Constructing an adjunction between algebras and differential graded algebras

Question

Fix a ring R. I am looking for a construction of the adjunction between R-algebras and differential graded R-algebras. I am looking for a reference which constructs the left adjoint to the functor from differential graded algebras to R-algebras which sends $\text{A}$${}_{*}$ to the product $\Pi_{\text{n} \in \mathbb{N}} \text{A}_{\text{n}}$.

Progress:

(Constructing the monoidal category of complexes of R-modules).

Chain complexes have a tensor product which is the total complex $\oplus_{n \in \mathbb{N}}$ of the bicomplex $B_{* , *}$ C${}_{*}$ $\otimes$ D${}_{*}$ which is C${}_{n}$ $\otimes$ D${}_{m}$ is in (n,m) and where the map C${}_{n}$ $\otimes$ D${}_{m}$ $\rightarrow$ C${}_{n}$${}_{+1}$ $\otimes$ D${}_{m}$ gets a sign for even n.

(Constructing the monoidal category of differential graded $R$-algebras).

A differential graded $R$-algebra is a monoid-object in differential.

Answer 1

($R$ is commutative, of course.)

This functor does exist but it is probably not the functor you're looking for, in particular it is not the de Rham algebra. It is very strange. Here is a description of it: let $S$ be an $R$-algebra presented by generators $x_i$ subject to relations $R_k$. Then the left adjoint $F(S)$ is the dg $R$-algebra obtained by starting with the $R$-algebra presented by, for each generator $x_i$ of $S$, countably many generators $x_{i, j}, j \ge 0$ where $x_{i, j}$ is in degree $j$, subject to, for each relation $R_k$, countably many relations which can be succinctly described by saying that the formal power series

$$x_i(t) = \sum_j x_{i, j} t^j$$

should continue to satisfy $R_k$. Then one freely adjoins differentials to this thing. This is a very large and bizarre object already for $F(R[x])$, which represents your forgetful functor.

I believe the de Rham algebra is the left adjoint to a different functor, namely the functor from (graded-)commutative dg $R$-algebras to commutative $R$-algebras given by $A \mapsto A_0$.