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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?

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    $\begingroup$ Do you want a cofibrant replacement in the category of all dg-algebras or in the category of commutative dg-algebras? in the latter, this object is already cofibrant. $\endgroup$
    – the L
    Nov 1, 2016 at 8:17
  • $\begingroup$ @theL in the former category. $\endgroup$ Nov 1, 2016 at 8:33
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    $\begingroup$ See Theorem 3.21(2) of arxiv.org/pdf/1412.4229v4.pdf for an explicit construction. I don't think the result wil be pretty, and I am not sure about your second question of making it smaller. $\endgroup$
    – the L
    Nov 1, 2016 at 8:45

1 Answer 1

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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.

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