Timeline for Varieties as an introduction to algebraic geometry / How do professional algebraic geometers think about varieties
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21 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Oct 21, 2010 at 14:02 | comment | added | Harry Gindi | That is a good question. I guess I need to learn more geometry! | |
| Oct 21, 2010 at 12:22 | comment | added | Emerton | Dear Harry, Yes, I realize that you can define the structure sheaf as the sheaf of morphisms to $\mathbb A^1$, but as I asked above, to what end? | |
| Oct 21, 2010 at 11:01 | comment | added | Harry Gindi | Since your category of varieties embeds fully and faithfully, we can see then that there is an extremely easy way to define the structure sheaf on a variety. Simply send an open subset U to the ring $Hom_{Var}(U,A^1)$. This turns all of your varieties into legitimate ringed spaces whose structure sheaves are actually sheaves of functions! What more could you want? | |
| Oct 21, 2010 at 10:52 | comment | added | Harry Gindi | To see that this makes sense, it is enough to show that $Spec:CRing\to Top$ is faithful. Then since $A^1$ is affine, we can give a sheaf on any affine scheme $X$ by defining for any open affine subscheme $U\subseteq X$ the set $Hom_{Aff}(U,A^1)$. It's clear then that this forms a B-sheaf on the space $X$, and that when completed to a sheaf, turns $X$ into a locally ringed space, and $Hom_Aff(X,Y)=Hom_{Loc}(X,Y)$. Given any scheme, then, we see that the structure sheaf is simply just $Hom_{Loc}(-,A^1)$. That is, $U$-sections of the sheaf are literally just maps $U\to A^1$. | |
| Oct 21, 2010 at 10:39 | comment | added | Harry Gindi | Dear Emerton, The reason why I mentioned looking at maps into A^1 is that they bijectively classify global sections of the structure sheaf. In particular, the restriction maps of the sheaf become actual legitimate restriction maps of functions. This means that instead of constructing the structure sheaf by hand, we can always construct a B-sheaf on the affine opens simply defined by $Hom(U,A^1)$. Then to define morphisms of any schemes, it suffices to define what a morphism of affines is, but to do this, we simply define $Hom_{Aff}(Spec S, Spec R)$ to be $Hom(R,S)$. | |
| Oct 21, 2010 at 8:09 | comment | added | Emerton | ... coordinate rings, and so on.) | |
| Oct 21, 2010 at 8:08 | comment | added | Emerton | ... I would prefer to do some geometry first. (Sheaves really demonstrate their merit once you start taking their cohomology. But sheaf cohomology is a fairly sophisticated tool, and I would rather introduce and explore some geometric problems, before describing how to solve them using cohomology. For example, arguments with $\mathcal O(n)$ and its cohomology on projective space are much more easily motivated, and understood, once you have some experience with the classical idea of taking intersections with hyperplane or hypersurfaces, of arguing with graded pieces of homogeneous ... | |
| Oct 21, 2010 at 8:04 | comment | added | Emerton | Dear Harry, For $\mathbb C$-points in a $\mathbb Q$-variety, if your variety is a concrete closed subset (say of $\mathbb Q$-bar points) in affine space over $\mathbb Q$-bar, then you can take its Zariski closure in the $\mathbb Q$-Zariski topology (i.e. the topology defined using equations with coeffs. in $\mathbb Q$) in affine space over $\mathbb C$ to get the $\mathbb C$-points. I should add that (in the set-up I have in mind, of which the preceding gives a hint) you can define morphisms from a variety to $\mathbb A^1$, and show it is a Zariski sheaf, but to what end? ... | |
| Oct 21, 2010 at 7:27 | comment | added | Harry Gindi | Dear Emerton, By the way, please don't interpret this as me questioning your judgement. I'm just honestly curious about what advantages there are to teaching it thsi way. | |
| Oct 21, 2010 at 7:19 | comment | added | Harry Gindi | Dear BCnrd, That's fair. I'll defer to your expertise, especially because you've tried it before! | |
| Oct 21, 2010 at 7:17 | comment | added | Harry Gindi | Dear Emerton, How are you looking at the $\mathbf{C}$-points of a $\mathbf{Q}$-variety? If you're doing things classically, with things done in terms of equations and functions, won't you run into the same problems whether you're using sheaves or not? | |
