\(\newcommand{\fa}{\mathfrak{a}} \newcommand{\fm}{\mathfrak{m}} \newcommand{\fp}{\mathfrak{p}}\) \(\newcommand{\cC}{\mathcal{C}} \newcommand{\CO}{\mathcal{O}} \newcommand{\CV}{\mathcal{V}}\) \(\newcommand{\RR}{\mathbb{R}}\) \(\DeclareMathOperator{\CHom}{\mathcal{H}om} \DeclareMathOperator{\colim}{colim} \DeclareMathOperator{\et}{Ét} \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\Op}{Op} \DeclareMathOperator{\res}{res} \DeclareMathOperator{\Spec}{Spec}\)

Abstract. We use sheaves of sections to define schemes.

Prerequisites. Familiarity with category theory and manifolds is assumed. My article Categorical Coproducts and K-Theory gives a brief overview of some of these ideas.

Table of Contents

1. Sheaves
2. Zariski Topology
3. Schemes
4. References

1. Sheaves

Sheaves and stalks

  Let \(X\) be a topological space. Then associated to \(X\) is a category \(\Op(X)\), whose objects are the open subsets of \(X\) and morphisms are the inclusion maps \(\iota \colon U \hookrightarrow V\) when \(U \subseteq V\). A presheaf of \(\cC\) is a contravariant functor \(F \colon \Op(X) \to \cC\). Elements in \(F(U)\) are called sections (see Example 1.1). Inclusions are mapped to restrictions, which we denote \(F(\iota) = \res_{V, U}\) or \(F(\iota)(s) = s \mid_{U}\).

  We call a presheaf \(F\) a sheaf if a gluing condition is satisfied for every open set \(U\): For every open cover \(\{ U_i \}_{i \in I}\) of \(U\) and elements \(s_i \in F(U_i)\) such that \(s_i \mid_{U_i \cap U_j} = s_j \mid_{U_i \cap U_j}\), there exists a unique \(s \in F(U)\) such that \(s \mid_{U_i} = s_i\).

Example 1.1a.   Let \(\varphi \colon Y \to X\) be a continuous function. Associated to \(\varphi\) is a sheaf of sections \(\Gamma_\varphi\), which maps each open subset of \(X\) to the set of sections on \(X\):

and whose restriction maps are given by restricting the individual sections, i.e. \(\res_{U, V}(s) = s \mid_{V}\). It is clear that this is indeed a sheaf. This is the canonical example from which sheaves are derived.

  The stalk of a sheaf \(F\) at a point \(x \in X\) allows us to observe its local behavor at \(x\). It is defined as

where the colimit is taken over open sets containing \(x\). This is equivalent to

where \((U_1, s_1) \sim (U_2, s_2)\) if there exists a \(U_3 \subseteq U_1 \cap U_2\) such that \(s_1 \mid_{U_3} = s_2 \mid_{U_3}\). Elements of the stalk \((U, s)\) are referred to as germs.

Example 1.1b.   With \(\varphi\) as above, we see that \((\Gamma_\varphi)_x\) is the set of sections \(s \colon U \to Y\) with \(x \in U\) modulo if two sections agree on any neighborhood of \(x\).

  Bundling the stalks gives us the étale space over \(X\)

where \(F_x\) is mapped to \(x\). Recall that a local homeomorphism is also called an étale map. If we equip each stalk with the discrete topology, then \(p\) is étale: for any germ \((U, s)\), we have \(\coprod_{x \in U} \{s\}\) is homeomorphic to \(U\). Importantly, \(\Gamma_p(U) = F(U)\), which (with further proof) yields an equivalence between sheaves of sets on \(X\) étale maps with codomain \(X\).

Theorem 1.2.   There is an equivalence between the categories of sheaves on \(X\) and étale spaces over \(X\), which is given by sending a sheaf to its étale space and an étale space to its sheaf of sections.

