Somewhat general question that includes: "Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?" - MathOverflow

Somewhat general question that includes: "Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?"

2011-11-28T06:42:57Z

Question: Given two quasi-isomorphic dg commutative algebras (over a field of characteristic zero, if you like), to what extent do their various homological geometric data agree?

Example: Given a dg commutative algebra $A$, there is a dg Lie algebra $\operatorname{Der}(A)$ defined by understanding the notion of "derivation" internal to the category of dg vector spaces. If $A$ and $B$ are quasi-isomorphic dg commutative algebras, are $\operatorname{Der}(A)$ and $\operatorname{Der}(B)$ quasi-isomorphic dg Lie algebras? Note that any homomorphism $f: A \to B$ defines a (dg) vector space of "derivations relative to $f$", which is a bimodule for the Lie algebras $\operatorname{Der}(A)$ and $\operatorname{Der}(B)$ and receives maps as one-sided modules from each of these; one would expect these maps to be quasi-isomorphisms if $f$ is.

Remark: I intend my question to be somewhat open ended. As such, I would accept an answer that points me to the appropriate literature.

Answer by Vitali Kapovitch for Somewhat general question that includes: "Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?"

2011-11-28T14:25:24Z

I don't know the full answer but it's not true that if $A$ and $B$ are quasi-isomorphic then so are $Der(A)$ and $Der(B)$. The following is over $\mathbb Q$. The simplest counterexample is the minimal model of $S^2$. $S^2$ is formal so its minimal model is quasi-isomorphic to the one for its cohomology. Note that $H^*(S^2)$ has no odd derivations being evenly graded. Yet, its minimal model is $M= (\Lambda (x,y), d)$ with $ deg (x) =2, deg (y)= 3$ and $dx=0, dy=x^2$. You can directly see that $H_{odd}(Der(M))$ is not zero. Specifically, there is a closed non cohomologous to zero derivation $\theta$ of degree $-3$ with $\theta(x)=0, \theta(y)=1$. In general there is a natural map $H(Der(M))\to Der(H(M))$ which is onto for formal spaces but it need not be injective. This map looks like it should be the edge homomorphism in some spectral sequence as it commutes two functors (which is common for some natural spectral sequences like the Eilenberg-Moore spectral sequence for diff Tor) but I don't know what that spectral sequence should be. It is however true that for positively elliptic spaces such as $S^2$ (they are all formal) one has that $H_i(Der(M))\to Der_i(H(M))$ is an isomorphism for negative even $i$. Here a space is called (rationally) elliptic if it has finite dimensional total rational cohomology and homotopy. Basic examples are homogeneous spaces and fiber bundles built out of them. A space is called positively elliptic if it's elliptic and has positive Euler characteristic (e.g. $S^2$). This is equivalent to saying that its homotopy Euler characteristic is zero, i.e. the total rank of all even homotopy groups is the same as the total rank of odd homotopy groups. See the book by [Félix, Halperin and Thomas on the basics of elliptic spaces.][1] [1]: http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=CNO&s1=1802847&loc=fromrevtext

Answer by Justin Noel for Somewhat general question that includes: "Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?"

2011-12-01T23:17:18Z

If you have two cdgas which are cofibrant (so built out of free cdgas and their cones via an iterated sequence of pushouts) and quasi-isomorphic then their homological invariants agree (one of the properties of cofibrant models is that any quasi-isomorphism between them admits a homotopy inverse). Vitali's example illustrates the problem in general: the minimal model for $H^* (S^2)$ is cofibrant, but $H^* (S^2)$ with trivial differential is not.

To clarify things a bit further, as Tom pointed out in the comments above, when we talk about derivations we usually have a map $A\rightarrow B$ of simplicial commutative rings (in characteristic zero we can use commutative dgas) and a $B$-module $M$. Since you didn't mention this data I assumed we were taking $A$ to be the unnamed field of characteristic zero and $B$ to be augmented over $A$ so that we could take the module $M$ to be $A$ (I think this is a common situation). Now $Der_A (B;A)$ is contravariantly functorial in $B$ as an augmented $A$ algebra and takes homotopic maps to the same map. So if I have maps of augmented $A$ algebras $B\rightarrow C\rightarrow B\rightarrow C$ such that the composite of each two maps is homotopic to the identity (such as when we have a quasi-isomorphism between two cofibrant $A$-algebras) I can apply $Der_A (-,A)$ to the sequence and obtain an isomorphism between $Der_A(B,A)$ and $Der_A(C,A)$.

You might find it helpful to read Quillen's 1970 paper: On the (co-)homology of commutative rings. These ideas are explained and generalized there.