Talk:Gluing axiom

I'm missing the definition of a fine sheaf here (and/or on sheaf (mathematics)). I think I could manage to write it, but as I'm not a guru, I prefer leaving this to s.o. "serious"... — MFH: Talk 20:44, 13 May 2005 (UTC)

wording

I think this page strongly needs several improvements.

First, the "zeroeth section" isn't a "gentle introduction" at all. This page severely lacks a somehow understandable introduction for people less familiar with formalism. Usually I advocate for rigourous definitions instead of too much handwaving (if I think of polynomial or so), but here I think it's too abstract right from the beginning. "Gluing axiom" has a quite easily understandable meaning in usual categories, which could be given in the introduction, e.g. "... the G.A. is the abstract concept generalizing the property of... (functions can be extended to a global section)..."

Concerning the current wording:

... the gluing axiom is introduced to define what a sheaf F on a topological space X must satisfy...

why "is introduced to define", and not simply "states" or so?

Here Open(X) is the partial order of open sets of X ordered by inclusion maps;

I think it should read "the partially ordered set of...": the 'partial order itself is the relation, thus a subset of the cartesian product.

As phrased on the sheaf page, there is a certain axiom that F must satisfy...

this is not clear: what "certain axiom", and: "F must satisfy", in order to be what ?

   F(X) is the subset of F(U)×F(V) with equal image in F(W). 

"equal image": equal to what? image under what map?

Of course this is understandable if one already knows what is explained, but the page should be helpful for someone who does not know what it is. — MFH:Talk 13:22, 21 March 2006 (UTC)

colimits turn into limits?

It is not true that the sheaf functor turns colimits of all small diagrams D into limits of the corresponding diagrams. I fixed this in the article.

To see why this is so, take an open set W which is contained in the intersection of some two open sets U and V. the colimit of the diagram is $U\cup V$, regardless of what W is. We can even take W to be the empty set. So assume W is the empty set. The image of the diagram under F has F(U) and F(V) with arrows to the F(W), which is a set containing only the empty function. the limit of this diagram is the product of F(U) and F(V), which is in general different from $F(U\cup V)$ - the set of pairs of sections which agree on $U\cap V$.

Essentially we "threw out" the compatibility requirement by shrinking W...

Amitushtush (talk) 16:42, 9 April 2008 (UTC)

Question

I was wondering about the explicit formulation of the "gluing axiom" for a cosheaf, if we could call it so.Noix07 (talk) 17:25, 25 July 2014 (UTC)

What were you wondering about it? Ozob (talk) 23:49, 25 July 2014 (UTC)

Why you mean? I was thinking about the Haag-Kastler axioms, which says that in quantum field theory, a system is the data of a precosheaf with values in C*-algebras. This precosheaf may satisfy some more conditions, one in particular, called "additivity" is incompatible with the definition of cosheaf. I just wanted a comparison.

By the way, you could have a look at this question http://math.stackexchange.com/questions/878919/equivalent-definitions-of-the-gluing-axioms 82.243.57.92 (talk) 16:01, 27 July 2014 (UTC)

Gluing axiom for a cosheaf: I guess it's something like this

Let ${U_i}$ be a cover of $U$ and $s_i$ sections in $F(U_i)$ where $F$ is a precosheaf.

 -  All sections of $F(U)$ are sums of the $s_i$ (the sum comes from the coproduct $\coprod F(U_i) \rightarrow F(U) $ defined by the extension maps, hence kind of formal sum if there isn't really any sum in $F(U)$)

 - I'm really really not sure about this second condition: euh, maybe next time... Noix07 (talk) 17:20, 28 July 2014 (UTC)

Explicitness of B-sheaf language

I hit a speed bump where I was shifting context between extending a B-sheaf to a sheaf, as is described here, versus reconstructing a sheaf from the B-sheaf produced by restriction to a basis of open sets B, and as a result I didn't target a relevant issue with my edits requesting clarifications. My general clarity concerns are independent of that misunderstanding, and I can outline them by contrast with my expectations of the content as follows. However, the article's treatment seems different, and I haven't assessed the article complete enough to make the modifications, yet mind the existing idiom (So, is a limit over the one treatment's diagram the same description of the constructed section as the direct limit over the other treatment's diagram?).

Saying "full subcategory" without specifying the family of arrows that category would take from O(X) creates an unnecessary accessibility barrier - people with a basic understanding of point-set topology should be able to read the topological part of the definition and appreciate the analogy without knowing any category theory - which the comment on sets being a colimit of a cover would allow them to do. Perhaps it wouldn't make things any more clear to replace the contravariant functor idiom with the set-theoretic equivalent, but it would definitely be clearer if the subcategory's arrows were given set-theoretically.

