Homework for Algebraic Geometry M.Math

Unless otherwise stated a ring would mean a commutative ring with identity. If unspecified, $k$ denotes a field and $R$ a ring.

Class 1

1. Let $X$ be an algebraic subset of $\mathbb{A}^n$. Show that $Z(I(X))=X$.

2. Show that the ideal $(y(y-x^2))$ in the polynomial ring $k[x,y]$ is a radical ideal.

3. Let $S$ be an extension of $R$ and a finitely generated $R$-module. Assume $S$ is an integral domain. Show that $a\in R$ is a unit iff it is a unit in $S$.

4. (Hilbert Nullstellensatz) Suppose a finitely generated $k$-algebra $R=k[x_1,\ldots,x_n]$ is a field then $x_1,\ldots,x_n$ are algebraic over $k$. (Hint: Use induction on $n$. If $x_1$ is transcendental, $R=k(x_1)[x_2,\ldots,x_n]$ is a field, hence by induction hyp. $x_2,\ldots,x_n$ are algebraic over $k(x_1)$. Use it to show that there exist $f(x_1)$ such that $R$ is a finite module over $k[x_1,1/f(x_1)]$. Now use the above result to get a contradiction.)

Class 2

5. Let $\{I_l\}_l$ be a collection of ideals in $k[X_1,\ldots,X_n]$. Show that $\cap_l Z(I_l)=Z(I)$ where $I=(\cup I_l)$.

6. Let $X$ be an irreducible affine algebraic set over an algebraically closed field $k$ and $U\subset X$ be Zariski open and nonempty. Show that $U$ is dense in $X$.

Class 3

7. Let $X$ be an affine algebraic set. Show that $X$ is connected iff the only idempotents in the coordinate $\mathcal{O}_X$ are 0 and 1.

8. Let $X$ and $Y$ be affine algebraic sets over $k$. Show that any $k$-algebra homomorphism $\mathcal{O}_Y\to \mathcal{O}_X$ induces a morphism from $X\to Y$.

9. Let $R$ be a finitely generated $\mathbb{Z}$-algebra and $I$ be an ideal in $R$. Show that $\cap\{m\in mspec(R)| I\subset m\}=\sqrt{I}$. (Hint: Follow the same algorithm which was used for finitely generated $k$-algebra.)

Class 4

10. Let $f:X\to Y$ be a morphism of affine algebraic sets. Show that $f$ is continuous.

11. Let $X\subset \mathbb{A}^n_k$ be an affine variety over a field $k$ and $f\in \mathcal{O}_X$ be nonzero. Let $U_f=X\setminus V(f)$ be the standard open set of $X$ corresponding to $f$. Construct a closed irreducible subset $Y$ of $\mathbb{A}^{n+1}_k$ such the projection map from $Y$ to $\mathbb{A}^n_k$ (the first $n$ coordinates) defines a morphism from $Y \to X$ which is injective and the image of this map is $U$. Show that the coordinate ring of $Y$ is isomorphic $\mathcal{O}_X[1/f]$.

12. Consider the space curve $C\subset \mathbb{A}^3$ given by the image of the morphism $\phi: \mathbb{A}^1\to \mathbb{A}^3$ which send $t\to (t,t^2,t^3)$. Compute the ideal of definition $I(C)\subset k[x,y,z]$, the co-ordinate ring $\mathcal{O}_C$ and the ring homomorphism $\phi^{\#}:\mathcal{O}_C\to \mathcal{O}_{\mathbb{A}^1}$.

13. Let $A\subset B$ be rings and $S\subset A$ be a multiplicative set. Let $C$ be the integral closure of $A$ in $B$. Show that $S^{-1}C$ is the integral closure of $S^{-1}A$ in $S^{-1}B$.

Class 5

14. Let $X_a$ in $\mathbb{A}^2_k$ be the algebraic set defined by $x^2+y^2-a$. Show that for any nonzero $a,b\in k$, $X_a$ and $X_b$ are isomorphic algebraic subsets. Show that $X_0$ is not isomorphic to $X_1$.

