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 a free module (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.

See also rank of a coherent sheaf.

Rank of a vector bundle

As a simple special case of the above, a vector bundle is said to have rank nn if each fiber is a vector space of dimension nn.

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).

Rank of a functor

Recall that a cardinal number α\alpha is said to be regular if | iIX i||\bigcup_{i\in I} X_i |<α\alpha whenever |I||I|<α\alpha and |X i||X_i|<α\alpha for all iIi\in I.

A functor F:𝒜F:\mathcal{A}\to \mathcal{B} has rank α\alpha for some regular cardinal α\alpha if FF preserves α\alpha-filtered colimits. FF has rank when it has rank α\alpha for some regular cardinal α\alpha.

The properties of functors with rank are discussed in section 5.5 of Borceux (1994).

Rank of a Lie group

Reference

  • Francis Borceux, Handbook of Categorical Algebra vol. 2 , Cambridge UP 1994.

Revised on February 8, 2017 03:30:54 by Thomas Holder (2.247.247.90)