Algebraic Geometry [Lecture notes] by Karl-Heinz Fieseler and Ludger Kaup

By Karl-Heinz Fieseler and Ludger Kaup

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Unfortunately, there is in general no cover of X by open affine subspaces, so the resulting object is a priori just a k-ringed space. e. one-dimensional subspace, L ⊂ k n a separate ”point at infinity”, denoted ∞L . , zn ). , zn ] ∈ Pn , they are unique up to a common nonzero multiple λ ∈ k ∗ . , zi zi zi zi . z]. For n = 1 we thus get the ”projective line” P1 = k ∪ {∞} with ∞ := [0, 1]. For k = C the projective line P1 is (with respect to the strong topology) homeomorphic to the 2-sphere S2 , the ”Riemann sphere” of complex analysis.

If, say, g = g˜f we have g˜m h h = fm gm on Xg . 19. The algebra of regular functions O(U ) on a principal open subset U = Xf of an affine variety X is defined as O(U ) := O(X)f . 20. 1. A regular function on U = Xf without zeros is invertible: We may assume that it is of the form h|U with h ∈ O(X). But already the restriction of h to Xh ⊃ Xf is invertible! 2. The restriction O(X) −→ O(Xf ) is an injection if f ∈ O(X) is not a zero divisor. In particular that holds true automatically if O(X) is an integral domain.

1. The morphism ϕ is dominant, iff ϕ∗ : O(Y ) −→ O(X) is injective. 2. The morphism ϕ is a closed embedding, iff ϕ∗ : O(Y ) −→ O(X) is surjective. 3. A finite dominant morphism ϕ : X := Sp(A) −→ Y := Sp(B) is onto (:= surjective) and has finite fibers: For every y ∈ Y we have |ϕ−1 (y)| ≤ r, if the finite B-module A can be generated by r elements. 4. A finite morphism is closed. 36 Proof. The first statement as well as the implication ”=⇒” of the second statement is obvious. For the reversed one take Z := ϕ(X).

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