The reductive Borel–Serre compactification as a model for unstable algebraic K-theory

Abstract

Abstract Let A be an associative ring and M a finitely generated projective A -module. We introduce a category $${\text {RBS}}(M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>RBS</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and prove several theorems which show that its geometric realisation functions as a well-behaved unstable algebraic K-theory space. These categories $${\text {RBS}}(M)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>RBS</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>M</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> naturally arise as generalisations of the exit path $$\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>∞</mml:mi> </mml:math> -category of the reductive Borel–Serre compactification of a locally symmetric space, and one of our main techniques is to find purely categorical analogues of some familiar structures in these compactifications.

Locations

• Selecta Mathematica - View - PDF