Let $R$ be a ring and $R-\operatorname{Mod}$
the category of $R$-modules. The tensor product
functor $- \otimes_R -: (R-\operatorname{Mod}) \times
(R-\operatorname{Mod}) \to R-\operatorname{Mod}-R$
has an analogon in derived category of
the category of $R$-modules
$- \otimes^L -: D(R-\operatorname{Mod}) \times
D(R-\operatorname{Mod}) \to R-D(\operatorname{Mod}-R)$
We write suggestively for two $R$–
modules $M, N$ which represents certain
classes $[M], [N]$ in derived category
$$ \operatorname{Tor}^L_i (M, N)=
H_i(M \otimes_R N) $$
to remind that we can calculate
the $i$-th grade or representant of
$ \operatorname{Tor}^L_i (M, N)$
by taking an projective resolution
$P^{\bullet} \to M$ (resp. injective resolution
$N \to I^{\bullet}$), tensor the proj resolution
with $N$ (resp tensor the inj resolution with
$M$) and calculate $i$-th homology.
Now on this wiki site is clamed a statement which I don't understand. It is stated in terms of homotopy groups (!) that
$$ \pi_i(M \otimes_R^L N) = \operatorname{Tor}^L_i (M, N) $$
By defition $\operatorname{Tor}^L_i$ is just the notation for $i$-th derived part $R^i(- \otimes^L -)$ of the tensor functor in derived context. And I'm not sure which meaning do have here the
homotopy groups $\pi_i(-)$ and why the claim before
is true.
I guess that the homotopy groups $\pi_i(-)$ are involved
here since the derived category
$D(R-\operatorname{Mod})$ has natural structure of a
triangulated category and therefore
we can associate to it a homotopy category
$Ho(D(R-\operatorname{Mod}))$ where introducing
homotopy groups make sense.
This category contains also the notation of
$n$-sphere analoga and therefore formally
homotopy groups $\pi_i(X)$ make sense for every
$X \in D(R-\operatorname{Mod})$.
But nevertheless I not understand why after passing to homotopy
category of $D(\operatorname{Mod}-R)$ the identity
$$ \pi_i(M \otimes^L N) = \operatorname{Tor}^L_i (M, N) $$
holds?
Does it more generally for every derived functor
$F: D(R-\operatorname{Mod}) \to D(R-\operatorname{Mod})$
and every $X \in D(R-\operatorname{Mod})$ hold
$$ \pi_i(F(X)) = R^iF(X)? $$
If yes, how to show that?
Best Answer
As mentioned in the comments, this is due to the Stable Dold-Kan correspondence. A core principle of stable homotopy theory is that it contains the usual theory of rings and modules. If $R$ is a commutative ring, then the Eilenberg-Mac Lane spectrum $HR$ is a commutative ring spectrum. One one hand, you can define the homotopy category $\mathcal{D}(R)$ of derived $R$-modules in which quasi-isomorphisms become isomorphisms. On the other hand, you can define the homotopy category $\mathsf{Mod}_{HR}$ of modules over $HR$ in spectra, in which stable equivalences are isomorphisms.
Now the homotopy categories above are equivalent and homotopy groups correspond to homology groups. This is really a stabilization of the usual Dold-Kan correspondence. The stabilization of simplicial $R$-modules naturally provides a model for $HR$-modules and the stabilization of non-negative chain complexes provides a model for $\mathcal{D}(R)$.
Even better, the equivalence is symmetric monoidal at the level of homotopy categories endowed with their respective derived tensor product. This shows that algebras and modules in either categories correspond (without coherence data).
Schwede-Shipley formalized the equivalence at the level of model categories (being Quillen equivalent is something stronger than just having equivalent homotopy categories), see Theorem 5.1.6. Shipley then provides an explicit zig-zag of Quillen equivalences between the model categories and show they are weakly monoidally equivalent. This allows to show that the model categories of algebras and modules are equivalent, see here. You already encounter troubles for commutative algebras and prefer to consider $\mathbb{E}_\infty$-algebras (commutative and associative up to higher homotopies).
Even better, Lurie transfer the arguments of Schwede-Shipley and show that the corresponding $\infty$-categories between $HR$-modules and derived $R$-modules are equivalent as symmetric monoidal $\infty$-categories, see 7.1.2.13. This is a stronger result as before as now we can say that $\mathcal{O}$-algebras (and also their modules) correspond. Here $\mathcal{O}$ is (any!) $\infty$-operad. You can again transfer the arguments of Shipley in $\infty$-categorical settings to make the equivalence concrete on objects, see Corollary 4.4.