Hochschild homology and Grothendieck Duality

Transcription

1 Hochschild homology and Grothendieck Duality Leovigildo Alonso Tarrío Universidade de Santiago de Compostela Purdue University July, 1, 2009 Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

2 Broad outline 1 Co/homology of singular spaces 2 Bivariant Hochschild theory 3 Bivariant Hochschild homology and cohomology 4 Orientation and fundamental class This is joint work with A. Jeremías (USC) & J. Lipman (Purdue). Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

3 Outline 1 Co/homology of singular spaces 1 Co/homology of singular spaces Nonsingular spaces Bivariant theories 2 Bivariant Hochschild theory 3 Bivariant Hochschild homology and cohomology 4 Orientation and fundamental class Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

4 Co/homology of singular spaces Nonsingular spaces Co/homology of nonsingular spaces Let X be a nonsingular space (technically an orientable topological manifold to fix ideas think on the underlying space of a complex manifold). There are two theories, namely, homology and cohomology. H (X ) and H (X ) By Poincaré duality they convey the same information H i (X ) = H n i (X ) However, if the space is singular these isomorphisms do not hold anymore. One is tempted to regard cohomology as the main invariant because of its ring structure and disregard homology. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

5 Co/homology of singular spaces Bivariant theories The starting point This is not the right answer. MacPherson observed that cohomology and homology played different roles, both important, in the case of singular spaces 1 Homology supports characteristic classes 2 Cohomology is the ring of operations of homology It is desirable to get a general framework that makes sense of this observation. This framework was developed by Fulton and MacPherson and christened bivariant theories. They show how to make them work as a good language to express Riemann-Roch type theorems for singular spaces. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

6 Co/homology of singular spaces Bivariant theories Ingredients of a bivariant theory A bivariant theory is not exactly a functor, it has some features related to (weak) 2-categorical ideas but this has not been pursued. In short it consists on 1 an underlying category, a category C with some extra structure, 2 the category of values, a graded counterpart GrA of a monoidal, abelian category A (usually graded modules over some ring), 3 and a map (the theory) T : Arr(C) GrA, where Arr( ) denotes the class of arrows of a category, subject to a list of conditions. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

7 Co/homology of singular spaces Bivariant theories Structure of the underlying category I The category C is endowed with two classes: 1 a class of maps in C called confined maps, 2 a class of diagrams in C, oriented commutative squares, called independent squares X g X f d f Y g Y satisfying the following conditions. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

8 Co/homology of singular spaces Bivariant theories Structure of the underlying category II uc A The class of confined maps contains the identities and is stable for composition. uc B The class of independent squares contains all squares d such that f = f and g = g = id and is stable for vertical and horizontal composition. uc C In a square d, if f (or g) is confined then so is f (or g, respectively). In other words to be confined changes of base through independent squares. X g X f d f Y g Y Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

9 Co/homology of singular spaces Bivariant theories Operations for a bivariant theory Prod Given f : X Y and g : Y Z in C homomorphisms : T i (X f Y ) T j (Y g Z) T i+j (X g f Z) (i, j Z). PF For each f : X Y confined and g : Y Z in C a homomorphism f T : T i (X g f Z) T i (Y g Z) PB For an independent square d, homomorphisms X g X (i Z). f d f Y g g T : T i (X f Y ) T i (X f Y ) Y Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

10 Co/homology of singular spaces Bivariant theories Axioms for the operations of a bivariant theory I A 1 Product is associative: Given X f Y g Z h W in C and α T i (f ), β T j (g), γ T l (h) then (α β) γ = α (β γ) A 2 Push-forward is functorial: Given X f Y g Z h W in C and α T i (hgf ) with f and g confined, then (gf ) T (α) = g T f T (α) A 12 Product and push-forward commute: Given X f Y g Z h W in C and α T i (gf ), β T j (h), with f confined, then f T (α β) = f T (α) β Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

11 Co/homology of singular spaces Bivariant theories Axioms for the operations of a bivariant theory II A 3 Pull-back is functorial: Given independent squares h g X X X f d f d f Y Y Y and α T i (f ) then (gh) T (α) = h T g T (α) in T i (f ) h g Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

