An Introduction to Algebraic Geometry and Algebraic Groups - download pdf or read online

By Meinolf Geck

ISBN-10: 019967616X

ISBN-13: 9780199676163

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

Building at the history fabric from algebraic geometry and algebraic teams, the textual content offers 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, an intensive remedy of Frobenius maps on affine forms and algebraic teams, zeta capabilities and Lefschetz numbers for types over finite fields. specialists within the box will get pleasure from a number of the new techniques to classical results.

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

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11 Example Let G ⊆ Mn (k) be an algebraic set which is a (e) subgroup of SLn (k) for some e 1. Then G is a linear algebraic group. (a) Let Un (k) be the set of all upper unitriangular matrices in Mn (k), that is, we have A = (aij ) ∈ Un (k) if and only if aii = 1 for all i and aij = 0 for all i > j. Then Un (k) is an algebraic subset of Mn (k) and a subgroup of SLn (k). Thus, Un (k) is a linear algebraic group. As an algebraic set, we have Un (k) ∼ = kn(n−1)/2 and so Un (k) is irreducible of dimension n(n − 1)/2 (if |k| = ∞).

M} such that dim(V × W ) = |I| + |J | and k[Xi , Yj | i ∈ I, j ∈ J ]∩I(V ×W ) = {0}. Using (b), this implies k[Xi | i ∈ I]∩I(V ) = {0} and k[Yj | j ∈ J ]∩I(W ) = {0} and so dim(V ×W ) dim V +dim W . Conversely, let I ⊆ {1, . . , 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}.

Dp (fn−d )}). We shall assume that the labelling of the variables is chosen 44 Algebraic sets and algebraic groups such that det Dj (fi )(p) 1 i,j n−d = 0. (1) Now consider the matrix (where i is the row index and j is the column index) Mq := Dj (fi )(q) 1 i 1 j n−d n where q ∈ V . Consider the elements of kn as column vectors and let v ∈ kn , with components v1 , . . , vn . Then the ith component of the product Mq v ∈ kn is given by n Dj (fi )(q) vj = dq (fi )(v) (Mq v)i = where q ∈ V . (2) j=1 Our assumption (1) means that, if we write Mq = Aq Bq with Aq of size (n−d)×(n−d) and Bq of size (n−d)×d, then det Ap = 0.

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An Introduction to Algebraic Geometry and Algebraic Groups by Meinolf Geck


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