Derived Tensor Product via Resolutions by Complexes of Flat Modules (Part 1)

mappingcone

Let U be a topological space, O a sheaf of commutative rings, and A the abelian category of (sheaves of) O-modules. The standard theory of the derived tensor product, via resolutions by complexes of flat modules, applies to complexes in D(A).

A complex P ∈ K(A) is q-flat if for every quasi-isomorphism Q1 → Q2 in K(A), the resulting map P ⊗ Q1 → P ⊗ Q2 is also a quasi-isomorphism; or equivalently, if for every exact complex Q ∈ K(A), the complex P ⊗ Q is also exact.

P ∈ K(A) is q-flat iff for each point x ∈ U, the stalk Px is q-flat in K(Ax), where Ax is the category of modules over the ring Ox. (In verifying this statement, note that an exact Ox-complex Qx is the stalk at x of the exact O-complex Q which associates Qx to those open subsets of U which contain x, and 0 to those which don’t.)

For instance, a complex P which vanishes in all degrees but one (say n) is q-flat iff tensoring with the degree n component Pn is an exact functor in the category of O-modules, i.e., Pn is a flat O-module, i.e., for each x ∈ U, Pxn is a flat Ox-module.

A q-flat resolution of an A-complex C is a quasi-isomorphism P → C where P is q-flat. The totality of such resolutions (with variable P and C) is the class of objects of a category, whose morphisms are the obvious ones.

Every A-complex C is the target of a quasi-isomorphism ψC from a q-flat complex PC, which can be constructed to depend functorially on C, and so that PC[1] = PC[1] and ψC[1] = ψC[1].

Every O-module is a quotient of a flat one; in fact there exists a functor P0 from A to its full subcategory of flat O-modules, together with a functorial epimorphism P0(F) ։ F (F ∈ A). Indeed, for any open V ⊂ U let OV be the extension of O|V by zero, (i.e., the sheaf associated to the presheaf taking an open W to O(W) if W ⊂ V and to 0 otherwise), so that OV is flat,its stalk at x ∈ U being Ox if x ∈ V and 0 otherwise. There is a canonical isomorphism

ψ : F (V) → Hom (OV, F) (F ∈ A)

such that ψ(λ) takes 1 ∈ OV(V) to λ. With Oλ := OV for each λ ∈ F(V),

the maps ψ(λ) define an epimorphism, with flat source,

P0(F) := (⊕V openλ∈F(V) Oλ) → F,

and this epimorphism depends functorially on F.

We deduce then, for each F, a functorial flat resolution ··· → P2(F) → P1(F) → P0(F) → F

with P1(F) := P0 (ker(P0(F) → F), etc. Set Pn(F) = 0 if n < 0. Then to a complex C we associate the flat complex P = PC such that Pr := ⊕m−n=r Pn(Cm) and the restriction of the differential Pr → Pr+1 to Pn(Cm) is Pn(Cm → Cm+1) ⊕ (−1)m Pn(Cm) → Pn−1(Cm), together with the natural map of complexes P → C induced by the epimorphisms P0(Cm) → Cm (m ∈ Z). Elementary arguments, with or without spectral sequences, show that for the truncations τ≤mC, the maps Pτ≤m C → τ≤m C are quasi-isomorphisms. Since homology commutes with direct limits, the resulting map

ψC : PC = limm Pτ≤m C → limτ≤m C = C

is a quasi-isomorphism….

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s