What is an intuitive Geometrical explanation of a "sheaf?"
2018年2月9日 · As I understand it, a sheaf is a very broad concept, but is most often used when referencing a function that maps algebraic structures (like rings) to points on a manifolds. Is a sheaf just yet another type of function, but built specifically for algebraic geometry? What is so special about it?
Where to read about sheaves? - Mathematics Stack Exchange
2016年7月8日 · Still my suggestion would Sheaf Theory by B. R. Tennison. It is a 163 page book. But just for the introduction to sheaf you can just read the first two chapters of it, which is some 30 pages. The book is self content. It is very detailed (For example, he defines inductive limit in order to define stalk of a sheaf at a point).
Why is the structure sheaf (sheaf of rings) - Mathematics Stack …
2017年5月7日 · Why is the sheaf of rings $\mathcal{O}$ defined above a sheaf of rings? While it seems clear that $\mathcal{O}$ is a presheaf, I fail to see why it satisfies the extra axioms that make it a sheaf. Hartshorne says "it is clear from the local nature of the definition that $\mathcal{O}$ is a sheaf", but while trying to prove it formally, I couldn ...
What exactly is sheafification? - Mathematics Stack Exchange
2021年2月21日 · What definition do you have, and what you find confusing about it? Let me also point out that a sheaf is a presheaf satisfying certain conditions, and sheafification is the universal way to enforce these conditions on a presheaf so that if your presheaf was already a sheaf, nothing changes.
Reference for "It is enough to specify a sheaf on a basis"?
The verification that the extension is a sheaf is long and mostly a very, very tedious verification that everything really does commute. One can formulate the notion of a $\mathcal{C}$-valued (pre)sheaf on a base $\mathscr{B}^{op}\subset \mathcal{O}(X)^{op}$ for any reasonably complete category $\mathcal{C}$. The idea is exactly as it works ...
Sheaf cohomology: what is it and where can I learn it?
Incidentally, sheaf cohomology provides a very simple proof that de Rham cohomology agrees with ordinary cohomology (at least when you agree that ordinary cohomology is cohomology of the constant sheaf, here $\mathbb{R}$) because the de Rham resolution is a soft resolution of the constant sheaf $\mathbb{R}$, and you can thus use it to compute ...
Is a subsheaf indeed a sheaf? - Mathematics Stack Exchange
2019年3月30日 · The sheaf $\mathcal C$ on $\mathbb R$ of continuous real-valued functions has as subpresheaf $\mathcal C_b ...
Prove that sheaf hom is a sheaf. - Mathematics Stack Exchange
$\begingroup$ @user123412 Yes, in some sense you should. However, since the restriction maps on the morphisms are just "using the same maps but on a smaller domain" then naturality follows from the fact that any $\sigma$ is a morphism, and so satisfies naturality.
limits - Understanding the Formal Definition of Sheaf
2018年5月5日 · If the sheaf vector spaces can be different types among the faces. Or they all have to be the same type (#1 in first sheaf definition). If an abstract simplical complex is an open cover or related. The second sheaf definition says a sheaf is on an open cover, while the first says it's on an abstract simplical complex.
Why is the ideal sheaf - Mathematics Stack Exchange
2016年10月27日 · Hence we have found an open covering such that the sheaf assigns to each open set an object isomorphic to the space of holomorphic functions on that open set. That is, we have shown that this sheaf is locally free (of rank one).