Also, if U is covered by a family of basic open sets, it's only a colimit of some diagram in O(X), and what diagram, without hand-waving? The article seems to suggest the colimit would exist in O'(X), i.e. that whatever set is covered by basic open sets is itself a basic open set.

Of course, the Geometry of Schemes is about as abbreviated as the article in how it quantifies some of these things, but read in context it's all (eventually) clear, because it's all shorthand for things which were explicitly quantified during definitions within the neighboring few pages. No such context exists for this article, and the non-set-theoretic definitions and changes of category add extra room for unclarity. That makes the article hand-wavy. Mainly, the B-sheaf version of the gluing axiom isn't given, which makes it unfair for Spectrum of a ring to refer a reader to this article for the details of the extension of a B-sheaf to a sheaf.

The sheaf axiom, as presented in Geometry of Schemes pg. 12 is:

For each open covering $(U_{a})_{a\in A}$ of an open set $U\subseteq X$, if

$(f_{a}\in F(U_{a}))_{a\in A}$

is a collection of elements with the property that for every two a, b in A the restrictions of $f_{a}$ and $f_{b}$ to $U_{a}\cap U_{b}$ are equal, then there is a unique element f in F(U) whose restriction to $U_{c}$ is $f_{c}$ for all c in A.

So, given that:

Knowing $F(U)$ must be defined, $U$ must be a basic open set. Knowing each $F(U_{a})$ must be defined, each $U_{a}$ must also be a basic open set.

and that the following instruction is given (pg 16, Geom. Schemes) for modifying the sheaf axiom to the B-sheaf axiom:

The sections of $U_{a}$ and $U_{b}$ in B agree on the section of $U_{a}\cap U_{b}$ -> The sections of $U_{a}$ and $U_{b}$ agree on the section of any basic open set $V\subseteq U_{a}\cap U_{b}$.

the B-sheaf condition should be:

For each covering $(U_{a})_{a\in A}$ of a basic open set U by basic open sets, if

$(f_{a}\in F(U_{a}))_{a\in A}$

is a collection of elements with the property that for every two a, b in A the restrictions of $f_{a}$ and $f_{b}$ to any basic open set $V\subseteq U_{a}\cap U_{b}$ are equal, then there is a unique element $f$ in $F(U)$ whose restriction to $U_{c}$ is $f_{c}$ for all $c$ in $A$.

Geometry of Schemes, pg. 17:

$F(U)$ is the inverse limit of the F(V) w/ V basic over the diagram of restriction maps between the sections on basic open sets.

That is, for a sheaf of sets, we want the B-indexed elements $(f_{V})_{V\in B}$ of $\prod _{V\in B}F(V)$ such that if $V,W$ are basic open sets and $W\subseteq V$ then $res_{V,W}(f_{V})=f_{W}$.

The restriction maps from this inverse limit section are defined by the universal property of inverse limits. ᛭ LokiClock (talk) 06:20, 12 September 2014 (UTC)

I guess I'm confused as to what the issue is. B-sheaves are not complicated. What the article has and what you wrote above are not complicated and are equivalent (unless I made an error). I am not quite sure why you prefer the text you wrote above? If you think you can improve the article please do so.

Geometry of Schemes is an excellent book and a much better starting place than Hartshorne. However, it doesn't teach you enough technique to be a working algebraic geometer. In particular it omits all discussion of sheaf cohomology, which is a vital tool. (One area that Geometry of Schemes excels in, though, is that it has a clear statement of the "functor of points" idea. This makes many constructions much easier (e.g., Hartshorne's construction of fiber products is needlessly complicated because he can't use the universal property because he hasn't defined the functor of points), and it's very important for understanding moduli spaces and arithmetic questions.)

B-sheaves are also discussed in EGA 0_I and, in the context of sites, in SGA IV (the "comparison lemma"), but those references are more technical. Ozob (talk) 14:13, 12 September 2014 (UTC)

It's not that the subject matter is complicated, it's that the article doesn't define things with enough detail to actually calculate the things without having to read the author's mind a bit. Particularly, Geometry of Schemes says the open set is reconstructed as the direct limit over all inclusions between basic open subsets of the open set, while the article says it's a colimit over the basic open subsets of the set. There are extra instructions required to actually form the right object. There are potential alternate instructions - take the colimit over the inclusions of the basic open sets into the open set to be constructed. When it goes on to look at the sections of the B-sheaf, the same issue applies, but just because the corresponding instructions to the alternate suggestions of morphisms can't be followed doesn't mean the correct family of morphisms is understood from the text. Instead, there is just an obstruction to understanding because of the text's failure to document the writer's train of thought. Such ambiguities almost always arise when a construction which depends on a family of morphisms between objects is only described up to objects related to those morphisms. They're simply not syntactically completed expressions. Say a programmer is reading this, who has access to a colimit construct in their programming language. Will they be able to tell the computer the right thing to compute, or will they have to find a book which has a more detailed definition of the gluing axiom? For that matter, say a computer with advanced language processing capabilities is reading this. Confusing a syntactically incomplete expression for a syntactically completed one is an error, actually, and the two descriptions are not equivalent because of that difference in detail.