12 Co/homology of singular spaces Bivariant theories Axioms for the operations of a bivariant theory III Consider the diagram of independent squares X h X A 13 Product and pull-back commute: Given, in the diagram, α T i (f ), β T j (g), then f f h T (α β) = h T (α) h T (β) g Y h g Y A 23 Push-forward and pull-back commute: Given, in the diagram, α T i (gf ), with f confined, then Z h Z f T (ht (α)) = h T (f T (α)) Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

13 Co/homology of singular spaces Bivariant theories Axioms for the operations of a bivariant theory IV A 123 Projection formula: Given the diagram X g X f d f g Y Y Z and α T i (f ), β T j (hg), with d independent and g confined, then g T (g T (α) β) = α g T (β) h Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

14 Outline 2 Bivariant Hochschild theory 1 Co/homology of singular spaces 2 Bivariant Hochschild theory Data for a bivariant Hochschild theory Definition of a bivariant Hochschild theory Checking the compatibilities 3 Bivariant Hochschild homology and cohomology 4 Orientation and fundamental class Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

15 Bivariant Hochschild theory The underlying category Data for a bivariant Hochschild theory Our base scheme will be a noetherian scheme S. The basic category is S S := Sch fl tf (S), the category of flat finite type separated schemes over S, that we will denote simply S. Note that all maps within S are separated and finite type, but not necessarily flat. Structure: 1 The proper maps of S constitute the class of confined maps. 2 The class of independent squares of S is formed by those oriented fiber squares in C such that the bottom is a étale morphism. Observation The axioms uc A, uc B and uc C hold by standard considerations. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

16 Bivariant Hochschild theory Data for a bivariant Hochschild theory A reminder on quasi-coherent cohomological operations. To every scheme X in S we can associate its derived category D qc (X ) of complexes of sheaves with quasi-coherent homology. It is monoidal closed with the derived tensor product denoted L. Let f : X Y a map in S. We have the usual adjunction Lf Rf Another operation is the twisted inverse image f! : D qc (Y ) D qc (X ). In the proper case satisfies the adjunction Rf f! while in the étale case we have f! := f. That this notion makes sense as a pseudo functor and the basic properties of f! is a non trivial theory developed by Grothendieck, Hartshorne, Deligne and Verdier and put up to date and clarified recently by Conrad, Neeman and Lipman (among others). Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

17 The theory I Bivariant Hochschild theory Definition of a bivariant Hochschild theory To define the theory we need a derived category incarnation of the Hochschild complex. For each scheme X in S with structure map x : X S take the canonical diagonal embedding. We define the complex δ x = δ : X X S X, H X := Lδ Rδ O X its homologies are the sheafified Hochschild homology. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

18 The theory II Bivariant Hochschild theory Definition of a bivariant Hochschild theory As X is separated over S then δ is a closed embedding, therefore δ is exact and δ = Rδ. Note that the composition of δ and Lδ is not the derivative of the composition that is trivially the identity functor. To relate this to the familiar Hochschild homology, consider the string of isomorphisms H X = Lδ δ O X = δ 1 δ O X L δ 1 O X S X O X = O X L δ 1 O X S X O X In the affine case with X = Spec(A) and S = Spec(R), A is flat over R, its i th cohomology H i (H X ) = (H i (A L A R A A)) = (Tor A RA i (A, A)) corresponding to the usual identification HH i (A R) = Tor A RA i (A, A). Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

19 The theory III Bivariant Hochschild theory Definition of a bivariant Hochschild theory Let us recall some functorial properties of the Hochschild X f Y complex. Let f : X Y be a morphism in S and consider δ x d δ y the commutative square X X f f Y Y We have the following composition of natural transformations Lf Lδy δ y LδxL(f f ) δ y LδxLδ x Lf We apply it to O Y and obtain the canonical morphisms f : Lf H Y H X and its adjoint f : H Y Rf H X Satisfying transitivity, i.e. (gf ) = f Lf (g ) and (gf ) = g Rg (f ). Proposition If f : X Y is étale then f : Lf H Y H X is an isomorphism. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

