I am struggling to find references for explicit computations of things like symmetric algebras and resolutions in the dg context. (any pointers in this direction would be highly appreciated!) I have the following question.


Let k be a field of char zero and let A = k[x,y], the free commutative algebra on two (degree zero) generators. What is an explicit resolution/cofibrant replacement of A in the category of differential graded algebras over k?


(I really mean the category of dgas not cdgas)


If the standard resolution is huge, is there a way to cut it down to get something smaller and more tractable?



Resolutions by free Differential Graded Algebras
Let $A=k[x\_1,\dots,x\_m]$ be the algebra of commutative polynomials in $m$ variables over a field (or commutative ring) $k$, of arbitrary characteristic. Denote by $B=\bigwedge(x\_1^\*,\dots,x\_m^\*)$ the exterior algebra in $m$ variables $x^\*\_1,\dots,x^\*\_m$ dual to $x\_1,\dots,x\_m$. Let $C=B^\*$ be the dual vector space to $B$; it is naturally a coassociative coalgebra over $k$. In other words, $C$ is the exterior coalgebra cogenerated by the variables/elements $x\_1,\dots,x\_m$.


The algebra $B$ is naturally augmented with the augmentation morphism $b:B\to k$ taking the generators $x\_1^\*,\dots,x\_m^\*$ to zero. Let $B\_+\subset B$ denote the augmentation ideal, i.e., the kernel of $b$. Passing to the dual coalgebras, we have the coaugmentation morphism $c:k\to C$ with the cokernel $C\_+=B\_+^\*$. The vector space $B\_+$ is naturally an associative algebra without unit, and $C\_+$ is a coassociative coalgebra without counit.


Assuming that $A$ is ungraded, or in other words, when viewed as a graded algebra it sits entirely in degree $0$, the algebra $B$ should be graded so that its generators $x\_j^\*$ sit in the cohomological degree $1$. Then the cogenerators $x\_j$ of the coalgebra $C$ sit in the cohomological degree $-1$. Now we are doing the cobar construction, so the cohomological grading on $C\_+$ will be shifted forward by $1$, making the elements $x\_j\in C\_+$ sit in the cohomological degree $0$.


The cobar construction $Cob(C)$ is the free associative algebra generated by the cohomologically graded vector space $C\_+[-1]$. The differential $d$ on $Cob(C)$ is induced by the comultiplication in $C$ (with some sign rule). This should be the smallest possible choice of a cofibrant replacement of the algebra $A$ in the model category of associative DG-algebras over $k$.


In the case of $m=2$, the DG-algebra $Cob(C)$ is the free associative algebra with two generators $x\_1$ and $x\_2$ in cohomological degree $0$ and one generator $x\_{12}$ in cohomological degree $-1$. The differential is provided by the rule $d(x\_{12})=x\_1x\_2-x\_2x\_1$. Generally for an integer $m\ge1$, the DG-algebra $Cob(C)$ is a free associative algebra with $2^m-1$ generators.