Proof. See MAC LANE REFERENCE page 88 cor 3

  One application of the étale space is sheafification. Given a presheaf \(F\) on \(X\), we would like to find the smallest sheaf \(F^{sh}\) which \(F\) factors through. This means for any sheaf \(G\) on \(X\) and morphism (i.e. natural transformation) of presheaves \(f \colon F \to G\), there exists a morphism \(f^{sh} \colon F^{sh} \to G\) making the following diagram commute:

We can construct \(F^{sh}\) by taking the sheaf of sections of the étale space of \(F\), and call it the sheafification of \(F\).

Proposition 1.3. PROVE THIS IS MINIMAL

Morphisms of sheaves

  A morphism between sheaves \(f^{\#} \colon F \to G\) on a fixed space \(X\) is a morphism of functors, and hence a natural transformation. Explicitly, \(f^{\#}\) associates to each open subset \(U\) a morphism \(f^{\#}_U \colon F(U) \to G(U)\) such that the following diagram commutes:

We denote the set of morphisms \(\CHom(F, G)\).

  The direct image and inverse image functors allow us to define morphisms between sheaves on different spaces (among other things). Fix a continuous map \(\varphi \colon X \to Y\), and let \(F\) and \(G\) be sheaves on \(X\) and \(Y\), respectively. We call the sheaf on \(Y\)

the direct image of \(F\) by \(\varphi\). Accordingly, we call the sheaf on \(X\)

the inverse image of \(F\) by \(\varphi\). Here, we are taking the colimit over open sets containing \(\varphi(U)\), because \(\varphi(U)\) is not necessarily open.

  These are adjoint functors between the categories of sheaves on \(X\) and sheaves on \(Y\)

Since every right (resp. left) adjoint is left (resp. right) exact, we see \(\varphi_* F\) is left exact and \(\varphi^{-1}\) is right exact.

Lemma 1.4.   Let \(\varphi \colon X \to Y\) be continuous and \(G\) a sheaf on \(Y\). Then \((\varphi^{-1} G)_x \cong G_{\varphi(x)}\).

Proof. Indeed,

where the second equality follows from continuity. \(\blacksquare\)

  Hence, a morphism of sheaves \((f, f^{\#}) \colon (X, F) \to (Y, G)\) is given by a continuous \(f \colon X \to Y\) and a map \(f^{\#} \in \CHom(G, f_* F)\). Using the adjoint relation, we can also consider \(f^{\#}\) as an element in \(\CHom( f^{-1} G, F)\). We will often use the supressed notation \(f \colon F \to G\) to denote a morphism.

  Recall for a fixed morphism \(f \colon F \to G\) in an arbitrary category, we have the following notions:

1. We call \( f \) a *monomorphism* if for any \( g_1, g_2 \colon E \to F \) we have \( f \circ g_1 = f \circ g_2 \) implies \( g_1 = g_2 \);
2. We call \( f \) an *epimorphism* if for any \( h_1, h_2 \colon G \to H \) we have \( h_1 \circ f = h_2 \circ f \) implies \( h_1 = h_2 \);
3. We call \( f \) an *isomorphism* if there exists a \( f^{-1} \colon G \to F \) such that \( f^{-1} \circ f = \Id_F \) and \( f \circ f^{-1} = \Id_G \);

Proposition 1.5.   Let \(f \colon F \to G\) be a morphism of sheaves. Then \(f\) is a monomorphism (resp. epimorphism, isomorphism) if and only if it the induced map on each stalk is.

Proof. It is clear that if \(f\) is a monomorphism then the map induced on stalks is. So assume \(f\) induces a monomorphism on stalks . The cases of epimorphisms and isomorphisms follow similarily.

Corollary 1.6. The inverse image functor is exact.

