Commutative Algebra - summer term 2017
Assignment sheet 6
Prof. Dr. Mohamed Barakat, M.Sc. Kamal Saleh
Exercise 1. (Valuation rings, 4 points)
Let R be a valuation ring. Prove that the set Γ := {I|I ER} of ideals in R is totally ordered w.r.t. inclusion.
Exercise 2. (Singular points, 4 points)
Letk be an algebraically closed field and letf ∈k[x1, . . . , xn] be an irreducible polynomial.
A pointP = (a1, . . . , an) of the algebraic setVk(f) is callednon-singulariff not all formal derivatives∂f /∂xi vanish atP. LetR =k[x1, . . . , xn]/hfi, and let m=hx1−a1, . . . , xn− ani ⊂R be the maximal ideal corresponding to the point P. Then show:
P is non-singular ⇐⇒ Rm is a regular local ring.
Exercise 3. (8 points)
1. Let R be a local Noetherian integral domain with maximal ideal m and dimR = 1.
Then the following are equivalent:
(a) R is a DVR;
(b) R is principal ideal domain;
(c) mis principal;
(d) dimR/mm/m2 = 1;
(e) Any ideal h0i 6=I ER is of the formmn for some n ∈N0;
(f) There exists an element p ∈ R such that any ideal h0i 6= I ER is of the form hpni for somen ∈N0;
(g) R is normal;
(h) dimR/mmk/mk+1 = 1 for allk ≥1.
2. Let K be a field and p a prime number. Show thatK[x]hxi and Zhpi are DVRs.
Hand in until January 23th 12:00 in the class or in Box in ENC, 2nd floor, at the entrance of the building part D.
