Answer Happy

Let K be field, let R = R[x, y, z] and let JCR be the ideal (x² + y² + 2²-1)R (xR+ (z-1)R). n Sketch V(J) C R³ and find the dimension of each of the localizations (R/J)m where (a) m = (x - 1)R+yR+ zR, (b) m = xR+ (y − 1)R+ (z − 1)R, - (c) m = xR+yR+ (z − 1)R. Following is some useful hints (Only need prove above, do not need prove below) Definition 37.3. Let K be a field and R = K[₁,..., En]. (a) Let X K be an irreducible algebraic set, i.e., X = V(P) for some prime ideal PCR. We define the dimension of X, denoted dim X, to be tr-degk L where L is the field of fractions of the domain R/P, i.e., L= (R/P)(o). If X is not irreducible, we define dim X to be the maximal dimension of an irreducible component of X. (b) Let X C Kn be any algebraic set and let x E X. Write X as a union of its irreducible components X = X₁ U...UX. The dimension of X at the point x, denoted dim, X, is max{dim X₂|1 ≤ i ≤s and x € X;}. (c) The codimension of a algebraic set XCK" is n - dim X. Proposition 37.4. Let XCK" be an irreducible algebraic set and let ZX be an algebraic subset. Then dim Z<dim X. Theorem 37.5. Let X CK be an irreducible algebraic set. Write R = K[x₁,...,xn] and let fe R. Then (a) XnV(f) = X, or (b) XnV(f) = 0, or (c) dim XnV(f) = dim X - 1. Corollary 37.6. Let X Kn be an irreducible algebraic set and let ZX be a maximal proper irreducible algebraic subset. Then dim Z = dim X - 1