Exercise 16, § 5

Let $A$ be a noetherian ring. The singular locus of Spec(A) is called and is denoted Sing(A), the set of the p ∈ Spec(A) such that A_p is not regular. Put

Now prove that the following conditions are equivalent: (i) For every quotient ring B of A, Reg(B) is open in Spec(B). (ii) For every integral quotient ring B of A, Reg(B) is a nonempty open subset of Spec(B). (iii) For every integral quotient ring B of A, Reg(B) contains a nonempty open subset of Spec(B). (To prove (i) ⇒ (ii), note that one has (0) ∈ Reg(B). To prove (iii) ⇒ (i), show that for every p ∈ Spec(A) such that V(p) ∩ Reg(A) ≠ ∅, the ring A_p is regular (p. 94, exerc. 5). Using (iii) and constructing completely secant sequences, show that V(p) ∩ Reg(A) contains a nonempty open subset of V(p). Conclude using exerc. 15.)