15. Let $X$ be an algebraic variety and $f,g$ be regular function on $X$. Show that if $f(a)=g(a)$ for all $a$ in a nonempty open subset of $X$ then $f=g$.

16. Let $X$ be an affine variety. Show that the field of rational functions $K(X)$ is isomorphic to the fraction field of $\mathcal{O}(X)$.

17. Show that the maximal ideal of the local ring $k[x,y]_{(x,y)}/(y^2-x^3)$ is not principal.

Class 6

18. Let $X$ be a topological space and $\phi:\mathscr{F}\to\mathscr{G}$ be a morphism of sheaves. Show that $U\mapsto ker(\phi_U)$ is a subsheaf of $\mathscr{F}$.

19. Let $X$ be the underlying topological space of a variety and $A$ be an abelian group. Show that the constant presheaf with $A$ as coefficient is a sheaf.

Class 7

20. Let $U=\mathbb{A}_k^2\setminus\{(0,0)\}$ and $\mathcal{O}$ be the sheaf of regular functions on $\mathbb{A}^2_k$. Show that $\mathcal{O}(U)\cong k[x,y]$, the polynomial ring in two variable.

21. Let $f:X\to Y$ be a continuous map and $\mathcal{G}$ be a sheaf on $Y$. Show that for all $x\in X$ the stalk of the inverse image sheaf $f^{-1}G_x$ is isomorphic to $G_{f(x)}$.

22. Let $0\to \mathcal{F}_1\to \mathcal{F}_2\to \mathcal{F}_3$ be an exact sequence of sheaves on $X$. Let $U$ be an open subset of $X$. Show that the following sequence of abelian groups is exact: $$0\to \mathcal{F}_1(U)\to \mathcal{F}_2(U)\to \mathcal{F}_3(U)$$

Class 8

23. Let $R$ be a $k$-algebra where $k$ is a field. If $R$ is a finite dimensional $k$-vector space then show that $|spec(R)|\le dim_k(R)$.

24. (Power of prime ideals need not be primary): Let $R=k[x,y,z]/(z^2-xy)$. Show that $P=(\bar x, \bar z)$ is a prime ideal of $R$. Show that $P^2$ is not primary. Also show $\sqrt{P^2}=P$. (Hint: First show $\bar x\bar y \in P^2$)

25. (Minimal primary decomposition may not be unique) In the ring $k[X,Y]$ show that $(X^2,XY)=(X)\cap (X,Y)^2=(X)\cap (X^2,Y)$.

26. Let $A\subset B$ be integral extension. Show that dimension of $A$ and $B$ are same.

Class 9

27. A topological space is said to be quasi-compact if every open cover has a finite sub-cover. Show that (underlying space) an affine scheme is quasi compact. Find an example of a scheme which is not quasi-compact. (Hint: Every noetherian scheme is quasi-compact. So look for a non-noetherian example.)

28. Let $(X,\mathcal{O}_X)$ be a scheme. Show that there exist a unique morphism of schemes from $X$ to $Spec(\mathbb{Z})$.

Class 10

29. Let $k$ be an infinite field. Show that $f\in k[X_0,\ldots, X_n]$ is homogenous iff there exist $d\ge 0$ such that $f(\lambda a_0,\ldots, \lambda a_n)=\lambda^d f(a_0,\ldots,a_n)$ for all $\lambda, a_0, a_1, \ldots, a_n \in k$.

30. Let $X$ be an algebraic subset of $\mathbb{P}^n$. Show that $I=\{ f \in k[X_0,\ldots, X_n] : f(a_0,\ldots,a_n)=0 \forall (a_0,\dots,a_n)\in \mathbb{A}^{n+1} \text{ with } [a_0,\ldots, a_n]\in X\}$ is a homogenous ideal and equals $I(X)$.

31. Give a geometric proof of $X$ is irreducible iff $CX$ is irreducible. (Hint: Suppose $CX=X_1\cup X_2$ with $X_1$ ,$X_2$ closed. Let $p\in X$ then the line $Cp$ is contained in $X_1\cup X_2$ and; $Cp \cap X_1$ and $Cp\cap X_2$ are closed in $Cp$. Hence $Cp\subset X_i$ for some $i$.)

