Exploring Continuity in Modern Algebra: Origins and Applications

[pivoxa15]

Does the notion of continuity exist in modern algebra?

If so how do they arise?

 

[Hurkyl]

You can certainly have topological monoids, groups, rings, et cetera. And you can construct topological spaces related to algebraic structures.

 

[mathwonk]

see zariski topology, and the metric completion of a ring wrt an ideal.

 

[pivoxa15]

So it can happen in algebra but rare for pure algebra as opposed to algebra mixed with other maths disciplines like topology.

But would you say 'discreteness' plays a large role in modern algebra?

How would it sound if someone said they like modern algebra because of its discreteness which lacks in say analysis and topology?

 

[Hurkyl]

I would think you meant something more like "exact" than "discrete". I'm not sure what you consider "pure", but commutative algebra, at least, is very closely intertwined with algebraic geometry.

 

[mathwonk]

define as metric on a ring wrt a fixed maximal ideal m, where the distasnce between f and g is 1/2^n, where n the largest n such that f-g belongs to m^n.then in this metric, where m = the maximal ideal of the origin in the polynomial ring, a formal power series is the limit of the sequence of its partial sums. i.e. one can complete a polynomial ring wrt a maximal ideal, and get a power series ring.

this leads to the concept of the completion of a ring wrt an ideal. this concept helps recover local concepts in algebra,

"more local" than given by the zariski topology.

 

[matt grime]

How many angels can dance on the head of a pin? More semantic hair splitting. Go, on, pixova, define 'pure algebra'...

 

[mathwonk]

a basic philosophy is that to understand a thing, one wants to construct as many invariants of it as possible. so to understand a ring, one can consider obvious invariants such as its cardinality, or its group of units.

but one can also look at more complicated invariants such as the associated affine scheme, or in the case of a local ring, its completion wrt the metric defined above.

for the local ring of a smooth point on an algebraic variety this leads to a proof of the theorem in pure algebra that the ring is a ufd because its completion is so.

 

[HallsofIvy]

I would be inclined to say that all of those many situations in which you have both continuity (i.e. topology) and an algebraic structure are "analysis" rather than just algebra. In my (probably idiosycratic) opinion, analysis is the combination of algebra with topology.

 

[pivoxa15]

I would be inclined to say that all of those many situations in which you have both continuity (i.e. topology) and an algebraic structure are "analysis" rather than just algebra. In my (probably idiosycratic) opinion, analysis is the combination of algebra with topology.

So is that why Herman Weyl said something like "...algebra and topology fight for every branch of mathematics." So even a fundalmental discipline like analysis is a combo of algebra and topology.

 

[pivoxa15]

How many angels can dance on the head of a pin? More semantic hair splitting. Go, on, pixova, define 'pure algebra'...

I can't define pure algebra just as I can't define mathematics. But I can give some examples. Galois theory is pure algebra. Whereas topics in algebraic topology is not pure algebra. There may also me things in between.

 

[pivoxa15]

a basic philosophy is that to understand a thing, one wants to construct as many invariants of it as possible. so to understand a ring, one can consider obvious invariants such as its cardinality, or its group of units.

but one can also look at more complicated invariants such as the associated affine scheme, or in the case of a local ring, its completion wrt the metric defined above.

for the local ring of a smooth point on an algebraic variety this leads to a proof of the theorem in pure algebra that the ring is a ufd because its completion is so.

What do you mean by invariants? Never changing?

http://en.wiktionary.org/wiki/invariant

Do you mean its properties?

 

[mathwonk]

An invariant is almost another name for a functor. I.e. an object or quantity F(X) constructed from X is an invariant of [the isomorphism type of] X if F(X) and F(Y) are isomorphic or equal whenever X and Y are.

 

[matt grime]

Examples: (co)homology rings, fundamental groups, etc of topological spaces are invariant under homeomorphism (and homotopy).

The ultimate aim of mathematics for some people is to find a set of invariants that uniquely characterise objects up to whatever level of equivalence you are working with. See, e.g. compact surfaces, genus, homotopy, and the poincare conjecture, for example.

 

[Hurkyl]

Galois theory is pure algebra.

The Galois group is profinite, its topology plays a criticial role in the theory.

Finite profinite groups are discrete -- if you've only worked with finite Galois extensions, that would explain why you haven't seen this before. A useful example of an infinite Galois extension is $\mathbf{\bar{Q}} / \mathbf{Q}$: the algebraic numbers over the rationals.

 

[matt grime]

Whereas topics in algebraic topology is not pure algebra.

Why not? Yes, I am playing devil's advocate, but I'd like you to examine your own preconceptions about mathematics.

 

[mathwonk]

i believe herman weyl actually said more like: " the angel of topology and the devil of abstract algebra fight for the soul of every discipline in mathematics."

 

[HallsofIvy]

Oh, I like that!

Does the fact that he said "angel of topology" and "devil of abstract algebra" tell us that he was a topologist?

 

[mathwonk]

read some of weyls books. he is author of some outstanding algebra books. as well as books on riemann surfaces,...