Talk:Spectrum of a ring

WikiProject Mathematics (Rated Start-class, Mid-priority)

Spectrum of an operator

Can someone please elaborate on what the object K[T] is? The article mentions that a vector space, equipped with a linear operator on it, can be viewed as a module over K[T], but then subsequently only mentions the ring K[T]. If K[T] is just another name for the polynomial ring K[x], then how are the spectral properties of a particular operator T supposed to manifest in the spectrum of K[x]? Or is K[T] instead supposed to be something like "the polynomial ring mod out by the minimal polynomial of T"? — Preceding unsigned comment added by 129.97.226.227 (talk) 12:38, 10 June 2013 (UTC)

You should think of K[T] as living inside GL(V). Inside GL(V), there is a field K (determined by diagonal matrices) and an element T (given to us). The subring of GL(V) generated by K and T is K[T].
This kind of notation turns up in other places. For instance, it's analogous to the notation Q(i) for the field of rational numbers together with a square root of −1 (which is a complex number—note that we've implicitly chosen a copy of C containing Q).
General facts imply that K[T] is isomorphic to K[x] modulo the minimal polynomial of T. Indeed by the universal property of K[x], there is a homomorphism K[x] → K[T] that sends x to T. The map is obviously surjective, and the kernel is generated by the minimal polynomial of T by the definition of the minimal polynomial.
By the way, on Wikipedia talk pages it's conventional to write new comments at the bottom. Ozob (talk) 14:04, 10 June 2013 (UTC)

Removed example

Header added. —Nils von Barth (nbarth) (talk) 07:48, 7 December 2009 (UTC)

I removed the following example:

A special but quite typical case of an affine scheme is obtained as follows. Take a field K and n variables, x1,...,xn. Given m polynomials, p1,...,pm in these variables over K, there is a functor F from the category of commutative K-algebras to sets characterized by F(A)={(x1,...,xn) in An|p1=...=pm=0}. Then F is represented by Spec(B) where B is the quotient of K[x1,...,xn] by the ideal I generated by the pj.

Reasons:

• This uses functors, but the article hasn't mentioned the functor connection yet.
• The technical term "represented" is not explained. The functor F is not represented by Spec(B), but by B.
• A more useful example would describe Spec(B) in detail.

AxelBoldt 16:07, 18 Jan 2004 (UTC)

IMO, this article contains too much general rubbish. Just focus on the connections to geometry. Schemes are a generalisation of this setting and reside in a seperate article. agrosquid