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 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 idea 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 therapy of Frobenius maps on affine types and algebraic teams, zeta capabilities and Lefschetz numbers for kinds over finite fields. specialists within the box will get pleasure from the various new ways 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 historic comments.
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Extra resources for An Introduction to Algebraic Geometry and Algebraic Groups
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 , . . , Xn ] → A[V ]; the rank is taken over the ﬁeld of fractions of A[V ]. 5. The above results were developed into a purely algebraic setting. We now translate them into geometric properties of algebraic sets. 7 Taylor expansion Let V ⊆ kn be a non-empty algebraic set with vanishing ideal I(V ) ⊆ k[X1 , . . , Xn ]. Let us ﬁx a point p = (p1 , .
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 .
2 that this phenomenon can not occur when we consider algebraic sets in projective space. 7 Direct products of algebraic sets Let V ⊆ kn and W ⊆ km be (non-empty) algebraic sets, deﬁned by sets S ⊆ k[X1 , . . , Xn ] and T ⊆ k[Y1 , . . , Ym ], respectively. Identifying kn × km with kn+m and k[X1 , . . , Xn ], k[Y1 , . . , Ym ] with subrings of Regular maps, direct products, and algebraic groups 27 k[X1 , . . , Xn , Y1 , . . , Ym ], we see that V ×W = (v, w) ∈ kn+m f(v) = 0 for all f ∈ S, g(w) = 0 for all g ∈ T = V(S ∪ T ) ⊆ kn+m , where we use the identities f(v) = f(v, w) for all f ∈ S and g(w) = g(v, w) = 0 for all g ∈ T .