rank

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

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 *free* (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 finite.

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).

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.

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:

$rank S = \bigcup_{x \in S} (rank x)^+ ,$

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

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