Presented By: Student Commutative Algebra Seminar - Department of Mathematics
From K_0 to higher algebraic K-theory
Gahl Shemy
Originating from the German word “klasse” (class), both topological and algebraic K-theory operates
under the philosophy that studying certain isomorphism classes of objects over a space/ scheme/ ring/
category is a good way to study that space (/scheme/ ring / category). In the case of topological K-theory,
we’ll consider isomorphism classes of vector bundles over a space X, whereas in algebraic
K-theory we’ll look at finitely generated projective modules over a ring R. The Serre-Swan theorem will allow
us to reconcile these stories on the level of K_0. We’ll finish the talk with an algebraic description of K_1(R)
and K_2(R), as well as Quillen’s “+” construction for defining higher algebraic K-groups.
under the philosophy that studying certain isomorphism classes of objects over a space/ scheme/ ring/
category is a good way to study that space (/scheme/ ring / category). In the case of topological K-theory,
we’ll consider isomorphism classes of vector bundles over a space X, whereas in algebraic
K-theory we’ll look at finitely generated projective modules over a ring R. The Serre-Swan theorem will allow
us to reconcile these stories on the level of K_0. We’ll finish the talk with an algebraic description of K_1(R)
and K_2(R), as well as Quillen’s “+” construction for defining higher algebraic K-groups.
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