Outline Outline 1. Rigid Dualizing Complexes over Rings Residues and Duality for Schemes and Stacks 2. Rigid Residue Complexes over Rings Amnon Yekutieli 3. Rigid Residue Complexes over Schemes Department of Mathematics Ben Gurion University 4. Residues and Duality for Proper Maps of Schemes Notes available at http://www.math.bgu.ac.il/~amyekut/lectures 5. Finite Type DM Stacks Written: 19 Nov 2013 Some of the work discussed here was done with James Zhang several years ago. Amnon Yekutieli (BGU) Residues 1 / 39 Amnon Yekutieli (BGU) Residues 2 / 39 1. Rigid Dualizing Complexes over Rings 1. Rigid Dualizing Complexes over Rings There is a functor Q : C ( Mod A ) → D ( Mod A ) 1. Rigid Dualizing Complexes over Rings which is the identity on objects. The morphisms in D ( Mod A ) are all of the form Q ( φ ) ◦ Q ( ψ ) − 1 , where ψ is a quasi-isomorphism. All rings in this talk are commutative. Inside D ( Mod A ) there is the full subcategory D b f ( Mod A ) of complexes We fix a base ring K , which is regular noetherian and finite dimensional (e.g. a field or Z ). with bounded finitely generated cohomology. In [YZ3] we constructed a functor Let A be an essentially finite type K -ring. Recall that this means A is a localization of a finite type K -ring. In particular A is noetherian and Sq A / K : D ( Mod A ) → D ( Mod A ) finite dimensional. We denote by C ( Mod A ) the category of complexes of A -modules, and called the squaring . D ( Mod A ) is the derived category. It is a quadratic functor : if φ : M → N is a morphism in D ( Mod A ) , and a ∈ A , then Sq A / K ( a φ ) = a 2 Sq A / K ( φ ) . Amnon Yekutieli (BGU) Residues 3 / 39 Amnon Yekutieli (BGU) Residues 4 / 39
� � 1. Rigid Dualizing Complexes over Rings 1. Rigid Dualizing Complexes over Rings Suppose ( N , σ ) is another rigid complex. A rigid morphism φ : ( M , ρ ) → ( N , σ ) If A is flat over K then there is an easy formula for the squaring: is a morphism φ : M → N in D ( Mod A ) , such that the diagram Sq A / K ( M ) = RHom A ⊗ K A ( A , M ⊗ L K M ) . ρ � Sq A / K ( M ) M But in general we have to use DG rings to define Sq A / K ( M ) . φ Sq A / K ( φ ) A rigidifying isomorphism for M is an isomorphism σ � Sq A / K ( N ) N ρ : M ≃ − → Sq A / K ( M ) is commutative. in D ( Mod A ) . We denote by D ( Mod A ) rig/ K the category of rigid complexes, and A rigid complex over A relative to K is a pair ( M , ρ ) , consisting of a rigid morphisms between them. complex M ∈ D b f ( Mod A ) and a rigidifying isomorphism ρ . Here is the important property of rigidity: if ( M , ρ ) is a rigid complex such that canonical morphism A → RHom A ( M , M ) is an isomorphism, then the only automorphism of ( M , ρ ) in D ( Mod A ) rig/ K is the identity . Amnon Yekutieli (BGU) Residues 5 / 39 Amnon Yekutieli (BGU) Residues 6 / 39 1. Rigid Dualizing Complexes over Rings 2. Rigid Residue Complexes over Rings Rigid dualizing complexes were introduced by M. Van den Bergh 2. Rigid Residue Complexes over Rings [VdB] in 1997. Note that Van den Bergh considered dualizing complexes over a noncommutative ring A , and the base ring K was a field. Again A is an essentially finite type K -ring. More progress (especially the passage from base field to base ring) was The next definition is from [RD]. done in the papers “YZ” in the references. A complex R ∈ D b f ( Mod A ) is called dualizing if it has finite injective Warning: the paper [YZ3] has several serious errors in the proofs, dimension, and the canonical morphism A → RHom A ( R , R ) is an some of which were discovered (and fixed) by the authors of [AILN]. isomorphism. Fortunately all results in [YZ3] are correct, and an erratum is being Grothendieck proved that for a dualizing complex R , the functor prepared. Further work on rigidity for commutative rings was done by Avramov, RHom A ( − , R ) Iyengar, Lipman and Nayak. See [AILN, AIL] and the references is a duality (i.e. contravariant equivalence) of D b f ( Mod A ) . therein. Amnon Yekutieli (BGU) Residues 7 / 39 Amnon Yekutieli (BGU) Residues 8 / 39
