ( X ) (Application to Donaldson-Thomas theory). [Lecture 1] [LS] Lowrey-Sch¨ urg’s derived algebraic bordism d Ω ⇤ ( X ) of derived schemes X . [Lecture 6 - 8] [A2] Annala’s precobordism theory Ω ⇤ ( X ) of derive schemes X (Application to Conner-Floyd over a general base) [Lecture 12] [LeeP] Lee-Pandharipande’s algebraic cobordism with bundles ω ⇤ , ⇤ ( X ). Remark (A summary) [LeeP] [LP] Lecture 2 / Ω ⇤ ( X ) / MGL BM ω ⇤ , r ( X ) ω ⇤ , r ( k ) ⌦ ω ⇤ ( k ) ω ⇤ ( X ) 2 ⇤ , ⇤ ( X ) ⇠ ⇠ ⇠ = = = ⇠ Lecture 8 = d Ω ⇤ ( X ) 3 / 39

Bivariant derived algebraic cobordism Overview Bivariant theories Bivariant theories unify cohomology and homology theories and all the functorial properties. It is introduced by Fulton-MacPherson in [FM]. See Fulton [F] chapter 17- 18 for a version of Riemann-Roch for singular varieties. 4 / 39

w ' Bivariant derived algebraic cobordism Overview Bivariant algebraic cobordism [Y] Yokura’s universal bivariant theory in a general setting M ⇤ ( X ! Y ) [Lecture 9] [A1] Annala’s bivariant algebraic cobordism Ω ⇤ ( X ! Y ) using derived algebraic geometry (Application to Conner-Floyd theorem for singular varieties over a field of char. = 0) [Lecture 9-11] [Ann-Y] Annala-Yokura’s bivariant algebraic cobordism with bundles ω ⇤ , ⇤ ( X ! Y ) Remark M ⇤ ( X ! Y ) quotient theory quotient theory the same if r = 0 ω ⇤ , r ( X ! Y ) Ω ⇤ ( X ! Y ) 5 / 39

✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples Setup (The category V with confined morphisms and independent squares) Let V be a category with a final object ⇤ and all fibre products, which has a class of confined morphisms , which contains all isomorphisms and is closed under composition and pullback a class of independent squares which are fibre squares satisfying (i) whenever two smaller squares in / Y 0 / Z 0 / Z X 0 Z 0 / Y / Z Y 0 / Y or X / X X 0 are independent, then so is the outer square, (ii) all the squares of the forms are independent: f / Y 1 / X X X id ✏ id f ✏ f f / Y id / Y X Y where f can be any morphism. 6 / 39

/ ✏ ✏ Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples Example Suppose V is the category of quasi-projective schemes over a field k . We choose all the proper morphisms to be the class of confined morphisms, all the Tor-independent fibre squares to be the class of independent squares. Recall that a fibre square X 0 X / Y Y 0 is called Tor-independent if Tor O Y ( O X , O Y 0 ) = 0 i for i > 0. 7 / 39

✏ Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples Definition (Bivariant theory) A bivariant theory B ⇤ on such a category V with values in the category of graded abelian groups is an assignment to any morphism f : X ! Y a graded abelian group f B ⇤ ( X ! Y ) which is equipped with the following operations: g f (1) (Product) For morphisms X ! Y ! Z , we have a map � � • : B i ( X ! Y ) ⌦ B j ( Y f ! Z ) ! B i + j ( X gf g ! Z ) . g f (2) (Pushforward) For morphisms X ! Y ! Z with f confined , we have an induced pushforward map � � f ⇤ : B i ( X gf ! Z ) ! B i ( Y g ! Z ) . (3) (Pullback) For any independent square X 0 / X f 0 ✏ f g / Y Y 0 we have an induce pullback morphism ! Y ) ! B i ( X 0 f 0 g ⇤ : B i ( X f ! Y 0 ) . These operations are required to satisfy the following 8 axioms: U , A 1 , A 2 , A 3 , A 12 , A 13 , A 23 , A 123 8 / 39

/ ✏ ✏ ✏ Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples Definition (Bivariant theory (Axioms U, A 1 - A 3 )) id X U (Units) B has units, that is for all X 2 ob ( V ), there is an element 1 X 2 B ( X ! X ). For any morphism W ! X and any α 2 B ( W ! X ), we have α • 1 X = α . For any morphism X ! Y and any β 2 B ( X ! Y ), we have 1 X • β = β . For any morphism g : X 0 ! X , we have g ⇤ 1 X = 1 X 0 . A 1 (Product is associative) Given f g h X ! Y ! Z ! W � � � f g h with α 2 B ( X ! Y ), β 2 B ( Y ! Z ), γ 2 B ( Z ! W ), then we have ( α • β ) • γ = α • ( β • γ ) . A 2 (Pushforward is functorial) Given g f h X � ! Y � ! Z � ! W hgf with f , g confined and α 2 B ( X � ! W ), then we have ( g � f ) ⇤ ( α ) = g ⇤ ( f ⇤ ( α )) . A 3 (Pullback is functorial) Given independent squares X 00 X 0 / X f g / Y h / Y 0 Y 00 f ! Y ), we have the equality ( g � h ) ⇤ ( α ) = h ⇤ ( g ⇤ ( α )). and α 2 B ( X 9 / 39

