14 LEONID POSITSELSKI

(b) for any object Y ∈ H there exists an object F ∈ F together with a morphism

Y −→ F in H such that a cone of that morphism belongs to A.

Proof of Lemma. It is clear that the functor F/A ∩ F −→ H/A is surjective

on the isomorphism classes of objects under either of the assumptions (a) or (b).

To prove that it is bijective on morphisms, represent morphisms in both quotient

categories by fractions of the form X ←− X −→ Y in the case (a) and by fractions

of the form X −→ Y ←− Y in the case (b).

Proof of Theorem. Part (a): First notice that any projective left DG-mod-

ule M over a DG-ring A is flat. Indeed, one has HomZ(N ⊗A M, Q/Z)

HomA(M, HomZ(N, Q/Z)) for any right DG-module N over A, so whenever N

is acyclic, and consequently HomZ(N, Q/Z) is acyclic, the left hand side of this

isomorphism is acyclic, too, and therefore N ⊗A M is acyclic. So it remains to use

Theorem 1.4 together with Lemma 1.3 and the above Lemma. To prove part (b),

switch the left and right sides by passing to the DG-ring Aop defined as follows.

As a complex, Aop is identified with A, while the multiplication in Aop is given

by the formula

aopbop

=

(−1)|a||b|(ba)op.

Then right DG-modules over A are left

DG-modules over

Aop

and vice versa.

Now let us define the derived functor

TorA

: D(mod–A) × D(A–mod) −−→

k–modgr

for a DG-algebra A over a commutative ring k, where

k–modgr

denotes the cate-

gory of graded k-modules. For this purpose, restrict the functor of tensor product

⊗A : Hot(mod–A)×Hot(A–mod) −→ Hot(k–mod) to either of the full subcategories

Hot(mod–A)fl × Hot(A–mod) or Hot(mod–A) × Hot(A–mod)fl and compose it with

the cohomology functor H : Hot(k–mod) −→

k–modgr.

The functors so obtained

factorize through the localizations D(mod–A) × D(A–mod) and the two induced

derived functors D(mod–A) × D(A–mod) −→

k–modgr

are naturally isomorphic to

each other.

Indeed, the tensor product N ⊗A M by the definition is acyclic whenever one

of the DG-modules N and M is acyclic, while the other one is flat. Let us check

that the complex N ⊗A M is acyclic whenever either of the DG-modules N and M

is simultaneously acyclic and flat. Assume that N is acyclic and flat; choose a flat

left DG-module F over A together with a morphism of DG-modules F −→ A with

an acyclic cone. Then the complex N ⊗A F is acyclic, since N is acyclic; while the

morphism N ⊗A F −→ N ⊗A M is a quasi-isomorphism, since N is flat.

To construct an isomorphism of the two induced derived functors, it suﬃces to

notice that both of them are isomorpic to the derived functor obtained by restricting

the functor ⊗A to the full subcategory Hot(mod–A)fl × Hot(A–mod)fl. In other

words, suppose that G −→ N and F −→ M are morphisms of DG-modules with

acyclic cones, where the right DG-module G and the left DG-module F are flat.

Then there are natural quasi-isomorphisms G ⊗A M ←− G ⊗A F −→ N ⊗A F .

1.7. Restriction and extension of scalars. Let f : A −→ B be a morphism

of DG-algebras, i. e., a closed morphism of complexes preserving the multiplication.

Then any DG-module over B can be also considered as a DG-module over A, which

defines the restriction-of-scalars functor Rf : Hot(B–mod) −→ Hot(A–mod). This

functor has a left adjoint functor Ef given by the formula Ef (M) = B ⊗A M and

a right adjoint functor Ef given by the formula Ef (M) = HomA(B, M) (where