3

I don't know both of these subjects, but I was wondering if there was any topology in commutative algebra. I don't need any detailed answer (since I don't know any of them yet)...So would it be helpful to know topology when studying commutative algebra, or is there no connection between them?

Thank you in advance

  • I think generally the link tends to go the other way. We can use commutative algebra to tell us things about topology (a small part of the much larger field of algebraic topology). Algebraic geometry may have something to say about commutative algebra though (I'm not too sure on this as I have very little background in algebraic geometry). – Dan Rust Jul 09 '13 at 19:36
  • 5
    "… commutative algebra is a lot like topology, only backwards." - John Baez – Zev Chonoles Jul 09 '13 at 19:39
  • 2
    Yes! Definitely there is topology in commutative algebra. A common topological space in commutative algebra is the Zariski Topology, localisation of rings gives you pre-sheaf (another techniques in algebraic topology). You also have things like the Koszul complex, Hilbert syzygy theorem, where you use a lot of homological algebra. Derived functors such as Tor and Ext are also quite common in comm alg, and they are also techniques in algebraic topology. Direct limits, inverse limits etc... in my opinion, it takes an imaginative mind to apply algebraic topology to comm alg! – enoughsaid05 Jul 09 '13 at 19:53

1 Answers1

4

Yes, it is useful to know some general topology to make any headway in commutative algebra. A very powerful technique in commutative algebra is that of completion of a local ring. Such complete rings are often easier to deal with and geometrically they contain important local information.

The process of completion actually borrows amply from general topology. Also topology is the first instance where you start worrying about important properties like compactness, separatedness, properness etc. It will be very difficult to understand the application of some concepts of commutative algebra (in algebraic geometry I guess) without topology.

DBS
  • 935
  • 1
    An important part of commutative algebra/algberaic number theory is study of discrete valuation rings and valued fields. It is inconceivable that you will have any clear insight about such objects without being conversant in (arid) general topology. – DBS Jul 09 '13 at 20:13