I agree, and I had to switch to Hartshorne after a while because I realized I couldn't leverage a lot of Geometry of Schemes when it came to calculations, even for the beginning exercizes in that book. I didn't read Hartshorne's algebraic geometry book's chapters on cohomology, but when I read the beginning of his sheaf cohomology book it was too opaque, because I didn't have the initial philosophy of cohomology down - particularly, the meaning of derived functors and what the derived categories might represent geometrically. To get that spelled out I had to learn some group cohomology, which how I've been studying it has better parity with the semantics of sheaf cohomology, because it's developed from a basis of analogues of Grothendieck's six operations. ᛭ LokiClock (talk) 03:46, 16 September 2014 (UTC)

Given a basis of open sets $\mathcal{B}$, there is only one plausible colimit one could take. One constructs the category $\mathcal{O}$ of all open sets and their inclusions as usual, and then, for an open set U of the ambient space X, one takes the colimit over the full subcategory of objects in $\mathcal{B}$ contained in U.

Sometimes one can actually compute the sections of the sheaf over U in this manner, but it's never necessary or interesting. For example, if we are working with the structure sheaf and if our basis consists of open affine subschemes, then we are interested in $\varprojlim \Gamma (V,{\mathcal {O}}_{V})$, where V ranges over the open affine subsets of X contained in U. So, for example, suppose that $U=\mathbf {A} _{k}^{2}\setminus \{(0,0)\}$. The open affines contained in this are of the form $\operatorname {Spec} k[x,y]_{f}$, where f is a polynomial that vanishes at the origin. The global sections on such an open affine are of course just the corresponding polynomial ring, so we want to compute $\varprojlim k[x,y]_{f}$, where the limit is taken over all f vanishing at the origin. In fact all these rings are contained in the rational function field $k(x,y)$, and inclusions of rings correspond to containments of open sets, so this projective limit is just an intersection. Obviously the intersection contains $k[x,y]$. If f₀ vanishes at the origin and is not in $k[x,y]$, then there exists a g in $k[x,y]$ coprime to f which also vanishes at the origin: If k is infinite, one can take g to be of the form $ax+by$, and because $k[x,y]$ is a UFD at least one such linear form cannot divide f; if k is finite, then one makes the same argument over the algebraic closure and then replaces g by its norm. Therefore the intersection is $k[x,y]$, i.e., sections over U are just $k[x,y]$. This is an algebraic analog of the idea of a domain of holomorphy; but the calculation is never useful in practice.

Hartshorne's presentation of sheaf cohomology is not particularly easy, but I don't know of one that is. There are other sources for sheaf cohomology, but they're often focused on topological applications (like Bredon or Kashiwara–Schapira). There is, of course, EGA III and SGA IV, but those are (again) rather heavyweight and hard to learn from. I've heard that Liu's book is good but I've never looked at it. Hartshorne's Resdiues and Duality is a bad place to start; it's not an introduction at all, and the technology underlying duality has advanced a lot since then. Also it has lots of errors: Brian Conrad wrote a book, Grothendieck Duality and Base Change, where he goes through and systematically fixes the errors. (Brian told me once that he hated his own book. It's nothing more than checking that a lot of diagrams commute, but in order to make everything rigorous you basically have to invert the order in which Hartshorne presents things, making it very hard to read.)

I mentioned sheaf cohomology because it's really unavoidable in modern algebraic geometry. That said, however, the technical details of the theory are: While there are situations where one needs derived categories (and even fancier objects like dg-categories and ∞-topoi), most uses of sheaf cohomology in algebraic geometry are more like this: I have a line bundle, and I'd like to know that it has some sections; there's an exact sequence which tells me that the obstruction to this happening is in H¹; by Kodaira vanishing (or some appropriate variant) that H¹ vanishes, and therefore I have the desired sections, QED. The huge machine of sheaf cohomology is used mostly as a black box; you need to study the workings of the black box at least a little so that you know which uses are valid, but rare are the applications which care about the black box itself. I got a better handle on how to use sheaf cohomology from actually doing geometry than from all the effort I put into Hartshorne's chapter III. In that respect, his chapters IV and V are much better to learn the tools because he's actually doing geometry there. Ozob (talk) 14:39, 16 September 2014 (UTC)