Algebraic Geometry

Download e-book for kindle: A geometric introduction to K-theory [Lecture notes] by Daniel Dugger

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

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The bundle L is called the tautological line bundle over RP n . Do not confuse this with the canonical line bundle over RP n that we will define shortly (they are duals of each other). Note that when n = 1 the bundle L is isomorphic to the M¨ obius bundle on S 1 . (c) One may generalize the previous example as follows. Let V be a vector space and fix an integer k > 0. Consider the Grassmannian Grk (V ) of k-planes in V . Let η = {(W, x) | W ∈ Grk (V ), x ∈ W }. A GEOMETRIC INTRODUCTION TO K-THEORY 53 Projection to the first coordinate π : η → Grk (V ) makes η into a rank k vector bundle, called the tautological bundle over Grk (V ).

This is known to be a closed subset of Spec R, and to say that M is supported on Z is just the requirement that Supp M ⊆ Z. Let G(R)Z be the Grothendieck group of all finitely-generated R-modules that are supported on Z. Similarly, if C• is a chain complex of R-modules then Supp C is defined to be {P ∈ Spec R | H∗ (CP ) = 0}. We say that C• is supported on Z if Supp C ⊆ Z, or if CQ is exact for every Q ∈ / Z. Note that C• is supported on Z if and only if all the homology modules H∗ (C) are supported on Z.

Proof. We must again show that if A ∈ GLn (R) has det(A) = 1 then A ∈ E(R). For any fixed row of A, the ideal generated by the elements in that row contains det(A) and is therefore the unit ideal. Pick an element x of smallest degree in this row and then use column operations (and the Euclidean division property) to arrange all other elements in this row to be either zero or have degree smaller than x. By repeating this process, eventually the row will contain a unit. Do a signed transposition to switch this unit into position (1, 1), and then do row operations to clear out all other terms in the first column.

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