Timeline for Somewhat general question that includes: "Do quasi-isomorphic cdgas have quasi-isomorphic spaces of derivations?"
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| Dec 20, 2011 at 8:28 | comment | added | Justin Noel | @Theo: Thanks for the positive response. Might I suggest looking at $Der_A(B; B)$. This is the set of derivations of $B$ in your sense. Since this functor also respects homotopy equivalences (covariantly) in the $B$-module you can see that a homotopy equivalence of $A$-algebras $B\simeq C$ induces an isomorphism [ Der_A(B;B)\rightarrow Der_A(B;C)→Der_A(C,C).] | |
| Dec 20, 2011 at 5:39 | comment | added | Theo Johnson-Freyd | cdga is nothing more nor less than a commutative graded algebra with a distinguished degree-$(\pm 1)$ (depending on conventions) self-commuting derivation. So I think this is subtly different than your setting. That said, I entirely agree that for any given module, it makes sense to take derivations into that module, and there will be some functoriality. I will think about what types of homotopies are necessary to use this functoriality to do what I'd like. Thank you for the Quillen reference. | |
| Dec 20, 2011 at 5:35 | comment | added | Theo Johnson-Freyd | Hi Justin, Your clarification is helpful. My impression is that it is quite common in the homotopy literature to work with augmented algebras, but my impression has always been that this is unsatisfying. In particular, when I say "derivation" of a commutative ring $B$, I mean a linear (over whatever ground ring) map $d: B \to B$ such that for $a,b\in B$, we have $d(ab) = da\,b + a\,db$. This is correct, of course, for degree-$0$ maps, and the usual sign rules are standard (and best handled with internal category theory a la Deligne–Morgan, rather than explicit sign rules). For example, a | |
| Dec 4, 2011 at 11:32 | history | edited | Justin Noel | CC BY-SA 3.0 |
Responded to comment.
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| Dec 3, 2011 at 7:58 | comment | added | Theo Johnson-Freyd | I'm probably just being stupid --- is it clear that even if you have maps in both directions, that you can move the homology of Derivations (say) from one to the other? | |
| Dec 1, 2011 at 23:17 | history | answered | Justin Noel | CC BY-SA 3.0 |