Theory of Schemes in Algebraic Geometry

[dx]

In algebraic geometry, the central notion is that of a scheme. This is the replacement for the classical notion of a variety. A scheme is a topological space X equipped with a sheaf of rings. I just assume that this notion of a scheme replaces the idea of a variety? or is that notion still needed somehow?

 

[martinbn]

In algebraic geometry, the central notion is that of a scheme. This is the replacement for the classical notion of a variety. A scheme is a topological space X equipped with a sheaf of rings. I just assume that this notion of a scheme replaces the idea of a variety? or is that notion still needed somehow?

Replaces may be not the most accurate. May be it is better to say that it generalizes it. Also it is not enough to say a topological space with a sheaf of rings. The sheaf cannot be arbitrary, it has to be locally of certain type.

 

[dx]

Yes, maybe it doesn't replace. Also, see "panorama of pure mathematics", especially the chapters on commutative algebra and analytic geometry.

 

[dx]

Also, reading about bundle morphisms for SU(2) and SU(3) in Charles Nash's book, it reminded about so-called gerbes. I think they are a kind of stack. I have read that Deligne-Mumford stacks are some kind of generalization or some notion closely related to schemes. There is a 2-form field in string theory called the B field, and i think some mathematicians have suggested to treat is as a 'U(1) gerbe.'

 

[dx]

just now, my friend recommended this page to me about bundle gerbes:
https://ncatlab.org/nlab/show/bundle+gerbe

 

[fresh_42]

just now, my friend recommended this page to me about bundle gerbes:
https://ncatlab.org/nlab/show/bundle+gerbe

This is a lexicon like Wikipedia, only better. It contains basically the definitions, normally from a category point of view.

 

[fresh_42]

another basic question I have, is set theory about categories? what is the difference between a category and a set?

Sets form a category, but a category is not a set. A category is a family of objects and morphisms between them. Those families can be modules, groups, sets or whatever.

Are they interchangable words?

No.

that is one thing i imagined. and what does set theory have to do with a 'topos'? does that term 'topos' come under the subject of set theory?

A topos is a certain kind of categories. The category of sets is a certain kind of topos.

 

[dx]

yes, i thought we could just replace sometimes the word category with set? thanks for clearing that up. I was thinking about the following idea: there can be injections, bijections, surjections between sets. are these the things that category theory generalizes?

 

[fresh_42]

yes, i thought we could just replace sometimes the word category with set? thanks for clearing that up. I was thinking about the following idea: there can be injections, bijections, surjections between sets. are these the things that category theory generalizes?

It are the morphisms in the category set. E.g. in the category groups we can call them embedding, isomorphism and projection. But there are more morphisms than those in either of the examples.

 

[dx]

Yes, the set of morphisms between two objects A and B is denoted Hom(A, B). I was looking at the wikipedia pages on things like etale topos, Grothendieck pretopology and so on. This page is nice:
https://en.wikipedia.org/wiki/Topos

 

[mathwonk]

A category is a collection of objects of the same kind, together with morphisms between them when defined. The collection of all sets and all functions between them is a category. But there are so many objects of the same kind, that a category is usually not a set. E.g. the set of all sets is too big to be a set, at least in the set theoretic language I use, where "big" collections are called classes and only small ones are called sets.

To me, a variety is a reduced, irreducible, scheme of finite type, usually defined over an algebraically closed field, so is a very special type of scheme. In particular, on every open subset of a variety, the ring of sections of the structure sheaf is not just some arbitrary ring, but an integral domain of k valued functions containing the base field k. (See Mumford's redbook, chapter II.3) These are very close to those closed subsets of projective space defined by homogeneous prime ideals, or (Zariski) open subsets of them, and functions which are restrictions of quotients of homogeneous polynomials.

 

[dx]

In the subject of homological algebra, we have the exterior derivative operator d, which is a kind of 'coboundary'. The map d_n which maps an n dimensional region X_n to an n - 1 dimensional region X_{n - 1}, and so on. This sequence of mappings d is called a chain complex. Associated to such a chain complex or cochain complex are the homology and cohomology objects H_n = ker d_{n + 1}/ I am d_n. the kernel ker d_{n + 1} and image I am d_n are the closed and exact differential forms.

can someone give me some info about how this idea is used to obtain sheaf cohomology?[/SUB]

 

[dx]

In the wikipedia page above, they say that a topos is like the category of sheaves of sets on a topological space X. They also say that a topos is a category that behaves like Set. A presheaf on a category C is a contravariant functor from C to Set.

what is the connection between these?

 

[Infrared]

In the subject of homological algebra, we have the exterior derivative operator d, which is a kind of 'coboundary'. The map d_n which maps an n dimensional region X_n to an n - 1 dimensional region X_{n - 1}, and so on. [/SUB]

There is no requirement that $\dim X_n=n.$ And I wouldn't describe $X_n$ as a "region". In order to to take the quotient $\ker d/\text{im } d$, usually $X_n$ is an abelian group or a vector space (possibly with some extra decorations like a filtration).

can someone give me some info about how this idea is used to obtain sheaf cohomology?

In nice cases, sheaf cohomology agrees with Cech cohomology, which by definition is the cohomology of a chain complex: https://en.wikipedia.org/wiki/Čech_cohomology Technically it is the direct limit of such cohomologies (each associated to an open covering), but usually, there is a sufficiently fine open covering that can be used to compute the Cech cohomology.

 

[mathwonk]

The best source for Cech sheaf cohomology in algebraic geometry is probably the original paper of J.P.Serre in the Annals of Math. Here is a link to an English translation, or you can hunt down the French version by Serre himself.

https://achinger.impan.pl/fac/fac.pdf

Grothendieck's "derived functor" sheaf cohomology is described in various places, his EGA, Hartshorne's Algebraic Geometry, and Kempf's Algebraic geometry, as well as Godement's Theorie des Faisceaux. It originally proceeded along the lines of the general theory of Cartan-Eilenberg, using "injective resolutions", after Grothendieck proved the category of sheaves has "enough injectives" for the purpose. Kempf's version may be the shortest, at the cost of being somewhat terse. (The simplifications in Godement and Kempf take advantage of the fact that one does not need injective resolutions, which are quite complicated, but can get along with "acyclic", e.g. "flabby", resolutions which are quite easy to produce.) Mumford also has a nice direct discussion of the Cech process in volume 2 of his algebraic geometry book. Here is a link to a free copy (legitimately offered by the author).

https://www.dam.brown.edu/people/mumford/alg_geom/papers/AGII.pdf