Triangulated category We have a list of properties for distinguished - - PowerPoint PPT Presentation

Triangulated category

◮ We have a list of properties for distinguished triangles in K(A) and the same for D(A) (see [KS,§1.4]).

• 1. X

f

− → Y → Z → X[1] is a distinguished triangle iff Y → Z → X[1]

−f [1]

− − − → Y [1] is.

• 2. If X

id

− → 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]).

◮ Given morphisms X

f

− → Y and Y

g

− → 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 ′ Y Z Z ′ Y ′ X[1] Y [1] Z ′ Y ′ 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 as D+(A) ← K +(I)

K +(F)

− − − − → 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 QA the natural functor K +(A) → D+(A) and likewise for QB. Then there is a natural morphism of functor sI : QB ◦ K +(F) → RF ◦ QA given by sI(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 QA the natural functor K +(A) → D+(A) and likewise for QB. Then there is a natural transformation s : QB ◦ K +(F) → RF ◦ QA 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 αs : Hom(RF, G) → Hom(QB ◦ K +(F), G ◦ QA). Then (RF, s) is universal in the sense that αs is always an isomorphism. ◮ Customary to write RnF := Hn ◦ 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 ShR(X) be the category of sheaves

• f R-modules on X.

◮ Alternatively, you can have a variety X over some field and the sheaf

• f Z/ℓnZ-modules on the Grothendieck topology of X. And then

you have inverse limit of such and then you tensor Qℓ ... ◮ In any case, when you have a morphism X → Y , there is a push-forward f∗ : ShR(X) → ShR(Y ) by f∗(F)(U) = F(f −1(U)). ◮ Define1 f ∗ : ShR(Y ) → ShR(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∗ → idShR(X) and idShR(Y ) → f∗ ◦ f ∗. ◮ f ∗ is exact, and the highlight is that f∗ is only left exact.

1In the quasi-coherent sheaf setting this is almost always denoted f −1, and for f ∗

• ne 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+(ShR(X)) → D+(ShR(Y )) be the derived functor. ◮ As f ∗ is exact, it preserves quasi-isomorphisms and we have f ∗ : D+(ShR(Y )) → D+(ShR(X)), and it is again the left adjoint of Rf∗; HomD+(ShR(X))(f ∗F, F′) = HomD+(ShR(Y ))(F, Rf∗F′). ◮ And of course, Hn(X; F) := Rn(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+(ShR(Z)) → D+(ShR(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! : ShR(X) → ShR(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

• f 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+(ShR(X)) → D+(ShR(Y )). ◮ With some effort one again has R(g ◦ f )! = Rg! ◦ Rf!. In particular Rh! = Rf∗ ◦ j! in the last example. ◮ And again, Hn

c (X; F) := Rn(X → pt)!F.

Proper base change

◮ Suppose we have a Cartesian square X ′ X Y ′ Y

p f ′ f q

◮ 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! ∼ = f ′

! ◦ p∗ : ShR(X) → ShR(Y ′) is

an isomorphism ◮ Corollary. (Proper base change, II) Same assumption and we have a natural isomorphism of functors q∗ ◦ Rf! ∼ = Rf ′

! ◦ p∗ : D+(ShR(X)) → D+(ShR(Y ′)).