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 − xη 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