20 Bivariant Hochschild theory Definition of a bivariant Hochschild theory The theory IV We associate to each map f : X Y in S the graded module HH i (X f Y ) := Ext i X (H X, f! H Y ) (i Z) Note that Ext i X (H X, f! H Y ) = Hom D(X ) (H X, f! H Y [i]). We define the three operations: 1 The composition uses composition in the derived category. 2 The push forward denoted f H is defined through the functor Rf, the covariant behavior of the Hochschild complex for a composition X f Y g Z and the duality trace f. 3 The pull back, denoted g H is defined through the functor Lg, the contravariant behavior of the Hochschild complex and the base change isomorphism from duality g f! f! g. Its definition requires the étale condition on the independent squares. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

21 Bivariant Hochschild theory Checking the compatibilities Simple compatibilities Proposition The product is associative, i.e. A 1 holds. It follows from pseudo functoriality of the twisted inverse image. Proposition The push forward is functorial, i.e. A 2 holds. It follows from pseudo functoriality of f! and Rf, functorial properties of f and the covariant behavior of the Hochschild complex. Proposition The pull back is functorial, i.e. A 3 holds. It follows from pseudo functoriality of f! and Lg, functorial properties of the base change isomorphism from duality and the contravariant behavior of the Hochschild complex. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

22 Bivariant Hochschild theory Checking the compatibilities Double compatibilities I Proposition Products and push forward commute, i.e. A 12 holds. It follows from the fact that the pseudo functoriality of f! isomorphism and f are natural transformations. Proposition Products and pull back commute, i.e. A 13 holds. It follows from the naturality of the base change isomorphism from duality and the compatibility of the pseudo functoriality of f! and the map that expresses the contravariant behavior of the Hochschild complex. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

23 Bivariant Hochschild theory Checking the compatibilities Double compatibilities II Proposition Push forward and pull back commute, i.e. A 23 holds. For α HH i (X gf Z), to compare f H (hh (α)) and h H f H (α) we have to express them through the commutativity of the corresponding diagrams. Next we construct a diagram that uses the naturality of the base change isomorphism from duality, its compatibility with the pseudo functoriality of f!, and the compatibility of the contravariant and covariant behavior of H X together with the usual base change between () and (). Finally one appeals to the commutative diagram of natural transformations. It expresses the compatibility of f with both base-changes. h Rf f! h bch f Rf h f! bch! Rf f! h f Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

24 Bivariant Hochschild theory Checking the compatibilities The projection formula Proposition The projection formula holds, i.e. A 123 holds. The result follows from the commutativity of a rather complicated diagram in which some parts commute due to naturalities and properties of the behavior of the Hochschild complex as in the previous results. One concludes using the following key ingredient: Lemma In an independent square of S as before with f proper, the composition f! g g g f! bch g! g f! g g g f! is an isomorphism when restricted to D + qc(y ) and its inverse is f! g f f! g f f! f! f g f! g f! Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

25 Bivariant Hochschild theory Checking the compatibilities Summing up Theorem Let S be a base scheme and R := Γ(S, O S ). The triple (S, GrR-Mod, HH) formed by the underlying category S = Sch fl tf (S), the category of values GrR-Mod, the theory HH: Arr(S) GrR-Mod defined through the Hochschild complex is a bivariant theory in the sense of Fulton-MacPherson. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

26 Outline 3 Bivariant Hochschild homology and cohomology 1 Co/homology of singular spaces 2 Bivariant Hochschild theory 3 Bivariant Hochschild homology and cohomology Homology and cohomology Relation to Căldăraru s theory Relation to Hodge cohomology 4 Orientation and fundamental class Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

27 Bivariant Hochschild homology and cohomology Homology and cohomology Bivariant Hochschild cohomology modules Let X S. The bivariant Hochschild cohomology modules or bivariant Hochschild cohomology is defined as The cup product HH i (X ) := HH i (X id X ) = Ext i X (H X, H X ) : HH i (X ) HH j (X ) HH i+j (X ) is associated to the composition X id X id X. There are pull back homomorphisms f H : HH i (X ) HH i (X ) for every étale morphism f : X X. These properties give HH the structure of a ring-valued contravariant functor for étale morphisms. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

