Sheaf of Rings

Recall from algebraic topology in the study of fundamental groups, we construct continuous paths on \(I = [0, 1]\) and invoke the gluing lemma to define their path composition on \(I\). In turn, we know that the groups that arise from this construction are an approach to understanding the underlying topological space. What this more generally hints at is that we can understand a topological space by looking at (nice) functions on that space. Sheaves are a way of encoding that information.

Definition ([Vakil-TRS, Definition 2.2.2], [Gathmann-AG, Definition 3.13]): A presheaf \(\mathcal{F}\) of rings on a topological space \(X\) consists of the following data:

For each open \(U \subseteq X\), we associate a ring \(\mathcal{F}\). The elements of \(\mathcal{F}(U)\) are called the sections of \(\mathcal{F}\) over \(U\).
For each inclusion of open subsets \(U \subseteq V\) of \(X\), there is a restriction map \[\mathrm{res}_{V,U}:\mathcal{F}(V) \to \mathcal{F}(U).\] By convention, we typically write \(f|_U := \mathrm{res}_{V,U}(f)\) where \(f\) is a section of \(\mathcal{F}\) over \(V\).

The data is subjected to the two following axioms:

The map \(\mathrm{res}_{U, U}\) is the identity.
For each inclusion of open subsets \(U \subseteq V \subseteq W\) of \(X\), the following diagram commutes:

The definition of a presheaf is so general that any function-like object will induce a natural presheaf. In particular, our (i.e. my own) focus is specifically on examples from functions.

Example (constant maps form a presheaf, [Gathmann-AG, Example 3.15c]): Let \(X\) be a topological space. For every open subset \(U\) of \(X\), let \(\mathcal{F}(U)\) ring of the constant maps on \(U\). Then, under the usual restriction maps, \(\mathcal{F}\) is a presheaf.

The example above demonstrates that presheaves do not care about topologically local properties. Accordingly, presheaves are problematic because global properties are a strong requirement. For instance:

In algebraic geometry we are interested in rational functions. It would be ideal if \(f:X \to k\) was a rational function where \(X\) is an affine variety and \(\mathcal{k}\) is an algebraically closed field. We can significantly relax this requirement by looking at \(f\) that is locally a rational function — that is, \(f\) restricts to a rational function on a collection of open subsets that cover \(X\).
Even if the above was not our interest, a lot of desirable properties of functions in geometry and topology are local. Continuity and smoothness, for instance.

The above tells us that we need to consider specific kinds of presheaves: namely, sheaves.

Definition ([Vakil-TRS, Definition 2.2.6]): A presheaf \(\mathcal{F}\) is a sheaf if it satisfies the following axioms:

Identity axiom. For any open subset \(U\) of \(X\), if \((U_i)\) is an open cover of \(U\) and \(f_1, f_2 \in \mathcal{F}(U)\) such that \(f_1|_{U_i} = f_2|_{U_i}\) for all \(i\), then \(f_1 = f_2\).
Gluability axiom. If \((U_i)\) is any open cover of \(U\) along with \(f_i \in \mathcal{F}(U_i)\) such that \(f_i|_{U_i\cap U_j} = f_j|_{U_i\cap U_j}\) for all \(i, j\), then there is some \(f\in\mathcal{F}(U)\).

Example (sheaf of continuous functions): If \(X\) is a topological space and \(\mathcal{F}(U)\) is the ring of continuous functions on the open subset \(U \subseteq X\), then \(\mathcal{F}\) is a sheaf. The gluability axiom follows from the Gluing Lemma [Lee-ITM, Lemma 3.23].

Example (sheaf of smooth functions): In a more geometrically-oriented picture, if \(M\) is a smooth manifold and \(\mathcal{F}(U)\) is the ring of smooth functions on the open subset \(U \subseteq X\), then \(\mathcal{F}\) is a sheaf. The gluability axiom follows from the Gluing Lemma [Lee-ISM, Corollary 2.8].

Definition ([Gathmann-AG, Definition 4.1]): The pair \((X, \mathcal{O}_X)\) is called a ringed space if \(X\) is a topological space together with a sheaf of rings \(\mathcal{O}_X\). The sheaf \(\mathcal{O}_X\) is called the structure sheaf of the the ringed space. Following the convention of Gathmann, we assume that the structure sheaf is a sheaf of \(k\)-valued functions.

Definition ([Gathmann-AG, Defintion 4.3]): The map \(f:X \to Y\) is a morphism of ringed spaces provided that:

\(f\) is continuous,
for all open subsets \(U \subseteq Y\) and \(\phi\in\mathcal{O}_Y\), \[f^*\phi \in \mathcal{O}_X(f^{-1}(U))\] where \(f^*\phi\) is the pullback of \(\phi\) by \(f\).

References

[Gathmann-AG] Class Notes “Algebraic Geometry” (2021/22 version) by Gathmann.
[Lee-ITM] Introduction to Topological Manifolds (2nd edition) by Lee.
[Lee-ISM] Introduction to Smooth Manifolds (2nd edition) by Lee.
[Vakil-TRS] The Rising Sea: Foundations of Algebraic Geometry Notes (July 27, 2024 version) by Vakil.

chris uchizono

References