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Scheme Theory

Anonymous Tue 21 Jul 03:10:23 2020 No.11921294 View Reply Original Report

Quoted By: >>11921330

Yo, can someone give me an example of a scheme that isn't fucking dumb? Worst case I'm willing to chug 200 pages of categorical bullshit to get to real examples, but dear god something besides affine schemes, varieties, Proj bullshit and the line with 2 points (or any nonsense that is just impossible to apprehend without reducing to one of those) would be great for my intuition.

Also, I'm willing to accept the fibered product, especially in light of the notion of S-valued points, but intuition for what it actually is which is given in a way that doesn't reduce to handwaving about it being the appropriate patching of information given by the pullback of spectra / tensor products of modules in constructing sheaves of the same (i.e., as an object which purports to be "geometric") would be great.

Inb4 EGA didn't have any pictures

El Arcón

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El Arcón Tue 21 Jul 2020 03:15:36 No.11921308 Report

Quoted By:

Fractional distance scheme.
https://ufile.io/0rkqyt30

>Fractional Distance: The Topology of the Real Number Line with Applications to the Riemann Hypothesis
>https://vixra.org/abs/1906.0237

Anonymous

Anonymous Tue 21 Jul 2020 03:28:54 No.11921330 Report

Quoted By: >>11921507

>>11921294
redpill me on sheaf theory

Anonymous

Anonymous Tue 21 Jul 2020 04:44:33 No.11921507 Report

Quoted By: >>11921516 >>11921551

>>11921330
Consider just a manifold for a sec. One way to typify it is of course as something that locally looks like real or complex space (in the sense of whatever morphism you think is appropriate, I'm thinking the latter type of manifold and holomorphisms), and as such comes concomitant with functions into that space. A fair question is, how much information can we recover about that space from these functions? If we're thinking a bit more generally, what about differential forms on this space? We have all of this information which fundamentally interfaces with the constitution of this space, especially if we restrict functions and differential forms with some degree of differentiability, so why not formulate techniques for apprehending a space based on its functions? Sheaves are really the based way to do this: by associating an appropriate ring of functions to every open set, you of course capture in one move (especially if i.e. you assign to each open set its holomorphic functions) an object which contains really powerful information about this space at every level, and at every place. This is not just true concretely in the sense of functoriality guaranteeing some notion of comparison between open sets, but also in a very strict algebraic sense: we have simultaneously that A. for some very general classes of examples, the maps between the associated set of functions that a sheaf gives to the entirety of a manifold and that same set for another manifold actually determines a shit ton of information about the what the sheaf itself has to be (and thus naturally about what the space has to look like, via the permissible functions given what can happen globally) and B. we have the explicit algebraic capability to, at any point of our manifold, take all equivalences classes of functions which agree in some open set of that point, i.e. the set of all local functional information about that point, make it a ring in the way you'd probably guess (continued)

Anonymous

Anonymous Tue 21 Jul 2020 04:49:55 No.11921516 Report

Quoted By: >>11921520 >>11921558

>>11921507
(treat these bits of info as functions on a small enough open set), and speak purely algebraically about all permissible ways for a function to look locally as well. These two things together allow you to pose questions like "how much local things on a manifold fail to be global things" in incredibly robust generality (this is the idea of sheaf cohomology), and legitimately can purely algebraically tell you a lot of stuff about a manifold. For example, the topological genus of a Riemann surface is equal to the dimension as a vector space (i.e. its certainly a C module) of the group of ways to assign a holomorphic function to each intersection of 2 open sets on our surface in such a way that on any intersection of 3 open sets, the alternating sum of the functions chosen for any 2 of them is 0 MODULO such choices which themselves are obtained by taking a choice of a function for each open set of the surface and then doing the same alternating sum, but for pairwise intersections.

Anonymous

Anonymous Tue 21 Jul 2020 04:55:24 No.11921520 Report

Quoted By: >>11921536

>>11921516
all this open set is of course made easier to deal with in terms of sheaf stuff of course, and much more powerful things can be said using it, though without sheaves you can essentially treat the last thing I said as essentially saying: the space of assignments of functional information which actually patch together to a function out of the triangulation of our Riemann surface (not a bad approximation to the surface itself) modulo the bits of functional information which did not patch together into an honest to god function out of the surface itself is precisely given by its genus, at least as a vector space.

