An Introduction to Algebraic Geometry and Algebraic Groups by Meinolf Geck

By Meinolf Geck

An available textual content introducing algebraic geometries and algebraic teams at complicated undergraduate and early graduate point, this publication 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 historical past fabric from algebraic geometry and algebraic teams, the textual content offers an advent to extra complex 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, a radical remedy of Frobenius maps on affine types and algebraic teams, zeta services and Lefschetz numbers for types over finite fields. specialists within the box will take pleasure in many 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 routines, in addition to bibliographical and historic feedback.

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N ) ∈ Zn 0 , we have αj = 0 for j ∈ J , and so γ + α ∈ C(J, τ ) ⊆ C(I) for any γ ∈ C(J, τ ). On the other hand, we have aX γ+α = X γ LT(f) ∈ (LT(I)), and so X γ+α is divisible by X β for some β ∈ M , contradicting γ +α ∈ C(I). Thus, we have deg a HPI (t) max 0 d n there exist 1 i1 < · · · < id n . such that I ∩ k[Xi1 , . . , Xid ] = {0} Conversely, if 1 i1 < · · · < id n are such that I∩ k[Xi1 , . . , Xid ] = {0}, then the natural map π : k[Xi1 , . . , Xid ] → k[X1 , . . , Xn ]/I induced by the inclusion k[Xi1 , .

Xn ]/I ∼ = k, we conclude that I is maximal and that I(V ) = I. So we have k[X1 , . . , Xn ]/I(V ) ∼ = k and, hence, a HFI(V ) (s) = 1 for all s 1. Thus, we have dim{v} = 0. 17, it follows that dim V = 0 whenever |V | < ∞. 4 of Cox et al. (1992). (b) Consider the twisted cubic C = V(X2 − X12 , X3 − X13 ) ⊆ 3 k and assume that |k| = ∞. 4 shows that k[X1 , X2 , X3 ]/I(C) ∼ = k[Y1 ], where Y1 is an indeterminate. Thus, we conclude that dim C = 1. An irreducible algebraic set of dimension 1 is called an affine curve; we have just seen that the twisted cubic is an affine curve.

Ym ] and ϕ is given by the polynomials f1 , . . , fm ∈ k[X1 , . . , Xn ], then g¯ ◦ ϕ is given by the polynomial g(f1 , . . , fm ), obtained by substituting Yi → fi for 1 i m. The assignment ϕ → ϕ∗ is (contravariant) functorial, in the following sense. If ϕ : V → W and ψ : W → Z are regular (where Z ⊆ kl is algebraic), then we have (ψ ◦ ϕ)∗ = ϕ∗ ◦ ψ∗ . Furthermore, for V = W , we have id∗V = idA[V ] . 4 Proposition Let V ⊆ kn and W ⊆ km be non-empty algebraic sets. Then the assignment ϕ → ϕ∗ defines a bijection ∼ {regular maps V → W } −→ {k-algebra homomorphisms A[W ] → A[V ]}.

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