Locally ringed spaces

  We conclude this section by discussing locally ringed spaces, which arise naturally in geometry. Let \(X\) be a topological space and \(\CO_X\) a sheaf of rings such that each stalk \(\CO_{X, x}\) is a local ring. Then we call the pair \((X, \CO_x)\) a locally ringed space. We denote the maximal ideal in \(F_x\) by \(\fm_x\). A morphism of locally ringed spaces \((f, f^{\#}) \colon (X, \CO_X) \to (Y, \CO_Y)\) is a map of sheaves of rings, such at each stalk, the map \(f_x \colon \CO_{Y, f(x)} \to \CO_{X, x}\) is a local ring homomorphism, i.e. \(f_x(\fm_{f(x)}) \subseteq \fm_x\). FIX INVERSE IMAGE FUNCTOR

Example 1.4.   Write \(M\) for a smooth (real manifold) of dimension \(n\), and consider the sheaf of smooth sections of the trivial bundle \(M \times \RR \to M\), which we denote \(\cC^{\infty}_M\). It associates to each open subset \(U\) the ring of smooth functions \(s \colon U \to \RR\). Notably, \((M, \cC^{\infty}_M)\) is a localy ringed space. Indeed, at any point \(x \in M\), a germ \((U, f)\) is equivalent to a smooth function \(f \colon U \to \RR\). The maximal ideal \(\fm_x\) in \(F_x\) is then the set of functions which vanish at \(x\), i.e. \(f(x) = 0\).

  Write \(A\) for a subset of \(\RR^n\) and \(\cC^{\infty}_A\) its sheaf of smooth functions. The following proposition shows that we could equivalently define a smooth manifold as a locally ringed spaced which is locally isomorphic to an affine space \((A, C^{\infty}_A)\).

Proposition 1.7.   Let \(M\) and \(N\) be smooth manifolds. If there exists a homeomorphism \(f \colon M \to N\) such that \(\cC^{\infty}_N \to \cC^{\infty}_M\) is a local isomorphism of locally ringed spaces, then \(M\) and \(N\) are diffeomorphic.

Proof. Prove this!

Corollary 1.8.   The category of smooth manifolds is equivalent to the category of locally ringed spaces locally isomorphic to affine space.

2. Zariski Topology

  Write \(R\) for a commutative ring with unit. An ideal \(\fp\) is called prime if it satisfies the following equivalent conditions:

1. \( R / \fp \) is a domain;
2. \( ab \in \fp \) implies \( a \in \fp \) or \( b \in \fp \).

We denote the set of prime ideals \(\Spec R\) and call it the prime spectrum. For instance, every maximal ideal is prime, and the zero ideal \((0)\) is prime.

  For any ideal \(\fa\) we let \(\CV(\fa)\) be the set of prime ideals containing \(\fa\):

The following proposition shows that these form the closed sets for what we call the Zariski topology on \(\Spec R\).

Proposition 2.1.   The sets \(V(\fa)\) form the closed sets for a topology on \(\Spec R\).

Proof. To show arbitrary intersections, let \(\fa_i\) be a (possibility infinite) family of ideals, and observe

Here, \(\sum_i \fa_i\) is the ideal generated by finite sums of elements from the \(\fa_i\). Likewise for finite unions we consider a finite family \(\fa_1, \dots, \fa_n\), and observe

Finally, we see that \(\CV(\varnothing) = \Spec R\) and \(\CV(R) = \varnothing\), so the empty set and entire space are both closed. \(\blacksquare\)

Remark 2.2.   Any ring homomorphism \(f \colon R \to S\) induces a continuous map \(f^{\#} \colon \Spec S \to \Spec R\) given by \(f^{\#}(\fp) = f^{-1}(\fp).\) This is one reason to consider prime ideals instead of, for instance, maximal ideals, as the inverse image of a prime ideal is again prime.

  The Zariski topology has a basis of the form

for \(r \in R\). Indeed, if \(U = \Spec R \setminus V(\fa)\) is open, then

We call the \(D(r)\) distinguished open sets.

  The Zariski topology can also be described using localization. Let \(S\) be a multiplicatively closed subset of \(R\), i.e. \(a, b \in S\) implies \(ab \in S\). Then we define the localization of \(R\) by \(S\) as the set of formal quotients

equipped with the operator

  One important instance of a multiplicatively closed subset is that associated to a prime ideal \(S_{\fp} = R \setminus \fp\). We denote \(R_{\fp} = S_{\fp}^{-1} R\). Another is given by taking any \(f \in R\) and considering the multiplicative system \(S_f = \{f, f^2, f^3, \dots, \}\). We denote \(R_f = S_f^{-1} R\). We have

and

INCLUDE HOW THIS RELATES TO COLIMITS

  We conclude this section by listing some properties and examples of the Zariski topology.

