rank in 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) i∈I(w_i)_{i \in I} of NN is called a basis of NN (over AA) if for every x∈Nx \in N there is a unique collection (a i) i∈I(a_i)_{i \in I} of elements of AA such that a i=0a_i = 0 for all but finitely many i∈Ii \in I and x=∑ i∈Ia 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 a finite number.

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

• any nontrivial commutative ring KK,

• the group ring K(G)K(G) for KK any field (or nontrival commutative ring?) and GG any group,

• any Noetherian ring.

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 ℵ 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 ℵ 0=nℵ 0\aleph_0 = n \aleph_0 (applied to the columns).

Rank of a linear map

Given a linear map, hence a homomorphism of modules, its rank is the rank of its image-module.

Often this is considered for the case that the linear map is represented by a matrix and one speaks of the rank of a matrix.

Rank of a sheaf of modules

Let (X,𝒪)(X,\mathcal{O}) be a locally ringed space and ℰ\mathcal{E} a 𝒪\mathcal{O}-module. Then its rank at a point x∈Xx \in X is the vector space dimension of the fiber ℰ(x)≔ℰ 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 ℰ\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 ℰ 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 X→ℕX \to \mathbb{N}.

In the internal language of the sheaf topos Sh(X)\mathrm{Sh}(X), the rank of ℰ\mathcal{E} can internally quite simply be defined as the minimal number of elements needed to generate ℰ\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 X→ℕX \to \mathbb{N} (details at one-sided real number), this definition coincides with the usual one if ℰ\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=⋃ x∈S(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 a↦a∪{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 |⋃ i∈IX i||\bigcup_{i\in I} X_i |<α\alpha whenever |I||I|<α\alpha and |X i||X_i|<α\alpha for all i∈Ii\in I.

A functor F:𝒜→ℬF:\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. A monad has rank (α\alpha) when its underlying endofunctor does.

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

Rank of a Lie group

• rank-nullity theorem

• rank of a Lie group

• rank of a coherent sheaf,

  degree of a coherent sheaf

  slope of a coherent sheaf

• stable coherent sheaf, Bridgeland stability condition

• rank-nullity theorem

References

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