TITLE: Applications of Descent?
QUESTION [5 upvotes]: The technique of faithfully flat descent, and, in the case of vector spaces, Galois descent has been used quite a bit in Algebraic Geometry. However, the question of whether, say, a given $k$-vector space $V$ arises from some $L$-vector space $W$ seems like it could be asked in a wide variety of settings. I'm wondering, in particular, if anyone has seen descent in modular representation theory.

REPLY [5 votes]: Descent does appear in modular representation theory (of finite groups) in a slightly surprising form. Indeed, descent techniques allow us to discuss extension from $Stab(kH)$ to $Stab(kG)$, where $H$ is a subgroup of $G$, of index prime to the characteristic of the field $k$, and where $Stab(-)$ denotes the stable category of modules modulo projectives. 
To understand how an extension problem turns out to be a descent problem, one should start by the following "monadicity" observation: There exists a commutative separable ring object (i.e. monoid) $A$ in the symmetric monoidal category $C:=Stab(kG)$ such that the category $A-Mod_C$ of $A$-modules in that category $C$ is equivalent to the stable category $Stab(kH)$, and this in such a way that restriction from $G$ to $H$ coincides with extension-of-scalars with respect to $A$.
So restriction to a subgroup is in fact a extension-of-scalars!
The condition that the index of $H$ in $G$ is prime to the characteristic exactly expresses the faithfulness of the ring object $A$. (Flatness is sort of built-in when you deal with tensor triangulated categories.) Hence extension from $H$ to $G$ is really faithful(ly-flat) descent!
One can then reason rather geometrically, pretty much as with (finite) etale extensions. There is even a Grothendieck topology and a stack hiding in the bushes... This is all explained in the preprint
"Stacks of group representations"
available on my publication page.