Proposition 2.3.   The Zariski topology makes \(\Spec R\) quasi-compact.

Proof. Suppose \(D(f_i)\) cover \(\Spec R\). Then \(\bigcap V(f_i) = \varnothing,\) which implies \(\sum (f_i) = R\). (If not, \(\sum (f_i)\) would be contained in a maximal, and hence prime, ideal). Therefore there exists a finite collection of \(a_i \in f_i\) such that \(a_1 + \cdots + a_n = 1\). Consequently,

and thus

yielding a finite subcover. \(\blacksquare\)

Proposition 2.4.   A point is closed in \(\Spec R\) if and only if it is a maximal ideal.

Proof. Suppose \(\fp \in \Spec R\) is closed. Then \(\CV(\fp) = \{ \fp \}\), meaning no other prime ideals contain \(\fp\), and in particular no maximal ideals properly contain \(\fp\). We conclude that \(\fp\) is maximal. \(\blacksquare\)

Proposition 2.5.   A subspace of \(\Spec R\) is irreducible if and only if it is of the form \(\CV(\fp)\) for \(\fp\) prime. INCLUDE \(\overline{\{\fp\}}\).

Proof. Let \(\CV(\fa)\) be a closed subspace. If \(\CV(\fa)\) contains multiple primes of height 1 above \(\fa\), then we can write \(\CV(\fa) = \bigcup \CV(\fp_i)\). Hence \(\CV(\fa)\) is irreducible if and only if \(\CV(\fa) = \CV(\fp)\) for some \(\fp\). FIX

Theorem 2.6. SPECTRAL SPACES

Example 2.7. Let \(k\) be a field. Then \(\Spec k = \{0\}\) with the trivial topology. Hence, any continuous map \(f \colon \Spec k \to \Spec R\) is determined by the image of \(\{0\}\).

Example 2.8. By definition

Example 2.9. Affine space

3. Schemes

  Any two fields have the same prime spectrum (\(\Spec k = \{0\}\)) while being not necessarily isomorphic. Therefore, we need to equip \(\Spec R\) with a sheaf to differentiate between nonisomorphic commutative rings as follows.

  Fix a commutative ring \(R\) with prime spectrum \(\Spec R\). Equip each \(R_{\fp}\) with the discrete topology. Consider the bundle

which sends elements in \(R_{\fp}\) to \(\fp\). We denote its sheaf of continuous sections by \(\CO_{\Spec R}\) and call the pair (\(\Spec R\), \(\CO_{\Spec R})\) an affine scheme.

Remark 3.1.   Any continuous function \(\varphi \colon X \to Y\) into a discrete space \(Y\) is locally constant. Indeed, choose an \(x \in X\) and write \(\varphi(x) = y\). By assumption \(\{y\}\) is open, so \(\varphi^{-1}(y)\) is an open neighborhood of \(x\) on which \(\varphi\) is constant. Thus, we could replace “continuous sections” by “locally constant sections” in our definition, which is the given definition in, for instance, Hartshorne's book CITE.

  We call a locally ringed space \((X, \CO_X)\) which is locally isomorphic to an affine scheme a scheme. PROPERTIES OF THE CATEGORIES OF SCHEMES. The following theorem shows that the category of rings is equivalent to a subcategory of the category of schemes.

Theorem 3.2.   Let \(R\) and \(S\) be commutative rings. The following are equivalent:

1. \( f \colon R \to S \) is an isomorphism of rings;
2. \( f^\# \colon (\Spec S, \CO_S) \to (\Spec R, \CO_R) \) is an isomorphism of locally ringed spaces.

Proof. Prove this!

4. References

1. Categorical Coproducts and K-theory