Lemma 22.3.5 (065X)—The Stacks project

Processing math: 100%

Table of contents
Part 1: Preliminaries
Chapter 22: Differential Graded Algebra
Section 22.3: Differential graded algebras
Lemma 22.3.5 (cite)

previous definition

numberstags

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 (1)

Comment #283 by arp on September 06, 2013 at 07:20

Typo: In the statement of the lemma, the tensor product in $\text{Tot}(A^\bullet \otimes_A B^\bullet)$ should be over $R$ not $A$ .

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (1)

Post a comment