5

Let $R$ be a $DG$ algebra over $A$, i.e, a $\mathbb{Z}$-graded $A$- algebra with a derivation $d$. For example, if $R$ is an $A$-algebra, then any chain complex $C^{\bullet}$ of $R$-modules with a product structure is a $DG$ algebra over $A$ (I think).

Since any $DG$-algebra $R$ over $A$ is, in particular, an $A$-algebra we should be able to define the $R$-module $\Omega_{R/A}$. On the other hand, it seems like the definition of $\Omega_{R/A}$ should take into account the $DG$-algebra structure on $R$.

What is definition of $\Omega_{R/A}$? For example, given a chain complex $C^{\bullet}$ of $R$-modules, how does one define $\Omega_{C^{\bullet}/A}$?

user7090
  • 5,679
  • 1
  • 26
  • 57

1 Answers1

1

$DG$-algebras are the algebra objects in the symmetric monoidal cocomplete linear category of $DG$-modules. So you can apply section 4.5 in my thesis (at least, in the commutative case, but the general case is similar).

Martin Brandenburg
  • 181,922
  • CDGAs are very different from DGAs. DGAs over a commutative ring $R$ model $\mathbb E_1$-$R$-algebras, while CDGAs over a commutative ring $R$ do not, unless $R$ is a $\mathbb Q$-algebra. – Yai0Phah Aug 04 '21 at 17:32