O. Definitions. (a) RnHomA ( , B) (A) = ExtA (A, B).

ThEOREM cal J. PROOF. Let n = 1. If R is regular, then J is generated by one nonzero divisor and so is free. Conversely, if J is projective then it is free. Hence J is generated by one element which is not a zero divisor so height (l);;;;' 1 (minimal primes consist of zero divisors). Therefore, by the principal ideal theorem, height J = 1 and R is regular. Let n> 1. If R is regular, select x E J - J2, and set R* By induction, p. d. ((J/xR)R') =n = R/xR. - 2. 26, p. d. ((J/xR)R) J(R*) =n - = J/xR.