**Date & Time:** November 09, 2011, 16:00-17:00.

**Venue:** Ramanujan Hall

**Title:** Multiple relative commutator formula

**Speaker:** Prof. Alexei Stepanov, University Of Illinois, At Urbana-Champaign

**Abstract:**
MULTIPLE RELATIVE COMMUTATOR FORMULA
ALEXEI STEPANOV

Let R be a commutative ring and n 3. The starting point of our research is a celebrated Suslin's theorem asserted that the elementary subgroup is normal in the general linear group of order n over R. After that Vaserstein and independently Borewich and Vavilov proved the standard commutator formulae:

[GLn(R); En(R; I)] = En(R; I) [En(R);GLn(R; I)] = En(R; I)

for any I of R. A common generalisation of both formulae above is a relative commutator formula [En(R; I);GLn(R; J)] = [En(R; I); En(R; J)]

where I; J are ideals in R (it is easy to show that [En(R; I); En(R)] = En(R; I)). If I+J = R, then [En(R; I); En(R; J)] = En(R; IJ), otherwise there are counterexamples to this equation (A.Mason). The subject of the talk is the proof of multiple relative commutator formula

[En(R; I1);GLn(R; I2); : : : ;GLn(R; Im)] = [En(R; I1); En(R; I2); : : : ;En(R; Im)] = [En(R; I1I2 : : : Imô€€€1); En(R; Im)]

which generalises all of the above. It was recently proved by R.Hazrat and Z.Zhang using \yoga of commutators".

I shall show the proof of the multiple relative commutator formula without any calculations. The proof is based on the Quillen{Suslin localisation method and one of the main concern of the talk is a statement of the localisation principle in general settings, in categorical terms.