ALGEBRAIC GEOMETRY (PART III) EXAMPLE SHEET 3 CAUCHER BIRKAR (1) Let X be a scheme and Z a closed subset of X. Show that there is a closed immersion f : Y ! X which is a homeomorphism of Y onto Z. (2) Show that a scheme X is affine iff there are b1; : : : ; bn 2 OX (X) such that each D(bi) is affine and the ideal generated by all the bi is OX (X) (see [Hartshorne, II, exercise 2.17]). (3) Let X be a topological space and let OX to be the constant sheaf associated to Z. Show that (X; OX ) is a ringed space and that every sheaf on X is in a natural way an OX -module. (4) Let (X; OX ) be a ringed space. Show that the kernel and image of a morphism of OX -modules are also OX -modules. Show that the direct sum, direct product, direct limit and inverse limit of OX -modules are OX -modules. If F and G are O H F G O L X -modules, show that omOX ( ; ) is also an X -module. Finally, if is a O F F O subsheaf of an X -module , show that the quotient sheaf L is also an X - module. O F O O F ' (5) Let (X; X ) be a ringed space and an X -module. Prove that HomOX ( X ; ) F(X). (6) Let f :(X; OX ) ! (Y; OY ) be a morphism of ringed spaces. Show that for any O F O G ∗G F ' G F X -module and Y -module we have HomOX (f ; ) HomOY ( ; f∗ ). (7) Let (X; OX ) be a ringed space and F and G be OX -modules. Show that F ⊗ G 'F ⊗ G 2 ( OX )x x Ox x for any x X. (8) Give an example of a morphism f : X ! Y of Noetherian schemes and a coher- ent sheaf F on X such that f∗F is not coherent. (9) Let f : X ! Y be a finite morphism [see Example Sheet 2] of Noetherian schemes. Prove that for any coherent sheaf F on X, f∗F is coherent. Deduce that direct image of coherent sheaves are coherent under a closed immersion of Noetherian schemes. Date: Nov 2009. 1 2 CAUCHER BIRKAR ' ~ ~ (10) Let X = Spec A be an affine scheme. Show that HomA(M; N) HomOX (M; N) for any A-modules M; N. F ' ~ F (11) Let X = Spec A be an affine scheme. Show that HomA(M; (X)) HomOX (M; ) for any A-module M and OX -module F. (12) Let X be a scheme and 0 !F!G!E! 0 an exact sequence of OX - modules. Show that if two of the sheaves are quasi-coherent, then the third one is also quasi-coherent. (13) Let f : Spec B ! Spec A be a morphism of affine schemes such that B is flat over A. Show that if 0 !F!G!E! 0 is an exact sequence of quasi-coherent sheaves on Spec A, then 0 ! f ∗F! f ∗G! f ∗E! 0 is exact. (14) Let X be a Noetherian scheme and F a coherent sheaf on X. Assume that Fx O 2 is a free x-moduleL for some x X. Prove that there is a neighborhood U of x Fj ' n O such that U 1 U for some n. (15) Let X be an integral scheme and let K be the constant sheaf on X defined by the function field K of X. Show that we have a natural injective morphism OX !K O F F ! F ⊗ K which for any X -module induces a natural morphism φ: OX . We say that F is torsion if it is equal to the kernel of φ. Now show that a quasi- coherent sheaf F is torsion iff Fη = 0 where η is the generic point of X. (16) Let R be a DVR with fraction field K and X = Spec R. Show that to give an OX -module is the same as giving an R-module M, a K-vector space V and a ho- momorphism α: M ⊗R K ! V . Show that such an OX -module is quasi-coherent iff α is an isomorphism. L (17) Let X = Proj S where S = d≥0 Sd is a graded ring which is generated by S1 over S0. Prove that the sheaves OX (n) are invertible. (18) Let S and T be gradedL rings such that A := S0 = T0. Let X = Proj S, ⊗ × Y = Proj T , Z = Proj d≥0 Sd A Td, and W = X Spec A Y . Show that ! O ' ∗O ⊗ ∗O there is an isomorphism f : Z W such that f∗ Z (1) p X (1) OW q Y (1) where p and q are the projections W ! X and W ! Y . L (19) Let S = d≥0 Sd be aL graded ring which is generated by S1 over S0. Pick 2 N (n) n and let S = d≥0 Snd. Show that there is an isomorphism f : X = (n) Proj S ! Y = Proj S such that f∗OX (1) 'OY (d). (20) Give an example of a graded ring S and non-isomorphic graded S-modules M; N such that M~ and N~ are isomorphic. ALGEBRAIC GEOMETRY (PART III) EXAMPLE SHEET 3 3 (21) Let X be an integral scheme. Show that every invertible sheaf L on X is iso- morphic to OX (D) for some Cartier divisor D on X. Pn (22) Let X = k = Proj k[t0; : : : ; tn] where k is a field, and let Y be the closed sub- scheme defined by the ideal generated by a homogeneous polynomial F of degree d. Show that the ideal sheaf IY of Y is isomorphic to OX (−d)..