nLab
rank

Rank

Idea

The term ‘rank’ is used in many contexts to number levels within a hierarchy.

Rank of a module

Let AA be a ring and NN a module over AA. If AA is a field, then NN is a vector space and we speak of the dimension of NN; in the general case, we may speak of the rank.

A collection of elements (w i) iI(w_i)_{i \in I} of NN is called a basis of NN (over AA) if for every xNx \in N there is a unique collection (a i) iI(a_i)_{i \in I} of elements of AA such that a i=0a_i = 0 for all but finitely many iIi \in I and x= iIa iw ix = \sum_{i \in I} a_i w_i.

If NN has a basis it is called free (over AA). For many examples of AA (the invariant basis number rings), the cardinality #I# I only depends on NN and not on the choice of basis. It is called the rank of NN over AA, notation: rank A(M)rank_A(M). In any case, NN is called the free module of rank #I# I. If NN is a finitely generated free module then the rank is finite.

All of the following are invariant basis rings (source: Wikipedia):

Besides the trivial ring (over which any module is free with any set as basis), an example of a ring without invariant basis number is the ring of 0\aleph_0-dimensional square matrices (over any ring) in which each column has only finitely many nonzero entries (which allows multiplication to be defined). As a module over itself, this ring is free on any inhabited finite set, as may be shown by using the equation 0=n 0\aleph_0 = n \aleph_0 (applied to the columns).

Rank of a sheaf of modules

Let (X,𝒪)(X,\mathcal{O}) be a locally ringed space and \mathcal{E} a 𝒪\mathcal{O}-module. Then its rank at a point xXx \in X is the vector space dimension of the fiber (x) x 𝒪 xk(x)\mathcal{E}(x) \coloneqq \mathcal{E}_x \otimes_{\mathcal{O}_x} k(x) over the residue field k(x)k(x).

If \mathcal{E} is of finite type, then the rank at xx can equivalently be defined as the minimal number of elements needed to generate the stalk x\mathcal{E}_x as a 𝒪 x\mathcal{O}_x-module (by Nakayama's lemma). In this case, the rank is a upper semicontinuous function XX \to \mathbb{N}.

In the internal language of the sheaf topos Sh(X)\mathrm{Sh}(X), the rank of \mathcal{E} can internally quite simply be defined as the minimal number of elements needed to generate \mathcal{E} (taken as an element of the suitably completed natural numbers, i.e. the poset of inhabited upper sets). Under the correspondence of internal inhabited upper sets in Sh(X)\mathrm{Sh}(X) and upper semicontinuous functions XX \to \mathbb{N} (details at one-sided real number), this definition coincides with the usual one if \mathcal{E} is of finite type; see this MathOverflow question.

Hereditary rank of a pure set

Every pure set within the von Neumann hierarchy appears first at some level given by an ordinal number; this number is its hereditary rank.

We may define this rank explicitly (and recursively) as follows:

rankS= xS(rankx) +, rank S = \bigcup_{x \in S} (rank x)^+ ,

where \bigcup is the supremum operation on ordinals (literally the union for von Neumann ordinals) and () +(-)^+ is the successor operation (which is aa{a}a \mapsto a \cup \{a\} for von Neumann ordinals).

Revised on February 8, 2013 19:44:59 by Ingo Blechschmidt (137.250.162.16)