ADVANCES COMMUTATIVE ALGEBRA
A. BERNARDI
1. Exercises
Exercise 1. Let N ⊂ M be a submodule. Show that M is Noetherian if and only if N and M/N
are Noetherian.
Exercise 2. Let X, Y ⊂ A2 be the plane curves defined by {y = x2 } and {xy = 1} respectively.
Let f be any irreducible quadratic polynomial in k[x, y], and let Z be the conic defined by f . Show
that A(Z) is isomorphic to A(X) or A(Y ). Which one is it when?
Exercise 3. Let I ⊂ K[x, y, z] be the ideal I = (x2 + y, x2 + z) and let X ⊂ A3 be the affine
algebraic set X = Z(I). Let X ⊂ P3 be the projective closure of X. Show that the homogeneous
ideal I(X) is not generated by the homogenization of x2 + y and x2 + z.
Exercise 4.
(1) Show that if X ⊂ Pn is an algebraic projective set consisting of d points and
X is not contained in a line, then X may be described as the zero locus of polynomials of
degree d − 1 and less.
(2) Show that if X ⊂ Pn is an algebraic projective set consisting of d ≤ 2n points in general
position, then X may be described as the zero locus of quadratic polynomials.
Exercise 5.
(1) Prove that the set X = {[t30 , t0 t21 , t31 ] s.t. [t0 , t1 ] ∈ P1 } is a projective set of
2
P and that I(X) = (x0 x22 − x31 ).
(2) Prove that the rational normal curve of degree d in Pd (parameterized by x0 = td0 , x1 =
td−1
t1 , . . . , td1 ) is defined by the vanishing of the minors of the matrix
0
x0 x1 . . . xd−1
.
x1 x2 . . .
xd
1