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;
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