1. DERIVED CATEGORY OF DG-MODULES 17
Theorem. One has Hi(A) = 0 for all i 0 if and only if any object X D
can be included into a distinguished triangle Y −→ X −→ Z −→ Y [1] in D with
Y D 1 and Z D 0.
Proof. “Only if”: suppose that the left DG-module X = A over A can be
included into a distinguished triangle Y −→ X −→ Z −→ Y [1] in D with Y D
0
and Z D
−1
(in the obvious notation). Then HomD(X, Z)
H0(Z)
= 0, hence
X is a direct summand of Y in D, so
Hi(X)
= 0 for i 0.
“If”: let M be a DG-module representing the object X. Let P be the direct
sum of left DG-modules A[−i] over A taken over all the elements x
Hi(M)
for
all i 1. There is a natural closed morphism of DG-modules P −→ M; set M1
to be its cone. One has Hi(P ) = 0 for all i 0 and the maps Hi(P ) −→ Hi(M)
are surjective for all i 1. Hence the maps Hi(M) −→ Hi(M1) are isomorphisms
for all i 0, a monomorphism for i = 0, and zero maps for i 0. Repeating the
same procedure for the DG-module M1, etc., we obtain a sequence of DG-module
morphisms M −→ M1 −→ M2 −→ · · · , each morphism having the properties
described above. Set N = lim

Mi; then there is a closed morphism of DG-modules
M −→ N over A such that the maps
Hi(M)
−→
Hi(N)
are isomorphisms for all
i 0 and a monomorphism for i = 0, and one has
Hi(N)
= 0 for all i 0. Set
Q = cone(M N)[−1]; then one has
Hi(Q) = 0 for all i 0. This provides the
desired distinguished triangle.
Remark 1. There is a much simpler proof of the above Theorem applicable
in the case when
Ai
= 0 for all i 0. However, unlike in the situation of (the
proof of) Theorem 1.8(b), it is not true that any DG-ring with zero cohomology
in the negative degrees can be connected by a chain of quasi-isomorphisms with
a DG-ring with zero components in the negative degrees. For a counterexample,
consider the free associative algebra A over a field k generated by the elements x, y,
and η in degree 0, z in degree 1, and ξ in degree −1, with the differential given by
d(x) = d(y) = d(z) = 0, d(ξ) = xy, and d(η) = yz. One can check that Hi(A) = 0
for i 0. The nontrivial Massey product ξz of the cohomology classes x, y,
and z provides the obstruction.
Remark 2. The construction of the above Theorem can be extended to arbi-
trary DG-rings in the following way. Given a DG-ring A, denote by D
0
D =
D(A–mod) the full subcategory formed by all the DG-modules M over A such that
Hi(M) = 0 for i 0 and by D 1 D the minimal full subcategory of D containing
the DG-modules A[−i] with i 1 and closed under infinite direct sums and the
following operation of countably iterated extension. Given a sequence of objects
Xi D 1 and a sequence of distinguished triangles Yi −→ Yi+1 −→ Xi+1 −→ Yi[1]
in D with Y0 = X0, the homotopy colimit cone(
i
Yi
i
Yi) of Yi should also
belong to D
1.
Then the same construction as in the above proof provides for
any X D a distinguished triangle Y −→ X −→ Z −→ Y [1] with Y D
1
and
Z D
0.
This can be generalized even further in the spirit of Remark 1.8.3, by
considering an arbitrary triangulated category D admitting infinite direct sums, and
a set of compact objects C D such that C[−1] C in the role of the DG-modules
A[−i] with i 1. The subcategory D 1 is then generated by C using the opera-
tions of infinite direct sum and countably iterated extension, and the subcategory
D 0 consists of all object X D such that HomD(C, X) = 0 for all C C. If
one does not insist on the subcategories D 0 and D 1 being closed under shifts in
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