A differential characterization of multiplicity sequences over arbitrary fields
William C. Brown · Rocky Mountain Journal of Mathematics · 1981
Introduction.Let (0, m, k) denote an excellent, local, domain of Krull dimension one with maximal ideal m and residue class field k.We assume that 0 is equicharacteristic and geometrically unibranched.Let (9 denote the integral closure of 0 in its quotient field K{(9), and let y : (9 = (9 0 -> &i -> @2 -+ " ' ~* ®n ~* 0 t> e tne blow up sequence of 0 in (9.Here the notation has been chosen to mean that (9 n is the last nonregular local ring in the blow up sequence of (9 (If 0 is regular, we write Sf as Sf\ (9 = (9).Let (9jm and 0,-/m,-, / = 0, . . ., n, denote the residue class fields of & and (9i respectively.Set/, = [0/m: (9jm t ].Finally, let Z>?(0) denote the ^-module of q-th order ^-differentials on Q.In [1] and [2], K. Fischer showed that if (9 is complete, and k is algebraically closed, then for all q > 1, 7)|(0) uniquely determines the multiplicity sequence {ju((9i)} of S?.In this paper, we shall prove a similar result when (9 is not necessarily complete, and k is not necessarily algebraically closed.Specifically, we shall show that if {(9, m, k) is an excellent, local, domain of Krull dimension one, of equal characteristic and geometrically unibranched, then, for all q sufficiently large (q > 1), D^(Ö) and the residue class sequence {/ 0 , ...,/"} uniquely determine the multiplicity sequence {ju((9 t )} of £?.We shall also give an example which shows that D$(0) by itself does not determine the multiplicity sequence of y.We shall assume that the reader is familiar with the contents of [1] and [4].We shall use much of the notation from those two papers.In particular, ju((9) will denote the multiplicity of a local ring (9, and X(M) will denote the length of an ^-module M, and K(A) will denote the total quotient ring of any ring ,4.Now let {(9, m, k) be as above.We shall explain why we must assume (9 is geometrically unibranched instead of just unibranched.In the theory that we shall present here (as well as that in [1]) the module I{Ôj(9) is the object which plays the principal role in determining {fi{(9 l )}.Because of the good functorial properties the module D%((9) enjoys, we would like to continue to deal with a class of rings in which 7(0/0) = D q G ((9) for all q > 1.If k is not algebraically closed, then a unibranched domain (0,