Editing Commutative ring (section)

== Homological notions ==
Several deeper aspects of commutative rings have been studied using methods from [[homological algebra]]. {{harvtxt|Hochster|2007}} lists some open questions in this area of active research.

=== Projective modules and Ext functors ===
Projective modules can be defined to be the [[direct summand]]s of free modules. If ''R'' is local, any finitely generated projective module is actually free, which gives content to an analogy between projective modules and [[vector bundle]]s.{{refn|See also [[Serre–Swan theorem]]}} The [[Quillen–Suslin theorem]] asserts that any finitely generated projective module over ''k''[''T''<sub>1</sub>, ..., ''T''<sub>''n''</sub>] (''k'' a field) is free, but in general these two concepts differ. 
A local Noetherian ring is regular if and only if its [[global dimension]] is finite, say ''n'', which means that any finitely generated ''R''-module has a [[resolution (homological algebra)|resolution]] by projective modules of length at most ''n''.

The proof of this and other related statements relies on the usage of homological methods, such as the
[[Ext functor]]. This functor is the [[derived functor]] of the functor
{{block indent|1= Hom<sub>''R''</sub>(''M'', &minus;). }}
The latter functor is exact if ''M'' is projective, but not otherwise: for a surjective map {{nowrap|''E'' &rarr; ''F''}} of ''R''-modules, a map {{nowrap|''M'' &rarr; ''F''}} need not extend to a map {{nowrap|''M'' &rarr; ''E''}}. The higher Ext functors measure the non-exactness of the Hom-functor. The importance of this standard construction in homological algebra stems can be seen from the fact that a local Noetherian ring ''R'' with residue field ''k'' is regular if and only if
{{block indent|1= Ext<sup>''n''</sup>(''k'', ''k'') }}
vanishes for all large enough ''n''. Moreover, the dimensions of these Ext-groups, known as [[Betti number]]s, grow polynomially in ''n'' if and only if ''R'' is a [[local complete intersection]] ring.{{sfn|Christensen|Striuli|Veliche|2010|ps=}} A key argument in such considerations is the [[Koszul complex]], which provides an explicit free resolution of the residue field ''k'' of a local ring ''R'' in terms of a regular sequence.

=== Flatness ===
The [[tensor product]] is another non-exact functor relevant in the context of commutative rings: for a general ''R''-module ''M'', the functor
{{block indent|1= ''M'' ⊗<sub>''R''</sub> &minus; }}
is only right exact. If it is exact, ''M'' is called [[flat module|flat]]. If ''R'' is local, any finitely presented flat module is free of finite rank, thus projective. Despite being defined in terms of homological algebra, flatness has profound geometric implications. For example, if an ''R''-algebra ''S'' is flat, the dimensions of the fibers
{{block indent|1= ''S'' / ''pS'' = ''S'' ⊗<sub>''R''</sub> ''R'' / ''p'' }}
(for prime ideals ''p'' in ''R'') have the "expected" dimension, namely {{nowrap|dim ''S'' &minus; dim ''R'' + dim(''R'' / ''p'')}}.