10 LEONID POSITSELSKI

defined as certain representing objects of DG-functors. The finite total object Tot

can be also expressed in terms of iterated cones, so it is well-defined whenever cones

exist in a DG-category DG, and it is preserved by any DG-functors.

A DG-functor DG −→ DG is said to be fully faithful if it induces isomorphisms

of the complexes of morphisms. A DG-functor is said to be an equivalence of DG-

categories if it is fully faithful and every object of DG admits a closed isomorphism

with an object coming from DG . This is equivalent to existence of a DG-functor in

the opposite direction for which both the compositions admit closed isomorphisms

to the identity DG-functors. DG-functors F : DG −→ DG and G: DG −→ DG

are said to be adjoint if for every objects X ∈ DG and Y ∈ DG a closed iso-

morphism of complexes HomDG (F (X),Y ) HomDG (X, G(Y )) is given such that

these isomorphisms commute with the (not necessarily closed) morphisms induced

by morphisms in DG and DG .

Let DG be a DG-category where (a zero object and) all shifts and cones ex-

ist. Then the homotopy category

H0(DG)

is the additive category with the same

class of objects as DG and groups of morphisms given by HomH0(DG)(X, Y ) =

H0(HomDG(X, Y )). The homotopy category is a triangulated category [12]. Shifts

of objects and cones of closed morphisms in DG become shifts of objects and cones of

morphisms in the triangulated category H0(DG). Any direct sums and products of

objects of a DG-category are also their directs sums and products in the homotopy

category. Adjoint functors between DG-categories induce adjoint functors between

the corresponding categories of closed morphisms and homotopy categories.

Two closed morphisms f, g : X −→ Y in a DG-category DG are called homo-

topic if their images coincide in H0(DG). A closed morphism in DG is called a

homotopy equivalence if it becomes an isomorphism in

H0(DG).

An object of DG

is called contractible if it vanishes in

H0(DG).

All shifts, twists, infinite direct sums, and infinite direct products exist in the

DG-categories of DG-modules. The homotopy category of (the DG-category of)

left DG-modules over a DG-ring A is denoted by Hot(A–mod) =

H0DG(A–mod);

the homotopy category of right DG-modules over A is denoted by Hot(mod–A) =

H0DG(mod–A).

1.3. Semiorthogonal decompositions. Let H be a triangulated category

and A ⊂ H be a full triangulated subcategory. Then the quotient category H/A

is defined as the localization of H with respect to the multiplicative system of

morphisms whose cones belong to A. The subcategory A is called thick if it coincides

with the full subcategory formed by all the objects of H whose images in H/A vanish.

A triangulated subcategory A ⊂ H is thick if and only if it is closed under direct

summands in H [53, 39]. The following Lemma is essentially due to Verdier [52];

see also [3, 11].

Lemma. Let H be a triangulated category and B, C ⊂ H be its full triangulated

subcategories such that HomH(B, C) = 0 for all B ∈ B and C ∈ C. Then the natural

maps HomH(B, X) −→ HomH/C(B, X) and HomH(X, C) −→ HomH/B(X, C) are

isomorphisms for any objects B ∈ B, C ∈ C, and X ∈ H. In particular, the

functors B −→ H/C and C −→ H/B are fully faithful. Furthermore, the following

conditions are equivalent:

(a) B is a thick subcategory in H and the functor C −→ H/B is an equivalence

of triangulated categories;