Triangulated category ◮ We have a list of properties for distinguished triangles in K ( A ) and the same for D ( A ) (see [KS, § 1.4]). f 1. X − → Y → Z → X [1] is a distinguished triangle iff − f [1] Y → Z → X [1] − − − → Y [1] is. id 2. If X − → X → 0 → X [1] is a distinguished triangle. 3. Any commutative diagram X Y X ′ Y ′ in K ( A ) (resp. D ( A )) can be completed to a morphism of distinguished triangles X Y Z X [1] X ′ Y ′ Z ′ X ′ [1]
Triangulated category, II We have a list of properties for distinguished triangles in K ( A ) and the same for D ( A ) (see [KS, § 1.4]). f g ◮ Given morphisms X − → Y and Y − → Z . We have distinguished triangles X → Y → M ( f ) → X [1], Y → Z → M ( g ) → Y [1] and X → Z → M ( g ◦ f ) → X [1]. They should be related. The last property states: 4. Writing Z ′ = M ( f ), X ′ = M ( g ) and Y ′ = M ( g ◦ f ), there exists a distinguished triangle Z ′ → Y ′ → X ′ → Z ′ [1], so that these morphisms make the following diagrams commute: Y ′ X ′ Z ′ Y ′ Y Z Z ′ Y ′ X [1] Y [1] X [1] X ′ Y [1] Z Z ′ [1] Y ′ X ′ ◮ An additive category with a functor of shift (like X �→ X [1]) satisfying properties in this and last slide is called a triangulated category .
Derived functors ◮ Suppose we have a covariant functor of abelian categories F : A → B . We have an induced functor K ( F ) : K ( A ) → K ( B ) that preserved distinguished triangles. This does not extend to D ( A ) → D ( B ) unless F is exact . ◮ Now suppose F is left exact, A has enough injectives and let I be the full subcategory of injectives. Then the natural map K + ( I ) → D + ( A ) is an equivalence of category (see e.g. [KS, Prop. 1.7.7]). Thus one may define RF : D + ( A ) → D + ( B ) by defining it K + ( F ) as D + ( A ) ← K + ( I ) → K + ( B ) → D + ( B ). − − − − ◮ Remark: We actually don’t really need enough injectives, but rather enough F -injectives; see [KS, Def. 1.8.2]. ◮ The functor RF preserves distinguished triangles because K + ( F ) does. ◮ Denote by Q A the natural functor K + ( A ) → D + ( A ) and likewise for Q B . Then there is a natural morphism of functor s I : Q B ◦ K + ( F ) → RF ◦ Q A given by s I ( X ) = K + ( F )( X ) → K + ( F )( I ) for any injective resolution X → I (i.e. a quasi-isomorphism for which I ∈ K + ( A ))
Derived functors, II ◮ The functor RF : D + ( A ) → D + ( B ) preserves distinguished triangles because K + ( F ) does. ◮ Denote by Q A the natural functor K + ( A ) → D + ( A ) and likewise for Q B . Then there is a natural transformation s : Q B ◦ K + ( F ) → RF ◦ Q A given by s ( X ) = K + ( F )( X ) → K + ( F )( I ) for any injective resolution X → I (i.e. a quasi-isomorphism for which I ∈ K + ( A )). Gotta do this for morphism and prove independence of choice of I ... ◮ In fact, ( RF , s ) has the following universal property: Let G : D + ( A ) → D + ( B ) be any functor that preserves distinguished triangles. The natural transformation s gives a homomorphism of abelian groups α s : Hom( RF , G ) → Hom( Q B ◦ K + ( F ) , G ◦ Q A ). Then ( RF , s ) is universal in the sense that α s is always an isomorphism. ◮ Customary to write R n F := H n ◦ RF , as an additive functor from D + ( A ) to B .