Anonymous

Anonymous Tue 21 Jul 2020 05:02:24 No.11921536 Report

Quoted By:

>>11921520
this is confusing without sheaves, so I can explain further if you want. But basically, statements which look like this, but are algebraically much more nice (and which if I explained them like this in tandem with a chalkboard and a mutual understanding of sheaves would actually be intelligible, though I apologize because I'm still not explaining this last theorem well anyway) abound in anything that purports to be geometry nowadays, or least anything that is not discrete. If you're down with algebra, and you care about shapes in the abstract, sheaves are just the way to deal with em: you can derive what I just said without em, but there are wide classes of incredible things you can only say with them, and indeed I'm only able to give you that actual geometric description at the end (as space of ways to specify intesection-wise a map out of triangulation modulo space obstructions to specifying a function open set-wise) because I've sat with the algebra for long enough to really be ok with this nonsense ass description, and indeed to parse the algebra into it.

tl;dr, they let you say fucking incredible things, let you apprehend spaces better than just dealing with the spaces themselves, and in general are just fuckin cool.

Anonymous

Anonymous Tue 21 Jul 2020 05:09:04 No.11921551 Report

Quoted By: >>11921577

>>11921507
Any good intro suggestions? I assume any intro would simultaneously be a book on alg geometry, but I don't know classical algebraic geometry so when I start reading it I get lost under the abstractions (like, I don't see where this modern take on geometry connects with the classic curve stuff).

Anonymous

Anonymous Tue 21 Jul 2020 05:10:36 No.11921558 Report

Quoted By:

>>11921516
oh also, this local information stuff is a local ring with maximal ideal the functions which evaluate to 0 around it, and you can evaluate functions at a point as follows: first, find their local description at that point (the ring of local descriptions is called a stalk, and a functions local info a germ). In particular for our example, this will be some taylor series if we take local coordinates identifying our point with 0 in C^n or whatever, and the maximal ideal with reference to these coordinates will just be generated by (x). Then, the germ of our function modulo (x) will precisely give us its evaluation, at least in with reference to our prescribed local coordinates (i.e. with 0 the local coordinate of our point). So we can even do evaluation of functions purely algebraiclly, as is explicitly made the case in schemes (though for them the field into which we are quotienting is no longer constant).

Anonymous

Anonymous Tue 21 Jul 2020 05:18:25 No.11921577 Report

Quoted By: >>11921580 >>11921599

>>11921551
yea real understandable hours, its kind of a mess on this front (even just varieties end up being treated as irreducible reduced finite type schemes over some ground field lmao which is mostly machinery you only really need to make sure projective stuff works but w/e). i learned all the sheafy stuff from otto forster's riemann surfaces book to be honest, which has also tremendously helped me in trying to make sense of AG thus far, so if you like complex analysis, reading that book up to chapter 3 is a blessing; you could also start reading the stacks project stuff where it begins defining sheaves, and then just go until you are totally lost; that alone will get everything you need at least hinted at in your brain sheaf-wise, though won't get you comfortable with this stuff per se.

Anonymous

Anonymous Tue 21 Jul 2020 05:20:02 No.11921580 Report

Quoted By: >>11921587 >>11921599

>>11921577
its also kinda fucked that most books people cite for AG stuff really gloss over this; in this light, and in retrospect, ravi vakil's notes are pretty fuckin good at not doing so actually; his treatment of sheaves is really thorough, only lacking insofar as i wish it was slightly more geometrical, but really great so check that out, it won't poop on your brain by trying to slide AG shit in at the same time at least (and then iirc he follows sheaves with Spec which is lovely).

Anonymous

Anonymous Tue 21 Jul 2020 05:23:08 No.11921587 Report

Quoted By: >>11921599

>>11921580
also if you're looking for explicitly pre-AG stuff in particular, Eisenbud's book is pretty stellar tho brutally fuckin long lmao

Anonymous

Anonymous Tue 21 Jul 2020 05:29:23 No.11921599 Report

Quoted By:

>>11921587
>>11921577
>>11921580
Thanks, I'll check those out.

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