✏ ✏ ✏ / ✏ Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples Definition (Bivariant theory (Axioms A 12 - A 123 )) g ! W with f confined and α 2 B ( X gf f h A 12 (Product and pushforward commute) Given X ! Y ! Z ! Z ), h β 2 B ( Z ! W ), we have f ⇤ ( α • β ) = f ⇤ ( α ) • β . A 13 (Product and pullback commute) Given independent squares X 0 X f 0 ✏ f h 0 / Y Y 0 g h / Z Z 0 g f with α 2 B ( X ! Y ), β 2 B ( Y ! Z ), then we have h ⇤ ( α • β ) = h 0⇤ ( α ) • h ⇤ ( β ). A 23 (Pushforward and pullback commute) Given independent squares as in A 13 with f confined and α 2 B ( X gf ! Z ), then we have f 0 ⇤ ( h ⇤ ( α )) = h ⇤ ( f ⇤ ( α )). ! Y ), β 2 B ( Y 0 hg f A 123 (Projection formula) Given an independent square with g confined and α 2 B ( X ! Z ) g 0 / X X 0 f 0 ✏ f g / Y h / Z Y 0 then we have g 0 ⇤ ( g ⇤ ( α ) • β ) = α • g ⇤ ( β ). 10 / 39

Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples Example (Bivariant K -theory) Let V be the category of quasi-projective schemes over a field k with confined (proper) morphisms and independent (Tor independent) squares. For every morphism f : X ! Y 2 Mor ( V ), we recall that an f -perfect complex is a complex of quasi-coherent sheaves F • on X such that i ⇤ ( F • ) is a perfect complex on P for any p i factorization f : X ! P ! Y with i closed and p smooth. We define f K alg ( X ! Y ) as the free abelian group on the set of quasi-isomorphism classes [ F • ] of f -perfect complexes on X , modulo [ F • ] = [ F 0 • ] + [ F 00 • ] for each exact sequence 0 ! F 0 • ! F • ! F 00 • ! 0 of f -perfect complexes on X . 11 / 39

✏ Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples Example (Bivariant K -theory) ! Y ) of f -perfect complexes (to say K alg is a bivariant theory): f We need to give the following data on K alg ( X (1) (Product) For morphisms f : X ! Y and g : Y ! Z , we define ! Z ) ! K alg ( X gf g f • : K alg ( X ! Y ) ⌦ K alg ( Y ! Z ) by [ F • ] • [ G • ] := [ F • ⌦ O X Lf ⇤ G • ]. (2) (Pushforward) For morphisms f : X ! Y and g : Y ! Z with f confined (proper), we define f ⇤ : K alg ( X gf g ! Z ) ! K alg ( Y ! Z ) by f ⇤ [ F • ] := [ Rf ⇤ F • ] (3) (Pullback) For any independent square (Tor independent fibre square) g 0 / X X 0 f 0 ✏ f g / Y Y 0 we define ! Y ) ! K alg ( X 0 f 0 g ⇤ : K alg ( X f ! Y 0 ) by g ⇤ [ F • ] := [ Lg 0⇤ F • ]. 12 / 39

Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples Example (Bivariant K -theory) Let V be the category of quasi-projective schemes over a field k with confined (proper) morphisms and independent (Tor independent) squares. For every morphism f : X ! Y 2 Mor ( V ), we recall that an f -perfect complex is a complex of quasi-coherent sheaves F • on X such that i ⇤ ( F • ) is a perfect complex on P for any p i factorization f : X ! P ! Y with i closed and p smooth. Recall f K alg ( X ! Y ) is the free abelian group on the set of quasi-isomorphism classes [ F • ] of f -perfect complexes on X , modulo [ F • ] = [ F 0 • ] + [ F 00 • ] for each exact sequence 0 ! F 0 • ! F • ! F 00 • ! 0 of f -perfect complexes on X . Example Note that id K 0 ( X ) (K-theory of vector bundles) is isomorphic to K alg ( X ! X ). p G 0 ( X ) (G-theory of coherent sheaves) is isomorphic to K alg ( X ! ⇤ ). 13 / 39

✏ ✏ Bivariant derived algebraic cobordism Bivariant theories: Definition & Examples Example (Operational bivariant Chow groups) Let V be the category of quasi-projective schemes over a field k with confined (proper) morphisms and independent ( all fibre) squares. For any morphism f : X ! Y , we form the fibre diagram f 0 / Y 0 X 0 g 0 g f / Y X for each morphism g : Y 0 ! Y . We define f op CH p ( X ! Y ) f as follows. An element c 2 op CH p ( X ! Y ) is a collection of homomorphisms c ( k ) : CH k ( Y 0 ) ! CH k � p ( X 0 ) g for all morphisms g : Y 0 ! Y and all k , compatible with proper pushforward (C1), flat pullback (C2), and intersection products (C3). This means: 14 / 39