Push-forward ◮ Let’s fix R a commutative ring with identity; in the end we will hide back to the case R = Q . ◮ For any topological space X , let Sh R ( X ) be the category of sheaves of R -modules on X . ◮ Alternatively, you can have a variety X over some field and the sheaf of Z /ℓ n Z -modules on the Grothendieck topology of X . And then you have inverse limit of such and then you tensor Q ℓ ... ◮ In any case, when you have a morphism X → Y , there is a push-forward f ∗ : Sh R ( X ) → Sh R ( Y ) by f ∗ ( F )( U ) = F ( f − 1 ( U )). ◮ Define 1 f ∗ : Sh R ( Y ) → Sh R ( X ): first take f ∗ F ( U ) to be the direct limit of sections of F on neighborhoods of f ( U ), and then sheafify ◮ The functor f ∗ is the left adjoint to f ∗ , i.e. we have natural transformations f ∗ ◦ f ∗ → id Sh R ( X ) and id Sh R ( Y ) → f ∗ ◦ f ∗ . ◮ f ∗ is exact, and the highlight is that f ∗ is only left exact. 1 In the quasi-coherent sheaf setting this is almost always denoted f − 1 , and for f ∗ one has to tensor the coordinate ring of the domain. Here we deal with sheaves of R -modules and don’t have this issue.
Derived push-forward f ∗ is exact, and the highlight is that f ∗ is only left exact. ◮ We have Rf ∗ : D + (Sh R ( X )) → D + (Sh R ( Y )) be the derived functor. ◮ As f ∗ is exact, it preserves quasi-isomorphisms and we have f ∗ : D + (Sh R ( Y )) → D + (Sh R ( X )), and it is again the left adjoint of Rf ∗ ; Hom D + (Sh R ( X )) ( f ∗ F , F ′ ) = Hom D + (Sh R ( Y )) ( F , Rf ∗ F ′ ) . ◮ And of course, H n ( X ; F ) := R n ( X → pt ) ∗ F . ◮ More basic facts: 1. We have f ∗ sends injective to injective; this is best proved via the left adjoint f ∗ . Consequently, R ( g ◦ f ) ∗ = Rg ∗ ◦ Rf ∗ . 2. If i : Z → X is a closed embedding, then i ∗ is exact and one has i ∗ : D + (Sh R ( Z )) → D + (Sh R ( X )). In this case Ri ∗ = i ∗ .
Proper push-forward ◮ Now suppose f : X → Y is a morphism of locally compact Hausdorff topological spaces. We may define f ! : Sh R ( X ) → Sh R ( Y ) by f ! F ( U ) := { s ∈ f ∗ F ( U ) | f | supp( s ) : supp( s ) → U is proper } . ◮ Apparently f ! = f ∗ is f is proper. ◮ One may check by straightforward topology that ( g ◦ f ) ! = g ! ◦ f ! . ◮ When j : U ֒ → X is an open immersion, we have j ! F is the extension of F by 0 to X . ◮ In particular j ! is exact, and if j : U → X is an open immersion while f : X → Y is a proper morphism and we are interested in h := f ◦ j , we can define h ! = f ! ◦ j ! = f ∗ ◦ j ! . This gives a definition of h ! for algebraic morphisms on ´ etale and various Grothendieck topology. ◮ We have j ! exact. In general f ! is only left exact and we have again Rf ! : D + (Sh R ( X )) → D + (Sh R ( Y )). ◮ With some effort one again has R ( g ◦ f ) ! = Rg ! ◦ Rf ! . In particular Rh ! = Rf ∗ ◦ j ! in the last example. ◮ And again, H n c ( X ; F ) := R n ( X → pt ) ! F .
Proper base change ◮ Suppose we have a Cartesian square p X ′ X f ′ f q Y ′ Y ◮ We always have natural transformation q ∗ ◦ f ∗ → f ′ ∗ ◦ p ∗ , and also q ∗ ◦ f ! → f ′ ! ◦ p ∗ ◮ Theorem. (Proper base change) Suppose the spaces are locally compact Hausdorff. Then q ∗ ◦ f ! ∼ ! ◦ p ∗ : Sh R ( X ) → Sh R ( Y ′ ) is = f ′ an isomorphism ◮ Corollary. (Proper base change, II) Same assumption and we have a natural isomorphism of functors q ∗ ◦ Rf ! ∼ ! ◦ p ∗ : D + (Sh R ( X )) → D + (Sh R ( Y ′ )). = Rf ′
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