Kevin Buzzard has a word of advice about the "generic point": John Baez wrote: We can think of elements of a commutative ring R as functions on certain space called the "spectrum" of R, Spec(R). So this is the set of all _prime_ ideals of R, right? Not just the maximal ones? So... So, the Riemann sphere is a scheme! Well, you have to throw in a mystical extra "generic point" if you really want to make it a scheme :-) Corresponding to the zero ideal. My impression is that most non-algebraic geometers think that the generic point is either confusing or just plain daft. But believe me, it's a *really* good idea! For decades in the literature in algebraic geometry people were using the word "generic" to mean "something that was true most of the time"---in fact a "generic point" is probably another really good example of a phantom! For example a meromorphic function on the Riemann sphere that wasn't zero would be "generically non-zero" to people like Borel and Weil, and if you asked them for a definition they would say that it just meant something like "the zero locus in the space had a smaller dimension than the whole space" or "the zero locus was nowhere dense in any component of the space" or something, and of course people could even make rigorous definitions that worked in particular cases and so on, but then Grothendieck came along with his "generic point", corresponding to the zero ideal [note to sub: check to see whether the idea was in the literature pre-Grothendieck!] and suddenly a function that was "generically non-zero" was just a function which was non-zero on the generic point! Such a cool way of doing it :-) Kevin If you get stuck on my puzzle "what does it mean for an n-ary operation to commute with an m-ary operation?", let me just show you what it means for a binary operation f to commute with a ternary operation g. It means: g(f(x_1,x_2), f(x_3,x_4), f(x_5,x_6)) = f(g(x_1,x_2,x_3), g(x_4,x_5,x_6)) I hope this example gives away the general pattern. If this is confusing, look at the case where we start with a ring R and take as our n-ary operations the "n-ary R-linear combinations" (x_1, ..., x_n) |-> r_1 x_1 + ... + r_n x_n with r_i in R. Here an example of a binary operation is addition: (x_1, x_2) |-> x_1 + x_2 while every unary operation is multiplication by some element of R: x1 |-> r x1 To say "addition commutes with multiplication by an element of R" means that r(x_1 + x_2) = rx_1 + rx_2 This is just the distributive law so it holds for any ring R. But, for the unary operations to commute with each other, we need R to be commutative, since this says: r(s x_1) = s (r x_1) (In the calculations I just did, we can either think of the x_i as elements of a specific R-module, or more abstractly as "dummy variables" used to describe the ring R as a generalized ring in Durov's sense - what Lawvere calls an algebraic theory.) For more discussion, go to the n-Category Cafe: http://golem.ph.utexas.edu/category/2007/12/geometric_representation_theor_14.html ----------------------------------------------------------------------- Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twfcontents.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html