Grade Schemes and Grade Functions

In recent years, two concepts similar to /?-sequences have appeared, essential sequences and asymptotic sequences. This work explores the general nature of such sequences. Introduction. In the last three years, two concepts similar to that of Ä-sequences have appeared, asymptotic sequences and essential sequences. Motivated by these examples, this paper will study the general nature of such sequences, which we will refer to as grade sequences. We will generally restrict our attention to a fixed Noetherian ring R and all of its locahzations. For any ideal Is in any localization Rs, we will assign to Is a finite subset of A(IS) of spec Rs, the function A having certain nice properties to be specified in the next section. Foremost among these nice properties is the following. Suppose x1,...,x„ are in Is and xx £ A(ORs), and for i = l,...,n 1, xt+1 Í U [P e A((xx,...,x¡)Rs), and also suppose there is a prime Ps g A((xx,...,xn)Rs) with Is c Ps. Now suppose yx,... ,ym in Is behaves similarly. We will then insist that n = m. If so, A will be called a grade scheme on R, and to A we will associate a so-called grade function / defined by f(Is) = n, n as above. The most well-known instance of such an A and / is, of course, A(IS) = Ass(Rs/Is), and f(Is) is the grade of Is. As we will use the term 'grade' more generally, this famous special case will herein be called classical grade. In the next section, we will also give essential grade and asymptotic grade as examples of what we are studying. As mentioned earlier, we will usually just be looking at a grade scheme or grade function on a fixed ring R, and we will often custom build a grade function to exist on a given ring R. Certain grade functions, primarily classical, essential, and asymptotic grade, as well as the height function, automatically exist on all Noetherian rings, and we will refer to these as natural grade functions. The two-fold purpose of this paper is to study grade functions in the abstract, as well as to investigate which properties of the known natural grade functions are abstract properties, as opposed to being specific to the example at hand. In §1, we make precise our definitions, and give the various natural examples. An important point made in our examples is that two different grade schemes can give rise to the same grade function. We give some useful technical lemmas concerning Received by the editors February 10, 1984. 1980 Mathematics Subject Classification. Primary 13E99; Secondary 13B99.