By Daniel Dugger

Show description

Read or Download A geometric introduction to K-theory [Lecture notes] PDF

Similar algebraic geometry books

Geometric Models for Noncommutative Algebra

The quantity relies on a direction, "Geometric types for Noncommutative Algebras" taught by means of Professor Weinstein at Berkeley. Noncommutative geometry is the learn of noncommutative algebras as though they have been algebras of services on areas, for instance, the commutative algebras linked to affine algebraic kinds, differentiable manifolds, topological areas, and degree areas.

Arrangements, local systems and singularities: CIMPA Summer School, Istanbul, 2007

This quantity includes the Lecture Notes of the CIMPA/TUBITAK summer season university preparations, neighborhood platforms and Singularities held at Galatasaray collage, Istanbul in the course of June 2007. the quantity is meant for a wide viewers in natural arithmetic, together with researchers and graduate scholars operating in algebraic geometry, singularity conception, topology and similar fields.

Algebraic Functions and Projective Curves

This booklet presents a self-contained exposition of the speculation of algebraic curves with out requiring any of the necessities of recent algebraic geometry. The self-contained remedy makes this significant and mathematically important topic available to non-specialists. even as, experts within the box could be to find a number of strange subject matters.

Riemannsche Flächen

Das vorliegende Buch beruht auf Vorlesungen und Seminaren für Studenten mittlerer und höherer Semester im Anschluß an eine Einführung in die komplexe Funktionentheorie. Die Theorie Riemannscher Flächen wird als ein Mikrokosmos der Reinen Mathematik dargestellt, in dem Methoden der Topologie und Geometrie, der komplexen und reellen research sowie der Algebra zusammenwirken, um die reichhaltige Struktur dieser Flächen aufzuklären und an vielen Beispielen und Bildern zu erläutern, die in der historischen Entwicklung eine Rolle spielten.

Extra resources for A geometric introduction to K-theory [Lecture notes]

Example text

48 DANIEL DUGGER We know K0 (D, S) ∼ = G(M | S −1 M = 0). The condition S −1 M = 0 just says that M is a torsion module. Consider the evident map G(D/P ) → G(M | S −1 M = 0) j: P =0 where the direct sum is over all nonzero prime ideals and where the map just forgets that a module is defined over D/P and instead regards it as a D-module. 15 and its proof). Note that each D/P is a field, and so G(D/P ) ∼ = Z. If M is a torsion D-module then MP is a torsion DP -module. Since DP is a discrete valuation ring, this means that MP has finite length.

This function is additive, and in fact it is clearly the universal additive invariant for bounded complexes of finitely-generated projectives. The usual Euler characteristic is χ(P• ) = χt (P• )|t=−1 . 5) χt (ΣP• ) = t · χt (P• ). If we differentiate χt with respect to t then we obtain χt (P• ) = jtj−1 [Pj ]. Clearly this is also an additive invariant of complexes. The invariant we called χ is just χt (P• )|t=−1 . 6) χt (ΣP• ) = χt (P• ) + t · χt (P• ), and consequently χ (ΣP• ) = χ(P• ) − χ (P• ).

Given a map β : (S −1 R)n → (S −1 R)n , choose a u ∈ S such that the standard matrix representing uβ has entries in R. Consider the assignment uβ u β → F (β, u) = [Rn −→ Rn ] − n[R −→ R] ∈ K0 (R, S). 16). It remains to show that the above formula does indeed define a homomorphism. We first show that F (β, u) does not depend on the choice of u. It suffices to prove that F (β, tu) = F (β, u) for any t ∈ S; for if u is another choice for u then 46 DANIEL DUGGER we would have F (β, u) = F (β, u u) = F (β, u ).

Download PDF sample

Rated 4.31 of 5 – based on 32 votes