Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.2 9/18/84; site faron.UUCP Path: utzoo!watmath!clyde!burl!ulysses!unc!mcnc!decvax!linus!faron!wdh From: wdh@faron.UUCP (Dale Hall) Newsgroups: net.math Subject: Re: Euler's formula and arithmetic dimensions Message-ID: <401@faron.UUCP> Date: Wed, 27-Nov-85 17:37:55 EST Article-I.D.: faron.401 Posted: Wed Nov 27 17:37:55 1985 Date-Received: Fri, 29-Nov-85 10:56:27 EST References: <1095@enea.UUCP> <64@brl-tgr.ARPA> Reply-To: wdh@faron.UUCP (Dale Hall) Organization: The MITRE Coporation, Bedford, MA Lines: 52 Keywords: Spec(whatever) In article <64@brl-tgr.ARPA> gwyn@brl-tgr.ARPA (Doug Gwyn ) writes: > >Tell us more about Spec(Z), whose elements are apparently just the >positive prime integers. What is this "algebro-geometric spectrum"? > >I have to object to calling every factor algebra a "dimension". Last point first: this isn't an episode of "here's something new! Let's call it a dimension." The notion of dimension for commutative rings (as the maximum length of a chain of prime ideals) is due to Krull (German mathematician, early 20th century [?]), and is relevant to what one would ordinarily think of as the dimension of a topological space. The construction of Spec(R), for a commutative ring R, proceeds as follows: Spec(R): the set of all prime ideals P in R, with a basis of open sets given by the set-theoretic complements (Spec(R) \ V(x)), for x in R; here, the (closed) set V(x) is: V(x) = {Q in Spec(R)| x in Q}. For R = Z (integers), Spec(Z) = {pZ | p :prime}, where 0 is prime, but 1 isn't. The topology restricted to non-zero primes is discrete, but the prime 0 is dense in Spec(Z) - it's called a generic point. The dimension of the prime spectrum Spec(Z) is 1. If k is an algebraically closed field (take your favorite one), and R = k[X], the ring of polynomials in one indeterminate, then Spec(R) is called the affine line over k. Its regular points correspond to maximal ideals in k[X], which are in 1-1 correspondence with the elements of k, and there is again a generic point, given by the zero ideal in k[X]. (this case is similar to that of Spec(Z) - in fact, k[X] is itself similar to Z, as many of you will undoubtedly suspect). The dimension of Spec(k[X]) is also 1. This generalizes to the ring R = k[X ,X ,...,X ], the ring of polynomials in n 1 2 n indeterminates over k; the prime spectrum Spec(R) is then affine n-space over k, and, except for the generic point (guess what?), Spec(R) looks just like the space of n-tuples of elements of k. Then, it can start to get a little technical. (you like sheaves? we got sheaves. you like cohomology? you came to the right place, we got cohomology. how about some nice schemes? a few Chern classes, the Todd genus, and you got a very attractive Riemann-Roch theorem ...) Enjoy. References M. Nagata, Local Rings - Wiley H. Matsumura, Commutative Algebra - Benjamin I.R.Shafarevitch, Basic Algebraic Geometry - Springer Verlag R. Hartshorne, Algebraic Geometry - Springer Verlag