nLab
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 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: ${\mathrm{rank}}_A(M)$. 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).

Rank of a sheaf of modules

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

If $ℰ$ is of finite type, then the rank at $x$ can equivalently be defined as the minimal number of elements needed to generate the stalk ${ℰ}_x$ as a ${𝒪}_x$-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 $ℰ$ can internally quite simply be defined as the minimal number of elements needed to generate $ℰ$ (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 $ℰ$ 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:

$$\mathrm{rank}S=\bigcup _{x\in S}(\mathrm{rank}x{)}^{+},$$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).