28 Bivariant Hochschild homology and cohomology Homology and cohomology Bivariant Hochschild homology modules Let x : X S be the structural map. The bivariant Hochschild homology modules or bivariant Hochschild homology is defined as HH i (X ) := HH i (X x S) = Ext i X (H X, x! O S ) The cap product : HH i (X ) HH j (X ) HH i+j (X ) is given by composition for the morphisms X id X x S. Associated to the composition X f x X S, with f proper (confined), there are push forward homomorphisms f H : HH i (X ) HH i (X ) This gives HH the structure of a covariant functor (for proper maps) that is a module over HH and satisfies the projection formula by A 123 : f H (f H (β) α) = β f H (α) (f confined) Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

29 Bivariant Hochschild homology and cohomology Relation to Căldăraru s theory Relation to Căldăraru s Hoschschild homology for schemes According to Căldăraru it makes sense to define Hoschschild homology for smooth S-schemes X x S of relative dimension n, as follows HH Cal i (X ) = Hom D(X X ) (δ! O X, δ O X [i]) Unravelling Căldăraru s definition δ! O X = δ ω 1 X [ n]. Now we compute Hom D(X X ) (δ ω 1 X [ n], δ O X [i]) = Hom D(X ) (O X, ω X [n] δ! δ O X [i]) but ω X [n] = x! O S and so, x! O S δ! δ O X [i] = δ! O X X Lδ δ x! O S [i] = δ! δ x! O S [i] Therefore, using the adjunctions, HH Cal i (X ) = Hom D(X ) (O X, δ! δ x! O S [i]) = Hom D(X ) (Lδ δ O X, x! O S [i]) and Hom D(X ) (Lδ δ O X, x! O S [i]) = Hom D(X ) (H X, x! O S [i]) = HH i (X ) Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

30 Bivariant Hochschild homology and cohomology Relation to Căldăraru s theory Relation to Căldăraru s Hoschschild cohomology for schemes In the case of cohomology the situation is quite different. Căldăraru s definition (that in fact goes back to Kontsevich) is: Using the adjunctions, HH i Cal (X ) = Hom D(X X )(δ O X, δ O X [i]) Hom D(X X ) (δ O X, δ O X [i]) = Hom D(X ) (Lδ δ O X, O X [i]) Now we recall that O X is direct summand of Lδ δ O X. This gives us a split map HH i Cal (X ) = Hom D(X )(H X, O X [i]) Hom D(X ) (H X, H X [i]) = HH i (X ) Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

31 Bivariant Hochschild homology and cohomology The HKR isomorphism Relation to Hodge cohomology Let now S = Spec(k) where k is a field of characteristic 0. Assume that X is smooth over k. Let n = dim(x ). We have the following Theorem There is a canonical quasi-isomorphism H X = n Ω p X [p] The second complex is understood as a zero differential complex. It was stated originally for affine schemes but it can be globalized by compatibility with localization. The theorem is due to Hochschild, Konstant and Rosenberg. p=0 Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

32 Bivariant Hochschild homology and cohomology Computation of HH i (X ), I Relation to Hodge cohomology We start by recalling that in this case x! k = Ω n X [n]. (Note that k = O S ). Also, there is a perfect pairing Ω p X O X Ω n p X Ω n X And as a consequence Ω n p X = Hom X (Ω p X, Ωn X ). In our case so HH i (X ) = Hom D(X ) (H X, x! k[i]) = HomD(X ) ( HH i (X ) = n p=0 n p=0 Hom D(X ) (Ω p X [p], Ωn X [n + i]) Ω p X [p], Ωn X [n + i]) Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

33 Bivariant Hochschild homology and cohomology Computation of HH i (X ), II Relation to Hodge cohomology HH i (X ) = = = = n p=0 n p=0 n p=0 n q=0 Hom D(X ) (Ω p X [p], Ωn X [n + i]) H i RΓ(X, Hom X (Ωp X, Ωn X [n p])) H i RΓ(X, Ω n p X [n p])) H q+i (X, Ω q X )) (q := n p) Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

