By Walter Ferrer Santos, Alvaro Rittatore
This self-contained advent to geometric invariant thought hyperlinks the idea of affine algebraic teams to Mumford's thought. The authors, professors of arithmetic at Universidad de los angeles República, Uruguay, make the most the perspective of Hopf algebra concept and the speculation of comodules to simplify a few of the correct formulation and proofs. Early chapters evaluation necessities in commutative algebra, algebraic geometry, and the speculation of semisimple Lie algebras. insurance then progresses from Jordan decomposition via homogeneous areas and quotients. bankruptcy workouts, and a thesaurus, notations, and effects are incorporated.
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43. Let X be an algebraic k–variety and assume that Y ⊂ X is a closed subset endowed with the induced topology. 42, Y is said to be a closed subvariety of X, or simply a subvariety. 44. 37 and makes of Y a closed subvariety of X. See Exercise 31. 45. Let X, Y, Z be algebraic varieties, and let f : X → Z, g : Y → Z be morphisms. We define the fibered product of X and Y over Z as a triple (X ×Z Y, pX , pY ) where X ×Z Y is an algebraic variety and 40 1. ALGEBRAIC GEOMETRY pX : X ×Z Y → X, pY : X ×Z Y → Y are morphisms of varieties such that f ◦pX = g ◦pY , and satisfying the following universal property: For an arbitrary triple (W, q1 , q2 ) with W an algebraic variety and q1 : W → X and q2 : W → Y morphisms such that f ◦q1 = g ◦q2 , there exists a unique morphism h : W → X ×Z Y such that q1 = pX ◦h and q2 = pY ◦h.
10), we have that X = V(f1 , . . , fm ). Next we reverse the above construction and associate to an arbitrary subset of An an ideal in the polynomial ring k[X1 , . . , Xn ]. 3. Let X ⊂ An be a arbitrary subset. Call I(X) = f ∈ k[X1 , . . , Xn ] : f |X = 0 ⊂ k[X1 , . . , Xn ] . Notice that I(X) is an ideal of k[X1 , . . , Xn ]. Below we list — and leave as an exercise for the reader to prove — the basic properties of the maps I and V. See Exercise 5. 4. Consider an algebraically closed field k and the maps V and I defined above.
Fn ∈ k[X], g1 , . . ,gn = (x, y) ∈ X × Y : fi (x)gi (y) = 0 . ,gn is a basis for the topology of X × Y . 23 that (X × Y )P fi ⊗gi is isomorphic to the affine variety Spm k[X] ⊗ k[Y ] P fi ⊗gi . ,gn . 25. Let X be an affine algebraic variety. Then the diagonal map ∆ : X → X × X, ∆(x) = (x, x) is a morphism of affine varieties. Moreover, ∆(X) is closed in X × X. 34 1. ALGEBRAIC GEOMETRY Proof: The composition of a regular function α = fi ⊗gi : X ×X → k with ∆ yields the function α ◦ ∆ = fi gi : X → k.
Actions and Invariants of Algebraic Groups by Walter Ferrer Santos, Alvaro Rittatore