What do you guys think about near-rings? Endomorphism near-rings of non-abelian groups? Whoa.

Here's a (probably) basic commutative algebra question: Suppose I have two regular local rings (A,I) and (B,J) of the same dimension along with a homomorphism $\phi:A \to B$. Suppose $I = (x_0,\ldots,x_{n-1})$ and $J = (y_0,\ldots,y_{n-1})$. What conditions do I need on $\phi$ and/or $B$ (as an $A$-module) to ensure that $x_i^{-1}A/(x_0,\ldots,x_{i-1}) \otimes_A B = y_i^{-1} B/(y_0,\ldots,y_{i-1})$?

I should add, I want this to be true for $i=1,\ldots,n-1$

hi sean

:)

Hey y'all, so, is it true that the completion tower for a ring at an ideal is really the tot tower of a cosimplicial ring?

And in general, this is is why, in topology, we define "completions" cosimplicially?

@user101036 I think grothendieck wanted an abelian category as a target for his universal cohomology theory and people still don't have a good t-structure on the triangulated category of mixed motives (not even sure if people currently think one should exist)

hmm okay out of curiosity

what is the best model for K-homology

also what's up with this MSO-orientation for L-theory -- when we say L-theory in this context, is it just the cohomology theory represented by L of a point ? like, one side of the assembly map ?

@ArnavTripathy graeme segal has a lovely one in "K-homology theory and algebraic K-theory"

it's similar to the theorem "homology is homotopy groups of the infinite symmetric product"

the infinite symmetric product has points are formal sums of points with coefficients in N (mod the basepoint)

if two labelled points collide, then you "add" these perpendicular subspaces together by taking their common span

unfortunately that only gets you connective K-homology, so you've got to formally impose Bott periodicity if you want the periodic one

+1 graeme segal, that's great

@Jon: completions are cosimplicial because they're limits (e.g. think of the limit defining I-adic completion) and totalizing cosimplicial objects computes limits. dually, realizing simplicial objects computes colimits