Exercises 25 Exercises (1) Prove the inclusion √ I ⊂ I(V(I)) for an ideal I of C[X1,...,Xn]. (2) Provide the proof of Proposition 1.1.2. (3) Provide the proof of Proposition 1.1.3. (4) Prove that for a subset V of An C , V = V(I(V )) if and only if V is closed. (5) Determine the open sets in the Zariski topology on A1 C . Prove that this topology is not Hausdorff. (6) Determine the closed sets in A2 C = A1 C × A1 C with respect to the product topology (with A1 C endowed with the Zariski topology) and with respect to the Zariski topology. In particular, obtain that Δ(A1 C ) = {(x, x) ∈ A1 C × A1 C } is a closed set in the latter topology but not in the former. (7) Let V ⊂ An C , W ⊂ Am C be aﬃne varieties. By identifying An C × Am C with An+m, C we can consider the cartesian product V ×W as a subset of An+m. C Prove that V × W is an aﬃne variety and that there is an isomorphism C[V × W ] C[V ] ⊗ C[W ]. Hint: Assign to a pair (g, h) ∈ C[V ] × C[W ] the polynomial function f(x, y) = g(x)h(y) on V × W . This assignment induces a C-algebra morphism C[V ] ⊗ C[W ] → C[V × W ]. (8) Let V ⊂ An C , W ⊂ Am C be closed irreducible sets. Prove that V × W is closed and irreducible in An+m C with respect to the Zariski topology. Hint: If V × W is the union of two closed subsets Z1,Z2, then V = V1 ∪ V2, where Vi := {x ∈ V : {x} × W ⊂ Zi},i = 1, 2, are closed in V . (9) Provide the proof of the statements in Example 1.2.4. (10) Check that an aﬃne variety with its sheaf of regular functions is an abstract aﬃne variety. (11) Prove that A2 \ {(0, 0)} is not an aﬃne variety. (12) Prove that every automorphism of A1 (=isomorphism from A1 into it- self) has the form x → ax + b (a ∈ C∗,b ∈ C). (13) a) Prove that an ideal I of a graded ring R = ⊕d≥0Rd is homogeneous if and only if it is generated by homogeneous elements. Deduce that sums, products, intersections and radical of homogeneous ideals are again ho- mogeneous. b) Prove that if I is a homogeneous ideal of a graded ring R, then the quotient ring R/I is graded in a natural way.

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