By Meinolf Geck

ISBN-10: 019967616X

ISBN-13: 9780199676163

An obtainable textual content introducing algebraic geometries and algebraic teams at complex undergraduate and early graduate point, this ebook develops the language of algebraic geometry from scratch and makes use of it to establish the idea of affine algebraic teams from first principles.

Building at the heritage fabric from algebraic geometry and algebraic teams, the textual content presents an creation to extra complicated and specialized fabric. An instance is the illustration thought of finite teams of Lie type.

The textual content covers the conjugacy of Borel subgroups and maximal tori, the idea of algebraic teams with a BN-pair, a radical remedy of Frobenius maps on affine types and algebraic teams, zeta services and Lefschetz numbers for forms over finite fields. specialists within the box will get pleasure from a number of the new ways to classical results.

The textual content makes use of algebraic teams because the major examples, together with labored out examples, instructive routines, in addition to bibliographical and historic comments.

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Extra info for An Introduction to Algebraic Geometry and Algebraic Groups

Example text

Xn ], we have g(x) = f(x) for all x ∈ V if and only if g − f ∈ I(V ). Thus, ϕ uniquely determines an element of A[V ] = k[X1 , . . , Xn ]/I(V ). Conversely, for any residue class f¯ = f + I(V ) ∈ A[V ], we obtain a well-defined regular map ϕ : V → k such that ϕ(x) = f(x) for all x∈V. This discussion shows that A[V ] may also be regarded as the set of all regular maps V → k. For f ∈ k[X1 , . . , Xn ], we will usually identify the function f˙ : V → k with f¯ ∈ A[V ] and call A[V ] the algebra of regular functions on V .

Then we have dp (f1 ) = −2xX1 +X2 and dp (f2 ) = −3x2 X1 + X3 , and so Tp (C) = (1, 2x, 3x2 ) k ⊆ k3 . Thus, Tp (C) is a one-dimensional subspace for each p ∈ C. 11 Theorem Assume that k is an infinite perfect field, and let V ⊆ kn be an irreducible (and, hence, non-empty) algebraic set. (a) We have dim Tp (V ) dim V for all p ∈ V . Furthermore, the set of all p ∈ V with dim Tp (V ) = dim V is non-empty and open. (b) Let p ∈ V be such that dim Tp (V ) = dim V . Then there exists some f ∈ k[X1 , .

N} be such that dim V = |I| and k[Xi | i ∈ I] ∩ I(V ) = {0}. Furthermore, let J ⊆ {1, . . , m} be such that dim W = |J | and k[Yj | j ∈ J ] ∩ I(W ) = {0}. It is easily seen (for example, using an argument analogous to that in the proof of (b)) that then we also have k[Xi , Yj | i ∈ I, j ∈ J ] ∩ I(V × W ) = {0}. 14, we have dim(V × W ) |I| + |J |, as required. Now we are ready to define algebraic monoids and algebraic groups. 9 Definition Consider Mn (k) = kn×n as an algebraic set 2 (under the identification kn×n = kn ).

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An Introduction to Algebraic Geometry and Algebraic Groups by Meinolf Geck

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