Dimension Sequences for Commutative Rings

Let JR be a commutative ring with identity of finite (Krull) dimension n0, and for each positive integer /c, let nk be the dimension of the polynomial ring R = R[XU . . . , Xk] in k indeterminates over R. The sequence {wjiio * Ud the dimension sequence for R, and the sequence {di}fLl9 where dt = nt — ni_1 for each i, is called the difference sequence for R. We are concerned with a determination of those sequences of nonnegative integers that can be realized as the dimension sequence of a ring. Several restrictions on the sequences {wj and {dt} are known. For example, nk + 1 ^ nk+1 ^ 2nk + 1 for each positive integer fc [5], and n0 + k ^ nk ^ (n0 + l)(/c + 1) — 1 [4]. In particular, the only dimension sequence for a zero-dimensional ring is 0,1,2,... . Jaffard in [4] proved that the difference sequence for R is eventually a constant less than or equal to n0 + 1—that is, there is a positive integer k such that n0 + 1 ^ dk = dk+1 = -. Moreover, if R is an integral domain, then both the integer k and the value dk relate to the valuative dimension dimvR, defined to be sup {rank F|Fis a valuation o ver ring of R}. Jaffard proved that the eventual value of the difference sequence for JR is 1 if and only if R has finite valuative dimension. In fact, if dim^R = k < oo, then Jaffard proved that dt = 1 for i ^ k + 1 ; in [1], Arnold improved Jaffard's bound by 1 to i ^ /c, and this bound, in turn, cannot be improved in the general case. We are able to prove that the restrictions mentioned in the previous paragraph are all that are necessary in order that an increasing sequence {mJ-% of positive integers with nonincreasing difference sequence {mt — mi^1}f==1 should be the dimension sequence of an integral domain. It is known, however, that the difference sequence for an integral domain need not be nonincreasing, and hence we require additional restrictions to solve the general problem. We denote by ^ the set of sequences {mj£L0 of positive integers such that the corresponding difference sequence {tt}fLi, where tt = mi m ^ , satisfies the following conditions : (1) m 0 + 1 ^h^t2^.... (2) There is a positive integer k such that 1 ^ tk = tk+1 = tk+2 = • • •.