34 Bivariant Hochschild homology and cohomology Computation of HH i (X ), III Relation to Hodge cohomology We got a relationship between bivarant Hochschild homology and Hodge homology. HH i (X ) = H p (X, Ω q X ) p q=i H 0,0 H 1,0 H 0,1 H 2,0 H 1,1 H 0,2 H 2,1 H 1,2 H 2,2 With H p,q = H p (X, Ω q X ). Note: The sums of the columns of the Hodge diamond yields Hochschild homology. The sums of the rows of the Hodge diamond yields De Rham cohomology. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

35 Outline 4 Orientation and fundamental class 1 Co/homology of singular spaces 2 Bivariant Hochschild theory 3 Bivariant Hochschild homology and cohomology 4 Orientation and fundamental class Definition and meaning of the fundamental class Orientations in bivariant Hochschild Homology A few words on proofs Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

36 Orientation and fundamental class Definition and meaning of the fundamental class General definition of the fundamental class Let f : X Y be a map in S, we define a natural transformation c f : Lδ x Rδ x Lf f! Lδ y Rδ y Let Γ: X X Y be the graph of f, a closed immersion. The map c f is defined as the composition of two natural maps Lδ xδ x Lf a f LΓ Γ f! b f f! Lδ y δ y The map a f is defined through a non trivial map λ: δ x Lf L(id f ) Γ f! using the duality trace, pseudo functorialities and base change. The map b f is an isomorphism obtained using standard properties of the cohomological operations. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

37 Orientation and fundamental class Meaning of the fundamental class I Definition and meaning of the fundamental class If we apply c f to the sheaf O Y, note that Lf O Y = O X, therefore we have that c f (O Y ): Lδ x Rδ x O X f! Lδ y Rδ y O Y or, otherwise said c f (O Y ): H X f! H Y This can be interpreted as saying that the fundamental class is a twisted covariant functoriality of the Hochschild complex. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

38 Orientation and fundamental class Meaning of the fundamental class II Definition and meaning of the fundamental class Now we will look at the situation in which the morphism is the structural morphism of X, i.e. x : X S, here H S = O S. The fundamental class becomes c X : H X f! O S To grasp the significance of this map let us take n th homology. H n (H X ) = HH n (X ) and H n (f! O S ) = ω X, where ω X denotes the dualizing sheaf that can be charaterized through a universal property. We get c n X : HH n(x ) ω X Moreover, we may compose with the canonical map Ω n X HH n(x ) and obtain yet another version of the fundamental class c X : Ω n X ω X It is an isomorphism precisely when X is smooth over S. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

39 Orientation and fundamental class Orientations in bivariant Hochschild Homology Gysin maps in bivariant theories Let us discuss briefly Gysin maps (i.e. wrong-way functorialities) in bivariant Hochschild theory. Let θ HH i (X f Y ), with i Z. There are two Gysin morphisms 1 θ : HH j (Y ) HH j i (X ) (j Z). 2 θ : HH j (X ) HH j+i (Y ) (j Z and f confined) Defined by 1 θ (α) = θ α for α HH j (Y ) 2 θ (β) = f H (β θ) for α HH j (X ) (f confined). These operations satisfy properties like functoriality, compatibility with pull back and with push forward, and certain mixed relations that follow directly from the structure of bivariant theory. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

40 Orientation and fundamental class Orientations in bivariant Hochschild Homology Definition of orientation Let, for a moment, (C, GrA, T ) be any bivariant theory. Let F be a class of maps in C stable for composition and containing the identity maps. If for every map f : X Y in F there is given an element such that c(f ) T (X f Y ) 1 c(gf ) = c(f ) c(g) for X f Y g Z in F. 2 c(id X ) = 1 X T (X ) for all X C we say that c is a canonical orientation for the maps of F in the corresponding bivariant theory. Sometimes we call the maps in F the orientable maps of C. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

