By Ivan Arzhantsev
Lecture notes from LMU Munich (?) (version 26 Jul 2012)
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Extra info for Higher Algebra [lecture notes]
Sample text
A geometric interpretation of Noether’s Normalization. For any algebraic subset X ⊆ An there exists a linear subspace Ad ⊆ An and a linear projection φ : An → Ad such that the restriction φ|X : X → Ad is a finite morphism; in particular, it is a surjection with finite fibers. 53. An irreducible affine variety X is normal if the algebra K[X] is integrally closed. 54. The affine space An is normal. The following construction allows to reduce many problems on affine varieties to the case of normal varieties.
This defines a structure of a (commutative associative) K-algebra on D. 7. There is an isomorphism of K-algebras ∼ K[x1 , . . , xn , y1 , . . , ym ], K[x1 , . . , xn ] ⊗K K[y1 , . . , ym ] = jm jm xi11 . . xinn ⊗ y1j1 . . ym ↔ xi11 . . xinn y1j1 . . ym . More generally, for any K-algebra A we have a natural isomorphism of K-algebras ∼ A[y1 , . . , ym ]. A ⊗K K[y1 , . . 1. Consider the following homomorphism of K-algebras ψ : K[X] ⊗K K[Y ] → K[X × Y ], (ψ( Fi ⊗ Gi ))(x, y) := i Fi (x)Gi (y).
2) A prime divisor in X is a closed irreducible subvariety of codimension 1 (=an irreducible hypersurface). 3) A complete intersection in X is a closed subvariety Y I(Y ) = (h1 , . . , hk ) ⊆ K[X] and dim Y = dim X − k. such that 4) A set-theoretic complete intersection in X Y = Z(f1 , . . , fk ) ⊆ X such that dim Y = dim X − k. subvariety is a closed Let us give a topological description of dimension in terms of flags of irreducible subvarieties. 26. Let X be an irreducible Noetherian topological space.
Higher Algebra [lecture notes] by Ivan Arzhantsev
by Ronald
4.1