How Does Infinity Manifest in Algebraic Concepts and Applications?

[pivoxa15]

How often does the notion of infinity occur in algebra? It usually deals with finite objects and processes right? Are there examples of where the notion of infinity is needed in algebra?

 

[matt grime]

Yes, all over the place. Any algebraically cloed field has infinitely many elements. There is nothing that forces you to use only finite dimensional algebras/vector spaces or finitely generated rings/modules. Whilst one is usually interested in finite dimensionl or finitely generated objects, there is no barrier to using 'infinite' things, and in general the study of limits is far far richer in algebra than it is in analysis.

http://www.maths.bris.ac.uk/~maxmg/maths/introductory/limits.html

was something I wrote about viewing limits in the category theoretic sense (but it is highly informal), but never finished.

Anyway, 'infinite' stuff has varying degrees of use in algebra. Topologists have long since accepted the need to use infinite C-W complexes for instance.

 

[HallsofIvy]

Although I would be hesitant to say that "the infinite" is used in Linear Algebra or any other field. It is more correct to say that there exist an infinite number of vectors in any vector space or that a vector space may have infinite dimension- using "infinite" as an adjective, not a noun!

 

[mathwonk]

one takes infinite products, direct sums, direct limits, inverse limits, completions wrt metrics defined by powers of ideals, those are some of my favorites, but there are more homological constructs too, involving infinite complexes,...