32. Let $S\subset \mathbb{P}^n$, show that $V(I(S))$ is the Zariski closure of $S$.

Class 11

33. Show that: $k[y,z, y^{-1}]/(y-z^3)\cong k[x,z,x^{-1}]/(x^2-z^3)$.

34. Let $A=\oplus A_i$ be graded ring. Show that $1\in A_0$.

35. Let $A$ be a graded ring and $P$ be a homogeneous prime ideal. Show that the homogeneous localization $A_{(P)}$ is a local ring.

Class 12

36. Let $(X,\mathcal{O}_X)$ be a scheme. Show that $X_{red},\mathcal{O}_{X_{red}}$ as defined in the class is a scheme and the natural map $i:X_{red}\to X$ is a closed immersion.

37. Let $(X,\mathcal{O}_X)$ be integral scheme. Show that $X$ is irreducible.

Class 13

38. Let $X$ be a scheme and $f\in \mathcal{O}_X(X)$. Show that $X_f:=\{x\in X: f_x\in m_x\mathcal{O}_{X,x}\}$ is an open subset of $X$.

39. Let $X$ be a noetherian scheme. Show that $X$ has finitely many irreducible components.

40. Let $X$ be an irreducible scheme. Show that there exist a unique point $x$ in $X$ such that its closure $\overline{\{x\}}=X$. This is called the generic point of $X$.

41. Let $X$ be a reduced noetherian scheme. Show that there exist at least one and at most finitely points $x_1,\ldots, x_n$ in $X$ such that the stalk $\mathcal{O}_{X,x_i}$ are fields. (Hint: Show that stalk at a generic point of $X$ is a field and stalk at any other point is not a field.)

Class 14

42. Let $A$ be a ring, $X$ be a scheme and $\phi:A\to \mathcal{O}_X(X)$ be a ring homomorphism. Show that the mophism of structure sheaves on Spec($A$) (defined in the class) is a local map. (Hint: Translate the problem to the following (easy) statement in commutative algebra: Let $\phi:A\to B$ and $r:B\to C$ be ring homomorphisms with $(C,m)$ local ring. Let $P=\phi^{-1}(r^{-1}(m)$ and $S=A\setminus P$. Then $S^{-1}(r\circ \phi):A_P\to C$ is a local map.)
43. (Gluing of morphisms) Let $X$ and $Y$ be a scheme and $\{U_i\}$ be a collection of open subschemes and $U_{ij}=U_i\cap U_j$. Let $f_i:U_i\to Y$ be morphism of schemes suct that for all $i,j$ $f_i|_{U_{ij}}=f_j|_{U_{ij}}$. Then there exist a morphism of schemes $f:X\to Y$ such that $f|_{U_i}=f_i$.

Class 15

44. Let $X$ and $Y$ irreducible variety over an algebraically closed field $k$. Then $X\times_k Y$ is irreducible. (Hint: Suppose $X\times Y= V_1 \cup V_2$, $V_i$ closed subsets of $X\times Y$. Let $U_i=\{y\in Y: V_i\cap X\times\{y\}=X\times \{y\}\}$. Note $X\times \{y\}=(V_1\cap X\times\{y\}) \cup (V_2\cap X\times\{y\})$. Use $X$ irreducible to show that $U_1\cup U_2=Y$. Also $V_i\supset X\times U_i$ and hence $V_i\supset X\times \bar U_i$. But $Y$ is irreducible, hence $\bar U_1=Y$ or $\bar U_2=Y$.

45. Describe $Spec(\mathbb{C})\times_{Spec(\mathbb{R})}Spec(\mathbb{C})$ as a topological space.

46. Show base change of a finite morphism is finite.

Class 16

47. Let $X$ be a scheme of finite type over a field $k$ and $\bar k$ be the algebraic closure of $k$. Show that the base change $X_{\bar k}$ is irreducible implies $X_{K}$ is irreducible for all field extension $K/k$. (Here $X(K)$ denote all $K$-points of $X$.)