| Oct 21, 2010 at 7:17 | comment | added | BCnrd | Harry, in my undergrad differential geometry course at UM I only discussed the idea of a sheaf of R-valued functions (avoiding the word "sheaf"!) as a substitute for atlases, and some related notions for different ways to think about vector bundles. Bad idea to introduce general concepts of sheaves or non-closed points in undergraduate courses. One merit to discussing sheaf of k-valued functions is normalization of non-affine varieties, and requires no non-closed pts or general sheaf stuff, but I tried that experiment once with an undergrad a.g. course and it was very hard for the students. | |
| Oct 21, 2010 at 7:13 | comment | added | Emerton | ... classical varieties first. My feeling is that certain geometric ideas (Bezout's theorem, blowing up, families of varieties, etc.), which are at the heart of algebraic geometry, can be profitably introduced first in the simple context of varieties. Certainly one can use sheaf- and scheme-theoretic ideas to go further in the study of them, but at the beginning, having a lot of technical baggage can (in my view) obscure the simple geometric ideas. Once you understand the ideas, the technical baggage itself becomes much less of a burden, since one then knows its intent and purpose. | |
| Oct 21, 2010 at 7:08 | comment | added | Emerton | Dear Harry, Yes, one can do this, but there are already complications; if one wants sheaves of functions, but also wants to, e.g., study the $\mathbb C$-points of a variety defined over $\mathbb Q$, one has to worry a bit about what field the functions take values in. Alternatively, one just passes to the scheme picture, but then one has to discuss abstract sheaves, rather than sheaves of functions, which is more baggage. My view is that schemes are an extended metaphor --- and to understand the geometric intuitions for which they are a metaphor, it helps to understand ... | |
| Oct 21, 2010 at 6:31 | comment | added | Harry Gindi | Granted, the hard part is defining what kinds of functions are admissible at the local level, but then you can use the classical point of view to say that they are regular functions locally. | |
| Oct 21, 2010 at 6:25 | comment | added | Harry Gindi | Dear Emerton, at least in the undergraduate sequence I took here, we were introduced to structure sheaves when we defined differentiable manifolds. It is my understanding that Brian Conrad also discussed/defined/proved theorems about them when he taught the course. It doesn't really seem like that much of a stretch to extend that line of reasoning to look at maps into $A^1 k$. Just like with manifolds, you can define such maps at a locally, and define the structure sheaf globally by gluing affines. (Not that you don't already know this.) The intution carries over from manifolds exactly. | |
| Oct 21, 2010 at 5:56 | comment | added | Emerton | $\mathbb Q$, or the algebraic closure of $\mathbb C(x_1,x_2,\ldots,x_n,\ldots)$ over $\mathbb C$), then looking at $\Omega$-valued points of algebraic sets over $k$ implicitly involves looking at non-closed points of the associated scheme. The main advantage, in any event, is that you can get somewhere in finite time, and actually do some geometry. | |
| Oct 21, 2010 at 5:49 | comment | added | Emerton | You can organize the data in an actual structure without mentioning sheaves in any substantial way: for any field $k$, there is a well-defined category of quasi-projective algebraic sets, admitting (finite) products and fibre products, in which one can prove many geometric results, and whose construction doesn't require sheaves/locally ringed spaces/etc. (I am teaching this in my algebraic geometry course this quarter!) Also, the distinction between closed and non-closed points is less than might appear at first sight: if $\Omega$ is a very big extension of $k$ (think of $\mathbb C$ over ... | |
| Oct 21, 2010 at 5:16 | comment | added | Harry Gindi | Dear Emerton, is there really an advantage to the classical approach of only looking at closed points, not talking about the obvious sheaf-structure of the obvious sheaf of regular functions, and generally being coy about organizing the data in an actual structure? | |
| Oct 21, 2010 at 4:59 | history | answered | Emerton | CC BY-SA 2.5 |