An Introduction to Algebraic Geometry and Algebraic Groups by Meinolf Geck

By Meinolf Geck

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 idea of affine algebraic teams from first principles.

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

The textual content covers the conjugacy of Borel subgroups and maximal tori, the speculation of algebraic teams with a BN-pair, an intensive therapy of Frobenius maps on affine types and algebraic teams, zeta features and Lefschetz numbers for forms over finite fields. specialists within the box will take pleasure in a number of the new methods 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 comments.

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D (˜ x) k ⊆ kn for all x ∈ Vf . Let Ψ(x) be the n × d-matrix whose columns are given by ψ1 (˜ x), . . , ψd (˜ x). 13. Consequently, the image of Tx (V ) under dx ϕ is the k-span of the columns of Jx (ϕ)Ψ(x). As usual, the condition that the rank of Jx (ϕ)Ψ(x) be dim W can be expressed by the non-vanishing of certain determinants in the coordinates of x. But, for p ∈ V , we know that this condition is satisfied. Hence, the set of all x ∈ Vf for which Jx (ϕ)Ψ(x) has rank dim W is open. This yields the desired set U3 .

Then ρx (X) = Xx is an irreducible component containing 1 and so Xx = G◦ . Hence X is equal to the coset G◦ x−1 . Thus, the cosets of G◦ are precisely the irreducible components of G. Consequently, G◦ has finite index in G. Furthermore, for any x ∈ G, the two cosets G◦ x = ρx (G◦ ) and xG◦ = λx (G◦ ) are irreducible components with non-empty intersection; hence they must be equal. This shows that G◦ is a normal subgroup. (b) Let H ⊆ G be any closed subgroup of finite index. Let g1 , . . , gr ∈ G (with g1 = 1) be such that G is the disjoint union 32 Algebraic sets and algebraic groups r i=1 Hgi .

Zd are algebraically independent. So we have ker(α) = (f) as claimed. On the other hand, the image of α is just R. Hence we have R∼ = k[Y1 , . . , Yd+1 ]/(f). 4. 14. So we have dimK TR,K = d and (∗) yields (a). Finally, (b) follows from (a) and by applying (∗) to R = A. 6 Corollary Assume that k is a perfect field. Let V ⊆ kn be an irreducible algebraic set and assume that I(V ) ⊆ k[X1 , . . , Xn ] is generated by f1 , . . , fm . Then we have dim V = n − rank Di (fj ) 1 i n 1 j m , where Di denotes the partial derivative with respect to Xi and the bar denotes the canonical map k[X1 , .

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