Timeline for What is the Grothendieck group of the category of $\mathbf{Z}_p[G]$-modules?

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Jun 1, 2014 at 1:21	vote	accept	ya-tayr
May 31, 2014 at 15:34	comment	added	Johannes Hahn		Note that surprinsingly many modules are zero in the Grothendieck group of $\mathcal{O}[G]$ because of exakt sequences like $0 \to M \xrightarrow{p} M \to M/pM \to 0$ for any $M$ which is free as an $\mathcal{O}$-module. In particular: The projective $\mathbb{F}_q[G]$-modules (considered via the canonical map as $\mathcal{O}[G]$-modules) are zero in the Grothendieck group because they lift to projective and hence torsionfree $\mathcal{O}[G]$-modules.
May 31, 2014 at 14:59	history	edited	Jim Humphreys		edited tags
May 31, 2014 at 14:58	comment	added	Jim Humphreys		I've added a tag, since you assume $G$ is finite. Note that the irreducible $p$-adic representations of an arbitrary $G$ are poorly understood, so it's not clear how far the full Grothendieck group can be characterized. Also, the standard texts of Serre and Curtis-Reiner emphasize (following Brauer) the "intermediate" role of the Grothendieck group of f.g. projective modules instead: projectives for $\mathbb{F}_q [G]$ lift nicely to projective $\mathcal{O}[G]$-modules. What is your motivation?
May 31, 2014 at 10:45	answer	added	user26223		timeline score: 5
May 31, 2014 at 4:23	review	First posts
May 31, 2014 at 5:12
May 31, 2014 at 4:05	history	asked	ya-tayr	CC BY-SA 3.0