Editing Sheaf (mathematics) (section)

=== Sheaves ===
Given a presheaf, a natural question to ask is to what extent its sections over an open set ''<math>U</math>'' are specified by their restrictions to open subsets of <math>U</math>. A ''sheaf'' is a presheaf whose sections are, in a technical sense, uniquely determined by their restrictions.

Axiomatically, a ''sheaf'' is a presheaf that satisfies both of the following axioms:
# (''Locality'') Suppose <math>U</math> is an open set, <math>\{ U_i \}_{i \in I}</math> is an open cover of <math>U</math> with <math> U_i \subseteq U</math> for all <math> i \in I</math>, and <math>s, t \in \mathcal{F}(U)</math> are sections. If <math>s|_{ U_i} = t|_{ U_i}</math> for all <math>i \in I</math>, then <math>s = t</math>.
# ([[Gluing axiom|''Gluing'']]) Suppose <math>U</math> is an open set, <math>\{ U_i \}_{i \in I}</math> is an open cover of <math>U</math> with <math> U_i \subseteq U</math> for all <math> i \in I</math>, and <math>\{ s_i \in \mathcal{F}(U_i) \}_{i \in I}</math> is a family of sections. If all pairs of sections agree on the overlap of their domains, that is, if <math>s_i|_{U_i\cap U_j} = s_j|_{U_i \cap U_j}</math> for all <math>i, j \in I</math>, then there exists a section <math>s \in \mathcal{F}(U)</math> such that <math>s|_{U_i} = s_i</math> for all <math>i \in I</math>.<ref>{{Citation|title = The Geometry of Schemes | last1 = Eisenbud | last2 = Harris | first1 = David | first2 = Joe | publisher = Springer | location = New York, NY | isbn = 978-0-387-22639-2 | series = [[Graduate_Texts_in_Mathematics|GTM]] | date = 6 April 2006 | pages = 11–18}}</ref>

In both of these axioms, the hypothesis on the open cover is equivalent to the assumption that <math display="inline">\bigcup_{i \in I} U_i = U</math>.

The section ''<math>s</math>'' whose existence is guaranteed by axiom 2 is called the ''gluing'', ''concatenation'', or ''collation'' of the sections <math>s_i</math>. By axiom 1 it is unique. Sections ''<math>s_i</math>'' and ''<math>s_j</math>'' satisfying the agreement precondition of axiom 2 are often called ''compatible'' ; thus axioms 1 and 2 together state that ''any collection of pairwise compatible sections can be uniquely glued together''. A ''separated presheaf'', or ''monopresheaf'', is a presheaf satisfying axiom 1.<ref>{{Citation | last1=Tennison | first1=B. R. | title=Sheaf theory | publisher=[[Cambridge University Press]] | mr=0404390  | year=1975}}</ref>

The presheaf consisting of continuous functions mentioned above is a sheaf. This assertion reduces to checking that, given continuous functions <math>f_i : U_i \to \R</math> which agree on the intersections <math>U_i \cap U_j</math>, there is a unique continuous function <math>f: U \to \R</math> whose restriction equals the <math>f_i</math>. By contrast, the constant presheaf is usually ''not'' a sheaf as it fails to satisfy the locality axiom on the empty set (this is explained in more detail at [[constant sheaf]]).

Presheaves and sheaves are typically denoted by capital letters, <math>F</math> being particularly common, presumably for the [[French language|French]] word for sheaf, ''faisceau''. Use of calligraphic letters such as <math>\mathcal{F}</math> is also common.

It can be shown that to specify a sheaf, it is enough to specify its restriction to the open sets of a [[basis (topology)|basis]] for the topology of the underlying space. Moreover, it can also be shown that it is enough to verify the sheaf axioms above relative to the open sets of a covering. This observation is used to construct another example which is crucial in algebraic geometry, namely [[quasi-coherent sheaf|quasi-coherent sheaves]]. Here the topological space in question is the [[spectrum of a ring|spectrum of a commutative ring <math>R</math>]], whose points are the [[prime ideal]]s <math>\mathfrak{p}</math> in <math>R</math>. The open sets <math>D_f := \{ \mathfrak{p} \subseteq R, f \notin \mathfrak{p}\}</math> form a basis for the [[Zariski topology]] on this space. Given an <math>R</math>-module <math>M</math>, there is a sheaf, denoted by <math>\tilde M</math> on the <math>\operatorname{Spec}R</math>, that satisfies
:<math>\tilde M(D_f) := M[1/f],</math> the [[localization (commutative algebra)|localization]] of <math>M</math> at <math>f</math>.

There is another characterization of sheaves that is equivalent to the previously discussed.
A presheaf <math>\mathcal{F}</math> is a sheaf if and only if for any open <math>U</math> and any open cover <math>\{U_a\}</math> of <math>U</math>, <math>\mathcal{F}(U)</math> is the fibre product <math>\mathcal{F}(U)\cong\mathcal{F}(U_a)\times_{\mathcal{F}(U_a\cap U_b)}\mathcal{F}(U_b)</math>. This characterization is useful in construction of sheaves, for example, if <math>\mathcal{F},\mathcal{G}</math> are [[sheaf of abelian groups|abelian sheaves]], then the kernel of sheaves morphism <math>\mathcal{F}\to\mathcal{G}</math> is a sheaf, since projective limits commutes with projective limits. On the other hand, the cokernel is not always a sheaf because inductive limit not necessarily commutes with projective limits. One of the way to fix this is to consider Noetherian topological spaces; every open sets are compact so that the cokernel is a sheaf, since finite projective limits commutes with inductive limits.