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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

  1. 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

  2. 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”.

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. Let $R$ be a ring. Let $(A, \text{d})$ be a differential graded algebra over $R$. The opposite differential graded algebra is the differential graded algebra $(A^{opp}, \text{d})$ over $R$ where $A^{opp} = A$ as an $R$-module, $\text{d} = \text{d}$, and multiplication is given by

\[ a \cdot _{opp} b = (-1)^{\deg (a)\deg (b)} b a \]

for homogeneous elements $a, b \in A$.

This makes sense because

\begin{align*} \text{d}(a \cdot _{opp} b) & = (-1)^{\deg (a)\deg (b)} \text{d}(b a) \\ & = (-1)^{\deg (a)\deg (b)} \text{d}(b) a + (-1)^{\deg (a)\deg (b) + \deg (b)}b\text{d}(a) \\ & = (-1)^{\deg (a)}a \cdot _{opp} \text{d}(b) + \text{d}(a) \cdot _{opp} b \end{align*}

as desired.

Definition 22.3.4. 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 22.3.5. 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 22.3.6. 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 22.3.5. $\square$


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