Commutative Algebra Assigment - II
Due 21 October 2023
All rings are assumed to be commutative with identity.
Any result from the lectures may be used without proof unless you are asked to prove the result.
If you use results from any other sources, provide proofs for them.
Solutions should be rigorous and clearly presented. Ambiguous arguments will not be given marks.
Question 1. Let K be an algebraically closed field.
(1) Consider the polynomial map F : K2 −→ K, (x,y) 7→ y. Show that the image F(V ) of the affine variety V = V (xy − 1) is not an affine variety.
(2) Let F : Kn −→ Km be a polynomial map, and let W = V(g1,g2,...,gk) ⊆
Km be an affine variety. Show that the inverse image F−1(W) of W is V(g1 ◦F,g2 ◦F,...,gk ◦F), thus is an affine variety.
Question 2. Consider the affine variety V = V (x1x4 − x2x3) ⊆ K4 over an alge- braically closed field K.
(1) Show that V is irreducible.
(2) Let η : K[x1,x2,x3,x4] → K[V] be the natural surjective ring morphism,
and let f = η(x1) ∈ K(V ). What is the domain of definition of f? η(x2)
Question 3. Let V be an irreducible affine variety over an algebraically closed field K. Recall that for any point p ∈ V, the ring Op(V) := {f ∈ K(V) |
f is regular at p} is local. Denote by Mp the maximal ideal of Op(V ).
(1) Let V = V (y − x2) ⊆ K2 and p = (0, 0). Show that dimK Mp/Mp2 = 1.
(2) Same question as above, but with V = V (y2 − x3) ⊂ K2 and p = (0, 0), but
now show that dimK Mp/Mp2 = 2.
Question 4. Let M be a module over a ring R. For any f ∈ R which is not nilpotent, denote by Mf the localisation of M with respect to {1, f, f2, . . . }. Given a set of elements {f1,f2,...,fn} of R which generate the unit ideal, and has the property that fkifki ̸= 0 for any i,j and all ki,kj ∈ Z+, prove the following assertions.
ij
(1)Ifm∈Mismappedto0bythenaturalmapτi :M−→Mfi,v7→v1,for
each i, then m = 0.
(2) If mi ∈ Mfi are elements such that mi and mj are mapped to the same
element in Mfifj by the natural maps Mfi −→ Mfifj and Mfj −→ Mfifj for any i, j, then there exists m ∈ M such that τi(m) = mi for all i.
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