22.3 Differential graded algebras
Just the definitions.
Definition 22.3.1. Let R be a commutative ring. A differential graded algebra over R is either
a chain complex A_\bullet of R-modules endowed with R-bilinear maps A_ n \times A_ m \to A_{n + m}, (a, b) \mapsto ab such that
\text{d}_{n + m}(ab) = \text{d}_ n(a)b + (-1)^ n a\text{d}_ m(b)
and such that \bigoplus A_ n becomes an associative and unital R-algebra, or
a cochain complex A^\bullet of R-modules endowed with R-bilinear maps A^ n \times A^ m \to A^{n + m}, (a, b) \mapsto ab such that
\text{d}^{n + m}(ab) = \text{d}^ n(a)b + (-1)^ n a\text{d}^ m(b)
and such that \bigoplus A^ n becomes an associative and unital R-algebra.
We often just write A = \bigoplus A_ n or A = \bigoplus A^ n and think of this as an associative unital R-algebra endowed with a \mathbf{Z}-grading and an R-linear operator \text{d} whose square is zero and which satisfies the Leibniz rule as explained above. In this case we often say “Let (A, \text{d}) be a differential graded algebra”.
The Leibniz rule relating differentials and multiplication on a differential graded R-algebra A exactly means that the multiplication map defines a map of cochain complexes
\text{Tot}(A^\bullet \otimes _ R A^\bullet ) \to A^\bullet
Here A^\bullet denote the underlying cochain complex of A.
Definition 22.3.2. A homomorphism of differential graded algebras f : (A, \text{d}) \to (B, \text{d}) is an algebra map f : A \to B compatible with the gradings and \text{d}.
Definition 22.3.3. A differential graded algebra (A, \text{d}) is commutative if ab = (-1)^{nm}ba for a in degree n and b in degree m. We say A is strictly commutative if in addition a^2 = 0 for \deg (a) odd.
The following definition makes sense in general but is perhaps “correct” only when tensoring commutative differential graded algebras.
Definition 22.3.4. Let R be a ring. Let (A, \text{d}), (B, \text{d}) be differential graded algebras over R. The tensor product differential graded algebra of A and B is the algebra A \otimes _ R B with multiplication defined by
(a \otimes b)(a' \otimes b') = (-1)^{\deg (a')\deg (b)} aa' \otimes bb'
endowed with differential \text{d} defined by the rule \text{d}(a \otimes b) = \text{d}(a) \otimes b + (-1)^ m a \otimes \text{d}(b) where m = \deg (a).
Lemma 22.3.5. Let R be a ring. Let (A, \text{d}), (B, \text{d}) be differential graded algebras over R. Denote A^\bullet , B^\bullet the underlying cochain complexes. As cochain complexes of R-modules we have
(A \otimes _ R B)^\bullet = \text{Tot}(A^\bullet \otimes _ R B^\bullet ).
Proof.
Recall that the differential of the total complex is given by \text{d}_1^{p, q} + (-1)^ p \text{d}_2^{p, q} on A^ p \otimes _ R B^ q. And this is exactly the same as the rule for the differential on A \otimes _ R B in Definition 22.3.4.
\square
Comments (0)