Timeline for Spectrum of ring in algebraic geometry vs spectrum of Banach algebra
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Apr 9 at 9:40 | comment | added | Onur Oktay | You might like the exposition in Chapter 7 of Palmer's book books.google.ca/books?id=3k0lKh6QY-QC&pg=PA619 | |
| Apr 7 at 20:26 | comment | added | Robert Furber | In algebraic geometry, the maximal spectrum is not of much use for a local ring, and fails to distinguish between local rings and fields, so the prime spectrum is used (there are other good reasons as well). The reason we don't use the prime spectrum in Banach algebra theory is that in general there are a lot of weird prime ideals in $C(X)$ that don't have a geometric interpretation. Examples can be found in Gillman and Jerison's Rings of Continuous Functions. (Of course, if anyone knows a use for these ideals, other than as counterexamples, I'd be glad to hear it.) | |
| Apr 7 at 19:06 | comment | added | JJJ | The maximal ideals are the Zariski closed points of the prime spectrum. | |
| Apr 7 at 19:02 | comment | added | LSpice | It is also rather common for a "user's approach" to algebraic geometry to identify an affine variety $X$ over an algebraically closed field $k$ with its set of $k$-valued points, i.e., with the set of maximal ideals in $k[X]$. (For example, Borel - Linear algebraic groups takes this approach.) | |
| Apr 7 at 18:03 | history | asked | Ma Joad | CC BY-SA 4.0 |