48. Show that base change of a closed immersion is a closed immersion.

49. Let $Y$ be a separated $Z$-scheme. Then for any $Y$-scheme $X_1$, $X_2$, show that the canonical morphism $X_1\times_YX_2\to X_1\times_ZX_2$ is a closed immersion.

Class 17

50. (Separatedness and properness are local on the base) Let $f:X\to Y$ be a morphism of schemes (with $X$ noetherian). Let $\{Y_i\}$ be an open cover of $Y$ and $X_i=f^{-1}(Y_i)$.
Show that $f$ is separated iff $f|_{X_i}$ is separated for $i$.
Show that $f$ is proper iff $f|_{X_i}$ is proper for $i$.

51. Let $A\subset B$ be an integral extension of domains and $K$ be a field containing $B$. Let $R$ be a valuation ring with fraction field $K$ such that $A\subset R$. Show that $B\subset R$.

52. Show that finite morphisms are proper. (Hint: Use the above two problems and valuative criterion)

Class 18

53. Let $A\to B$ be a ring homomorphism. Show that $\mathbb{P}^n_B\cong \mathbb{P}^n_A\times_{Spec(A)}Spec(B)$. For a scheme $Y$, define $\mathbb{P}^n_Y=\mathbb{P}^n_{\mathbb{Z}}\times_{Spec(\mathbb{Z})}Y$.

54. For a ring $A$, consider the ring homomorphism between polynomial rings$A[z_{ij}; 0\le i \le n, 0\le j \le m]\to A[x_0,\ldots,x_n]\otimes_AA[y_0,\ldots,y_m]$ which sends $z_{ij}\to x_iy_j$. Show that this induces a morphism from $\mathbb{P}^n_A\times_{Spec(A)}\mathbb{P}^m_A\to \mathbb{P}^{nm+n+m}$ which is a closed immersion. (This is called the segre embedding)

55. Use the above to show that the composition of two projective morphisms is projective.

Class 19

56. Let $X$ be a noetherian connected nonsingular scheme. Show that $X$ is irreducible (and hence an integral scheme).

57. Let $f:X\to Y$ be a dominant $k$-morphism of proper integral curves over a field $k$. Show that $f$ is a finite morphism.

Class 20

58. Let $X$ and $Y$ be integral finite type scheme over $k$ and $f:X\to Y$ be a homeomorphsim such that for $x\in X$, there are $k$-isomorphsims $s_x:\mathcal{O}_{Y,f(x)}\to \mathcal{O}_{X,x}$ such that the induced map on the fraction fields is independent of the choice of $x$. Show that $X$ and $Y$ are isomorphic. (i.e. given $V$ open in $Y$ construct a ring homomorphism $f^{\#}:\mathcal{O}_Y(V)\to \mathcal{O}_X(f^{-1}(V))$ which induce $s_x$ at the level of stalks.)

59. Let $f:X\to Y$ and $g:Y\to Z$ be morphism of schemes such that $g\circ f$ is projective and $g$ is separated. Show that $f$ is projective.

Class 21

60. Let $X$ be the projective plane curve over a field $k$ given by the irreducible homogeneous polynomial $zy^2-x^2(x-z)$. Show that the normalization of $X$ is isomorphic to $\mathbb{P}^1$.

61. Let $k$ be algebraically closed field. Let $A$ be a finitely generated $k$-algebra and $m(z)$ be a monic polynomial in $A[z]$. Show that closed points of $Spec(A(z)$ are in bijection with $(a,t)$ where $a\in Spec(A)$ and $t\in Spec(k[z])$ are closed points and the morphism $Spec(A[z])\to Spec(A)$ is projection onto the first factor. Let $B=A[z]/m(z)$ then $Spec(B)$ is a closed subset of $Spec(A[z])$. Show that $(a,t)\in Spec(A[z])$ lies in $Spec(B)$ iff $\bar m(t)=0$ where $\bar m(z)$ is the image of $m$ in $(A/p_a)[z]$ where $p_a$ is the maximal ideal of $A$ corresponding to the point $a$.