Commutative and homological algebra?

[pivoxa15]

How is commutative algebra and homological algebra linked? Does one build on from the other or separate?

 

[matt grime]

Commutative algebra really means algebraic geometry. The two subjects are simultaneously completely independent and obviously linked. That might seem strange to say, but you will only think that if you believe maths to be linearly ordered.

 

[pivoxa15]

Commutative algebra really means algebraic geometry. The two subjects are simultaneously completely independent and obviously linked. That might seem strange to say, but you will only think that if you believe maths to be linearly ordered.

I know homological algebra and algebraic geometry are linked. So homological algebra must be linked with commutative algebra. It's like analytic continuation in complex analysis? Apparently differerent functions (or fields in this case) are linked somehow in the larger domain.

 

[matt grime]

Homological algebra is the study of (abstract) homology and cohomology theories. Originally it comes from algebraic topology. Varieties are topological spaces, so of course there is a 'link'. Moreover a homology theory will frequently give a commutative graded ring as its ouput. But this is not a facet of a deep relationship as you seem to crave - the algebra required to do homology is almost trivial, as is the topology needed in algebraic geometry.

I don't think that it is at all reasonable to imply that there is something deeply causal about this.

 

[pivoxa15]

It seems commutative algebra is older and more fundalmental (or closer to algebra if that makes sense) then homological algebra. Maybe it does since homological algebra originally came from algebraic topology as you pointed out, namely homology?

Study commutative algebra first?

 

[matt grime]

Why do you insist on having to study one before the other? Maths isn't linearly ordered.

 

[mathwonk]

matt is more expert than I on algebra and particularly homological algebra.

still let me make some remarks from my experience as a student of some top algebraists of the previous generation.

as you say, commutative algebra was developed greatly by zariski as an aid to algebraic geometry, to put it on a firm foundation, from the 50's.

then in the 60's eilenberg's students auslander and buchsbaum introduced homological algebra into commutative algebra and made fundamental advances, in particular the first proof that all regular local rings are ufd's, and that a regular local ring is characterized by having finiye homological dimension. this showed clearly a firm connection between commutative algebraic notions and homological ones..

this was taken up by serre, who pushed the subject further and wrote a basic treatise, algebre locale, multiplicites, and it continued from there,...

in particular intersection theory is very homological in nature now,.. and after that i have lost contact, with the new areas of the subject as advanced by matt, and jon carlson, and dave benson, and dan nakano, ...

the newer concepts include periodic resolutions,...ask matt for an update.