Exercises to Algebraic Geometry II

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1 We fix a principal ideal domain A for exercises Exercise 1.1. The Smith normal form I We say that a matrix M = (m ij ) Mat(n k, A) has Smith normal form, when m ij = δ ij m j and we have a descending chain of ideals (m 1 ) (m 2 ) (m min(n,k) ). We say that N Mat(n k, A) has Smith normal form M, if M has Smith normal form and there exist invertible matrices S GL k (A), and T GL n (A) such that M = T 1 NS. Two Smith normal forms M, M Mat(n k, A) are equivalent, if the ideals in the above chain coincide, that is (m i ) = (m i) for all i = 1... min(n, k). Show the uniqueness of the Smith normal form: Suppose N Mat(n k, A) has Smith normal forms M and M. Then M and M are equivalent. Exercise 1.2. The Smith normal form II Show the existence of a Smith normal form for any N Mat(n k, A). Furthermore, give the Smith normal form for the matrix N = Mat(3 3, Z) Hint: The idea is more or less an adaption of the Gaussian elimination algorithm to matrices with coefficients in A. More precisely: (1) Show that there exist matrices S GL k (A), and T GL n (A) such that N = T 1 NS has a smallest element n i 0 j 0. (Here smallest means (n ij ) (n i0 j 0 ).) If not use that A is a principal ideal domain, to find a transformation N T 1 NS: Take an element a of the matrix entries and suppose that there exists another matrix entry a such that neither (a) (a ) nor (a ) (a). Then we have (a ) = (a, a ) and a is a A-linear combination of a and a. Use this to find a transformation N T 1 NS with a as a matrix entry. Show that this process terminates. (2) Now proceed like in the Gaussian elimination algorithm to move the smallest element to n 11, make all the n 1i = 0 and n i1 = 0 for i 2 and proceed by induction on n + k. (3) It may be a good idea to start with the above example. Exercise 1.3. Structure theorem for finitely generated A-modules Prove that any finitely generated A-module V possesses an isomorphism V = A r A/(a 1 )... A/(a k ) with r, k uniquely given integers and a unique ideal chain (a 1 ) (a 2 ) (a k ). Exercise 1.4. Here we consider the ring A = Z[ 5]. Show that the ideal M = (2, 1 + 5) is not isomorphic to A, and give an isomorphism between M M and A A.

2 Exercise 2.1. Valuations on Q Let ν : Q G be a valuation of the rational numbers. (We assume ν to be surjective.) Show that either G is the trivial valuation (ν 0 (q) = 1 for all q Q) or isomorphic to the valuation ν p for a prime number p. Hint: Show that ν is given by its values on N Q. Show next that ν(n) 0 for all natural numbers n > 0. Suppose that ν is not trivial, show that the minimal n with ν(n) > 0 is a prime number. Then show ν(m) = 1 for all integers m (n). Then conclude the above statement for the unique decomposition into prime numbers. Exercise 2.2. Valuations on finite algebraic extensions L/Q We consider a finite field extension L/Q. Since any valuation of L defines a valuation on Q we obtain a map from the valuations of L to the valuations of Q. Show that this map is surjective but not injective. Hint: Consider the extension of Dedekind rings Z A = Z L, and the previous exercise. Exercise 2.3. Ostrowki s theorem I A norm on a field K is map K R with λ λ R 0 such that: λ = 0 λ = 0, λ µ = λ µ, and λ + µ λ + µ holds. The valuation is called non archimedean, if λ + µ max{ λ, µ } holds. Two norms are equivalent when they define the same topology on K. Classify all equivalence classes of non archimedean norms on Q. Hint: Show that for a non archimedean norm K R the function K R given by λ log λ defines a valuation on K. Exercise 2.4. Ostrowki s theorem II Show that there exists only one equivalence class of archimedean valuations on Q, the class of the absolute value.