41 Orientation and fundamental class Orientations in bivariant Hochschild Homology The fundamental class as a canonical orientation Back to bivariant Hochschild theory (S S, GrR-Mod, HH). Let f : X Y be a flat morphism in S Recall the fundamental class c f : H X f! H Y, i.e. c f HH 0 (X f Y ). We have that 1 c idx = id HX HH (X ). 2 Moreover: Theorem The fundamental class is transitive i.e. for flat maps X f Y g Z in S c gf = c f c g Corollary Flat morphisms constitute an orientable class of maps in bivariant Hochschild theory and the orientation is given by the fundamental class. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

42 Orientation and fundamental class Orientations in bivariant Hochschild Homology Canonical Gysin maps in bivariant Hochschild homology The fundamental class is an orientation for flat maps in bivariant Hochschild homology, therefore we have wrong way functorialities, i.e. Gysin maps, defined for a flat map f : X Y as follows 1 f H! : HH (Y ) HH (X ) defined by f H! = (c f ). 2 f H! : HH (X ) HH (Y ) defined by f H! = (c f ), where we assume in addition f confined. The properties of f H! and f H! follow from the structure of the bivariant theory, let us spell them. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

43 Orientation and fundamental class Orientations in bivariant Hochschild Homology Properties of the canonical Gysin maps I Let us discuss some properties of these morphisms CG1 Functoriality: Let X f Y g Z be flat maps in S Then 1 (gf ) H! (α) = (f H! g H! )(α) for α HH (Z). 2 (gf ) H! (β) = (g H! f H! )(β) for β HH (X ), if, in addition, f and g are confined. CG4 Mixed relations: Let X f Y be a flat map in S. Let α HH (X ), β HH (Y ) and γ HH (Y ). Then 1 f H! (f H (β) α) = β f H! (α), assuming f is étale. 2 f H (α f H! (γ)) = f H! (α) γ, assuming f is confined. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

44 Orientation and fundamental class Orientations in bivariant Hochschild Homology Properties of the canonical Gysin maps II CG3 Push forward: Let X f Y g Z, be flat maps in S, f is confined. Note that the following holds in HH (Y g Z) f H (c gf ) = f H (c f c g ) = f H (c f ) c g Then, we have the identities: 1 f H (c gf ) α = f H ((gf ) H! (α)) for α HH (Z). 2 g H (β f H (c gf )) = (gf ) H! (f H (β)) for β HH (Y ), where we assume moreover that f is étale An explicit computation of f H (c f ) would shed further light over these relations. What happened to CG2 (Pull back)? Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

45 Orientation and fundamental class Base change of the fundamental class Orientations in bivariant Hochschild Homology Let X g X f d f Y g Y be an independent square in S. Theorem The fundamental class is compatible with base change, i.e. g H (c f ) = c f Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

46 Orientation and fundamental class Orientations in bivariant Hochschild Homology Pull back for the canonical Gysin maps CG2 Pull back: Let d be an independent square as before X g X f d f Y g Y with f (and therefore f ) flat morphisms. By the previous theorem g H (c f ) = c f. Then it follows that 1 g H (f H! (α)) = f H! (g H (α)) for α HH (Y ), assuming g confined; 2 f H! (g H (β)) = g H (f H! (β)) for β HH (X ), assuming f confined. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

47 Orientation and fundamental class A few words on proofs On the proofs of the structure I The proofs of the existence of the bivariant Hochschild theory is based on showing the commutativity of certain diagrams. The constructions use properties of the cohomological operations, some formal properties and some further properties that have to be developed from scratch. The compatibilities need more and more complicated diagrams as we go from A 1 all the way down to A 123. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48

48 Orientation and fundamental class On the proofs of the structure II A few words on proofs The transitivity of the fundamental class amounts to saying that the diagram c f g f! c g δ xδ x f g f! δ y δ y g f! g! δ z δ z δ xδ x (gf ) c gf (gf )! δ z δ z commutes. This is achieved after decomposing it into diagram after diagram. This amounts to about 18 L A TEXpages. As Joe Lipman has remarked, advances in the problem of coherence in categories should provide a way to streamline the needed arguments. Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue July / 48