prob15 - Problem 0.1 Let X be a scheme of fintie type over...

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Unformatted text preview: Problem 0.1. Let X be a scheme of fintie type over a field k (not necessarily algebraically closed). (a) Show that the following three conditions are equivalent (in which case we say that X is geometrically irreducible ). (i) X × k ¯ k is irreducible, where ¯ k denotes the algebraic closure of k . (By abuse of notation, we write X × k ¯ k to denote X × Spec k Spec ¯ k . (ii) X × k k s is irreducible, hwere k s denotes the separable clo- sure of k . (iii) X × k K is irreducible for every extension field K of k . (b) Show that the following three conditions are equivalent (in which case we say X is geometrically reduced . (i) X × k ¯ k is reduced. (ii) X × k k p is reduced, where k p denotes the perfect closure of k . (iii) X × k K is reduced for all extension fields K of k . (c) We say that X is geometrically integral provided that X × k ¯ k is integral. Give examples of integral schemes which are neither geometrically irreducible nor geometrically reduced. Proof. Part a. Lemma 0.2. Suppose that X is a topological space and that { U i } is an open cover by pairwise intersecting irreducible subspaces. Then X is irreducible. Proof. Suppose that V ⊆ X is open and nonempty. Then I claim that V ∩ U i 6 = ∅ for all i . We know there exists some i for which this is true as V 6 = ∅ and the U i cover. But then for all j we have that U i ∩ U j 6 = ∅ is open in U i . As U i is irreducible, V ∩ ( U i ∩ U i ) 6 = ∅ . Therefore, V ∩ U j 6 = ∅ and this claim is proved. Now I claim that V is dense in X . For suppose that V is another nonempty open subspace of X . We have to show that V ∩ V 6 = ∅ . Fix an i . Then V and V meet U i in open subspaces. But U i is irreducible, so ( V ∩ U i ) ∩ ( V ∩ U i ) = V ∩ V ∩ U i 6 = ∅ . Thus, V ∩ V 6 = ∅ . So X is irreducible as every open subspace is dense. Lemma 0.3. Open immersion is stable under base change....
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This note was uploaded on 05/02/2010 for the course MATH 632 taught by Professor Staff during the Spring '08 term at University of Michigan.

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prob15 - Problem 0.1 Let X be a scheme of fintie type over...

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