Where should I search for resolutions?

Question

In my research, to test some conjectures or just to illustrate some facts, I often need to compute some explicit examples of derived functors (in the sence of Quillen's model categories). Mainly I work in the category of differential non-negatively graded algebras over some field $k$ of characteristic zero (denote this category $DGA^+_k$). I will regard that differentials have degree $-1$.

It seems like the main problem when computing these functors is to find an appropriate cofibrant resolution. To be more precise, for an algebra $A\in DGA^+_k$ a map $\phi\colon R\twoheadrightarrow A$ is a cofibrant resolution if $\phi$ is surjective in each degree quasi-isomorphism and $R$ is semi-free. Here $R\in DGA^+_k$ is called semi-free if it's underlying graded algebra $R_\#$ is free.

It's not a big deal to construct some resolution. For example, the cobar-bar adjunction gives a resolution $\Omega B(A)\twoheadrightarrow A$. But this resolution is "huge". So the problem is how to construct resolutions that are "small".

I know a couple of examples. For exapmle, the (commutative) polynomial algebra $A=k[x,y]$ has a resolution in $DGA^+_k$ of the form $R=k\langle x,y,t\rangle$, where $\deg x=\deg y=0$, $\deg t=1$ and $d(t)=xy-yx=[x,y]$.

Algebra $A=k[x,y,z]$ has resolution of the form $R=\langle x,y,z;\xi,\lambda,\theta;t\rangle$ with $\deg x,y,z=0$, $\deg \xi,\lambda,\theta=1$ and $\deg t =2$. The differential is defined by $d(\xi)=[y,z],d(\lambda)=[x,y],d(\theta)=[z,x]$ and $d(t)=[x,\xi]+[y,\theta]+[z,\lambda]$.

So my questions are the following.

1) Do you know any other nice examples of algebras with simple resolutions? What are resolutions of symmetric algebras, matrix algebras?

2) How can you come up with such nice resolutions? I understand that probably there is no general recipe, but maybe there are some techniques, or hints, how to do that.

I hope this question is appropriate to ask here.

Thanks a lot for your help!

Answer 1

One can try to reduce the computation to another easier computation in an abelian setting (e.g. Hochschild (co)homology). This is described in many places including Quillen's paper: On the (co-)homology of commutative rings.

In the abelian setting one often uses Koszul resolutions to compute $\mathrm{Tor}$ and $\mathrm{Ext}$ groups. For symmetric algebras $A$ on $n$-generators $x_i$, one tensors with an exterior algebra on $n$-generators $\sigma x_i$ to obtain a resolution of the base field. Tensoring this resolution on the left side with $A$ you will get a resolution of the $A$ as a module over the enveloping algebra $A\otimes A^\mathrm{op}$ (where on one side the action is trivial). Pulling back along the automorphism of $A\otimes A^{\mathrm{op}}$ defined by \begin{align*} x_i\otimes 1\mapsto & x_i\otimes 1\\ 1\otimes x_i \mapsto & x_i \otimes 1-1\otimes x_i \end{align*} takes $A$ as an interesting bimodule structure to one that is trivial on the right. For matrix algebras one can use the Morita invariance of Hochschild homology to reduce to the underlying algebra.

But more generically one can compute Andr\'e-Quillen (co)homology essentially the same way one computes (co)homology of CW-complexes. One just needs to use the dictionary: \begin{align*} S^n \leftrightarrow &k\langle x \rangle, |x|=n, dx=0 \\ D^{n+1} \leftrightarrow & k\langle x,\sigma(x)\rangle, |x|=n, |\sigma(x)|=n+1, dx=0, d\sigma(x) = x \end{align*}

You start by building your complex by picking a surjection from a free algebra and then 'coning' off relations via a pushout diagram. Unfortunately pushouts are generally big in the non-commutative setting.

Choosing null homotopies in the DGA you are resolving will give you an induced map from the pushout. Now just repeat the coning process.

From this inductive procedure will obtain a sequence of pushout diagrams of cofibrant objects in your category. After applying Andr\'e-Quillen (co)homology each of these pushout squares becomes a long-exact sequence where one of the terms we know by induction and the other term comes from our 'spheres' and have simple (co)homology. You can see how this is applied in the examples you mentioned.

Answer 2

There is a way to approach it going back to one of the most efficient general computational methods for associative algebras. Namely, if you know a "Groebner basis" of your algebra (a confluent presentation via generators and relations, if that sounds less technical?) then you can build a resolution out of it, sort of. This approach is explained in (sorry for this self-advertising) in the full generality of operads here, and a preprint predecessor of that paper which discusses first as a toy example the case of associative algebras (maybe preferable for you) is here.