2. Rigid Residue Complexes over Rings 2. Rigid Residue Complexes over Rings A rigid dualizing complex over A relative to K is a rigid complex ( R , ρ ) such that R is dualizing. For a prime ideal p ∈ Spec A we define We know that A has a rigid dualizing complex ( R , ρ ) . rig.dim K ( p ) : = rig.dim K ( k ( p )) , Moreover, any two rigid dualizing complexes are uniquely isomorphic in D ( Mod A ) rig/ K . where k ( p ) is the residue field. If A = K is a field, then its rigid dualizing complex R must be The resulting function isomorphic to K [ d ] for an integer d . We define the rigid dimension to be rig.dim K : Spec A → Z rig.dim K ( K ) : = d . has the expected property: it drops by 1 if p ⊂ q is an immediate Example 2.1. If the base ring K is also a field, then specialization of primes. rig.dim K ( K ) = tr.deg K ( K ) . For any p ∈ Spec A we denote by J ( p ) the injective hull of the A -module k ( p ) . This is an indecomposable injective module. On the other hand, rig.dim Z ( F q ) = − 1 for any finite field F q . Amnon Yekutieli (BGU) Residues 9 / 39 Amnon Yekutieli (BGU) Residues 10 / 39 2. Rigid Residue Complexes over Rings 2. Rigid Residue Complexes over Rings A rigid residue complex over A relative to K is a rigid dualizing complex ( K A , ρ A ) , such that for every i there is an isomorphism of A -modules � A ∼ K − i = J ( p ) . The algebra A has a rigid residue complex ( K A , ρ A ) . p ∈ Spec A It is unique up to a unique isomorphism in C ( Mod A ) res/ K . So we call it rig.dim K ( p )= i the rigid residue complex of A . A morphism φ : ( K A , ρ A ) → ( K ′ A , ρ ′ A ) between rigid residue complexes Let me mention several important functorial properties of rigid is a homomorphism of complexes φ : K A → K ′ A in C ( Mod A ) , such that residue complexes. Q ( φ ) : ( K A , ρ A ) → ( K ′ A , ρ ′ A ) is a morphism in D ( Mod A ) rig/ K . We denote by C ( Mod A ) res/ K the category of rigid residue complexes. Amnon Yekutieli (BGU) Residues 11 / 39 Amnon Yekutieli (BGU) Residues 12 / 39
� � � � � 2. Rigid Residue Complexes over Rings 2. Rigid Residue Complexes over Rings Suppose A → B is an essentially étale homomorphism of K -algebras. Now let A → B any homomorphism between essentially finite type K -algebras. There is a unique homomorphism of complexes There is a unique homomorphism of graded A-modules q B / A : K A → K B , Tr B / A : K B → K A , satisfying suitable conditions, called the rigid localization homomorphism . satisfying suitable conditions, called the ind-rigid trace homomorphism . The homomorphism q B / A induces an isomorphism of complexes It is functorial: if B → C is another algebra homomorphism, then B ⊗ A K A ∼ = K B . Tr C / A = Tr B / A ◦ Tr C / B . If B → C is another essentially étale homomorphism, then When A → B is a finite homomorphism, then Tr B / A is a q C / A = q C / B ◦ q B / A . homomorphism of complexes. In this way rigid residue complexes form a quasi-coherent sheaf on the The ind-rigid traces and the rigid localizations commute with each étale topology of Spec A . This will be important for us. other. Amnon Yekutieli (BGU) Residues 13 / 39 Amnon Yekutieli (BGU) Residues 14 / 39 2. Rigid Residue Complexes over Rings 2. Rigid Residue Complexes over Rings Example 2.2. Take an algebraically closed field K (e.g. K = C ), and let (cont.) The ind-rigid trace Tr A / K is the vertical arrows here: A : = K [ t ] , polynomials in a variable t . � ∂ A = ∑ ∂ m K − 1 Hom cont K ( � A = Ω 1 K 0 A = A m , K ) K ( t ) / K The rigid residue complex of A is concentrated in degrees − 1, 0 : m ⊂ A max Tr − 1 Tr 0 A / K = 0 � ∂ A = ∑ ∂ m A / K K − 1 K ( � A = Ω 1 K 0 Hom cont A = A m , K ) K ( t ) / K m ⊂ A max ∂ K = 0 K − 1 K 0 K = 0 K = K Note that for a maximal ideal m = ( t − λ ) , λ ∈ K , the complete local ring is � A m = K [[ t − λ ]] . The homomorphism Tr 0 A / K is � ∑ m φ m � The local component ∂ m sends a meromorphic differential form α to : = ∑ m φ m ( 1 ) ∈ K . Tr 0 the m -adically continuous functional ∂ m ( α ) on � A / K A m coming from the residue pairing : Taking α : = d t t ∈ Ω 1 K ( t ) / K , whose only pole is a simple pole at the ∂ m ( α )( a ) : = Res m ( a α ) ∈ K . origin, we have ( Tr 0 A / K ◦ ∂ A )( α ) = 1. The rigid residue complex of K is just K 0 K = K . We see that the diagram is not commutative; i.e. Tr A / K is not a Now consider the ring homomorphism K → A . homomorphism of complexes . Amnon Yekutieli (BGU) Residues 15 / 39 Amnon Yekutieli (BGU) Residues 16 / 39
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