Rank

Idea

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

Rank of a module

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

A collection of elements $(w_i)_{i \in I}$ of $N$ is called a basis of $N$ (over $A$) if for every $x \in N$ there is a unique collection $(a_i)_{i \in I}$ of elements of $A$ such that $a_i = 0$ for all but finitely many $i \in I$ and $x = \sum_{i \in I} a_i w_i$.

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

• the group ring $K(G)$ for $K$ any field (or nontrival commutative ring?) and $G$ 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 $\aleph_0$-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 $\aleph_0 = n \aleph_0$ (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 $(X,\mathcal{O})$ be a locally ringed space and $\mathcal{E}$ a $\mathcal{O}$-module. Then its rank at a point $x \in X$ is the vector space dimension of the fiber $\mathcal{E}(x) \coloneqq \mathcal{E}_x \otimes_{\mathcal{O}_x} k(x)$ over the residue field $k(x)$.

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

In the internal language of the sheaf topos $\mathrm{Sh}(X)$, 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 $\mathrm{Sh}(X)$ and upper semicontinuous functions $X \to \mathbb{N}$ (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 $n$ if each fiber is a vector space of dimension $n$.

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:

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

Rank of a functor

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

A functor $F:\mathcal{A}\to \mathcal{B}$ has rank $\alpha$ for some regular cardinal $\alpha$ if $F$ preserves $\alpha$-filtered colimits. $F$ has rank when it has rank $\alpha$ for some regular cardinal $\alpha$. A monad has rank ($\alpha$) 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

Related concepts

• rank-nullity theorem

References

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