3 Exercise 3.1. We consider the smooth affine curve C = V (y 2 x 3 + 6x). Extend the morphism (C \ {(3, 3)}) P 1 (x, y) (x 3 : y 3) from C \ (3, 3) to C and compute the value of the extension at the point (3, 3). Exercise 3.2. We consider the morphism k 2 \ {(0, 0)} P 1 k which assigns the point (x, y) (x : y). Show that for any point (a : b) P 1 (k) there exists a curve C K 2 such that (i) C contains (0, 0). (ii) ψ C\{(0,0)} extends uniquely to ψ : C P 1. (iii) ψ(0, 0) = (a : b). Exercise 3.3. We consider the two affine curves C = V (Y 2 X 3 X 2 ) ι ψ ι A 2 and C = V (Y 2 X 3 ) A 2. Show that neither C or C are nonsingular. Compute the normalizations ψ : C C and ψ : C C. Show that the morphism A \ {(0, 0)} ϕ P 1 given by ϕ(x, y) = (x : y) extends to C and C. Does it extend to C or to C? Give a geometric explanation. Exercise 3.4. Let X = Spec(k[X, Y ]), and suppose we are given nonzero functions f 0, f 1,..., f n k[x, Y ]. Show that there exist a finite set of closed points {P 1,..., P m } X such that the function ψ : X P n ψ(x) = (f 0 (x) : f 1 (x) : : f n (x)) is well defined on the open subset U = X \ {P 1,..., P m }.

4 Exercise 4.1. Let B be a base scheme. Is the functor representable? (B schemes) (sets) X O X (X) Exercise 4.2. Show that the Zariski topology on A 2 k = A1 Spec(k) A 1 k is not the product topology for the two projections from A 2 to A 1. Exercise 4.3. We consider the morphism X = Spec(k[x, t]/(x n f t)) Y = Spec(k[t]) given by the inclusion of rings k[t] t [t] k[x, t]/(x n t). We assume k = k of arbitrary characteristic. Determine the fibers X y for all points (including the generic one!) y of Y. Exercise 4.4. We consider the following morphism between two affine k-varieties X = A 3 f k A 3 k = Y f(x, y, z) = (xz, yz, z). Show that X Y X is not irreducible. Determine the irreducible components and give their dimensions.

5 Exercise 5.1. Let B be a finite A algebra. Show that the map Spec(B) Spec(A) is proper. Exercise 5.2. Let X and Y be two schemes over Z, and = U X be open. Suppose that (i) X is integral (in particular reduced), and (ii) Y is separated. Show that two morphisms f, g : X Y which coincide on U are identical. Give two examples which prove, that both conditions (i) and (ii) are necessary. Exercise 5.3. Which of the following ring morphisms define proper morphism of their spectra? (i) R C. (ii) A A[X] (iii) A A[X]/(X 2 7) (iv) k[x] k[x, Y ]/(XY 1) (v) k[x] k[x, Y ]/(X 2 Y ) (vi) k[x] k[x, Y ]/(X 2 + Y 2 1) Exercise 5.4. Let X be a variety over an algebraic closed field k. Decide, whether the following equivalence holds: X Spec(k) is proper O X (X) = k. Give a proof or a counterexample for each direction.

6 Exercise 6.1. the resolution by value (a) Let X/k be a scheme over an algebraically closed field k. For which schemes is the sheaf k X(k) which is given by U { maps from U(k) k} divisible? (b) We have a natural sheaf morphism O X k X(k) which assigns a regular function its values at the closed points. For which schemes X is this morphism injective? (c) What is the answer to the question in (b) when k is not algebraic closed? (d) For which irreducible k -schemes is O X k X(k) an injective resolution? (e) For which irreducible k -schemes is O X k X(k) a flabby resolution? Exercise 6.2. the constant resolution Let A be a domain with fraction field K. (a) Is K a divisible A-module? (b) Show that any quotient of a divisible module is divisible. (c) Show that for a principal domain A and a short exact sequence 0 M M M 0 we obtain for any A-module N a long exact sequence 0 Hom(M, N) Hom(M, N) Hom(M, N) Ext 1 (M, N) Ext 1 (M, N) Ext 1 (M, N) 0 Exercise 6.3. Interpretation of Ext 1 part I Let C be an abelian category with enough injectivities. We define Ext i (M, N) to be the ith derived functor of N Hom(M, N). Write down the long exact cohomology sequence. Show that any short exact sequence 0 M M M 0 defines an element η Ext 1 (M, M ). Show that η = 0 the sequence splits. Exercise 6.4. Interpretation of Ext 1 part II Take an injective resolution M I 0 d 0 I 1 d 1 I And a class η Ext 1 (M, M ). Show that η can be represented by a morphism [η] : M I 1 such that d 1 [η] = 0. Define a module M(η) := {(m, i) M I 0 d 0 (i) = [η](m )}. Show that the natural morphism M(η) M is surjective, and the kernel is isomorphic to M. Thus, we obtain a short exact sequence 0 M M(η) M 0. Show that this construction is the inverse of the construction in part one.

