Timeline for When does $M \otimes_{A} \pi_{0}(A) \simeq 0$ imply $M \simeq 0$?
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| Mar 8, 2012 at 22:22 | comment | added | Eric Wofsey | I'm considering this graded ring as (say) a dga with trivial differential. If you want to think of this as a spectrum, there's an equivalence between $A_\infty$-algebra spectra over $H{\mathbb Z}$ and associative dgas, and rationally an equivalence between $E_\infty$-algebras over $H{\mathbb Q}$ and commutative dgas over $\mathbb Q$. There is correspondingly an equivalence between categories of modules over $H{\mathbb Z}$-algebras and derived categories of dg-modules. | |
| Mar 8, 2012 at 22:07 | comment | added | Yosemite Sam | @eric: I am quite the beginner in this area, could you be so kind to clarify the counterexample you gave? What kind of object is A? To me it just seems a graded ring. | |
| Mar 8, 2012 at 20:00 | comment | added | Eric Wofsey | I should add that this argument doesn't require $M$ to be perfect, but does require $A$ to have only finitely many homotopy groups. Tom's argument works for arbitrary (connective) $A$ and perfect $M$ (so that you know $M$ has to have a smallest homotopy group). The statement is not true for arbitrary $M$ and arbitrary connective $A$. For instance, $A$ could be a polynomial ring generated by a class in degree 2 and $M$ could be $A$ with that class inverted. | |
| Mar 8, 2012 at 17:39 | history | answered | Eric Wofsey | CC BY-SA 3.0 |