In
mathematics, a
commutative ring ''R'' is catenary if for any pair of
prime ideals
:''p'', ''q'',
any two strictly increasing chains
:''p''=''p''
0 ⊂''p''
1 ... ⊂''p''
''n''= ''q'' of prime ideals
are contained in maximal strictly increasing chains from ''p'' to ''q'' of the same (finite) length. In a geometric situation, in which the
dimension of an algebraic variety
In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways.
Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commut ...
attached to a prime ideal will decrease as the prime ideal becomes bigger, the length of such a chain ''n'' is usually the difference in dimensions.
A ring is called universally catenary if all finitely generated algebras over it are catenary rings.
The word 'catenary' is derived from the Latin word ''catena'', which means "chain".
There is the following chain of inclusions.
Dimension formula
Suppose that ''A'' is a Noetherian domain and ''B'' is a domain containing ''A'' that is finitely generated over ''A''. If ''P'' is a prime ideal of ''B'' and ''p'' its intersection with ''A'', then
:
The dimension formula for universally catenary rings says that equality holds if ''A'' is universally catenary. Here κ(''P'') is the
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
of ''P'' and tr.deg. means the transcendence degree (of quotient fields). In fact, when ''A'' is not universally catenary, but