7 Exercise 7.1. We consider the Z-module M = n N Z, and its submodule M = {(a n ) n N a n (2 n )}. Let T 1 be the (2)-adic topology on M, and T 2 be the restriction of the (2)-adic topology on M to M. Which of the maps (M, T 1 ) (M, T 2 ), and (M, T 2 ) (M, T 1 ) is continuous? Exercise 7.2. Let A be a noetherian ring. For a finitely generated A-module M and some open subset U Spec(A) with U = X \ V (a), show that the following two definitions coincide H 0 a(m) := ker( M(X) M(U)), and H 0 a(m) := {m M a n m = 0 for n 0}. Use the first definition to define on a scheme X, with some closed subscheme Z, for any coherent sheaf F a presheaf H 0 Z F. Show that this presheaf is a sheaf. Exercise 7.3. Let A be a noetherian ring. For any ideal a A show that the functor M H 0 a(m) is left exact. Use injective resolutions, to define the right derived functors H i a(m). Show that for A = Z, a = (3), M = A the module H 1 a(m) 0. Exercise 7.4. Let X be a scheme with a closed subscheme Y. Show that F is flabby = H 0 ZF is flabby.

8 Exercise 8.1. We consider a k-rational point P P 1 k, and an integer m Z. Compute the Čech cohomology Ȟk (U, O P 1(mP )) using the standard open covering U = {U 0, U 1 }. Exercise 8.2. We consider the elliptic curve E P 2 k defined by E = V (Y 2 Z X 3 +XZ 2 ). Show that U 0 = {Y 0} E, and U 1 = {Z 0} E form an open affine cover. Compute the Čech cohomology Ȟm (U, O E ) for U = {U 0, U 1 }. Exercise 8.3. We consider the projective curve X P 2 k defined by X = V (g) for some homogeneous polynomial in (X, Y, Z) of degree d. Assume that E does not contain the point (1 : 0 : 0). Compute the dimension of the Čech cohomology groups Ȟ0 (U, O X ) and Ȟ 1 (U, O X ) with U = {{Y 0} X, {Z 0} X}. Exercise 8.4. Let X be a projective curve over some field k. Let A = O X O X with algebra structure defined by (a b) (a b ) := aa (ab + a b). Let X be the Spec(A). Note that the algebra morphisms A (a b) a O X defines an embedding X X. Show that X cannot be embedded into P 2 k.

9 Exercise 9.1. Let B = n 0 B n, a graded ring. We assume that A = B 0 is noetherian, B 1 is a finitely generated A-module, and B 1 generates B as a A-algebra. Let M be a finitely generated graded B-module. We define the r-truncation M [r] M to be the submodule given by M [r],n = M n for n r, and M [r],n = 0 for n < r. (i) Show that on X = Proj(B) we have M [r] = M. (ii) Suppose we have Ñ = M for two finitely generated graded B-modules. Show that N [r] = M[r] for some r 0. (iii) Suppose M [r] = 0 for some r Z. Show that the support of M in Spec(B) is contained in Spec(A). Exercise 9.2. The rational quartic curve We consider the following homogeneous ideal I in C = k[x, Y, Z, T ]. I = (XT Y Z, Y 3 X 2 Z, Z 3 Y T 2, XZ 2 Y 2 T ). Let B = C/I, and X = Proj(B). (i) Show that B is not a normal ring. (ii) Set B := k 0 Γ(O X (k)). Show that the natural morphism B ν B is the normalization of B. In particular ν is not an isomorphism. Hint: The rational quartic curve which appears in the title is the image of the morphism P 1 k P3 k given by (u : v) (u4 : u 3 v : uv 3 : v 4 ). Exercise 9.3. Show the following implication for a coherent sheaf F on P n A : F is globally generated = H n (F ( n)) = 0. Is the converse also true? (Hint: Consider the ideal sheaf of a point in P 2 k.) Exercise 9.4. Let k be an algebraically closed field. Show that for any coherent sheaf F on P 1 k we have an equivalence: F is globally generated H 1 (F ( 1)) = 0. Can you replace (P 1 k, O1 P (1)) by an arbitrary projective curve X and a very ample line bundle O X (1) on it?