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
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