Ring Theory 05: Structure Sheaf and Affine Schemes

Ring theory / 05

Structure sheaf and affine schemes

The structure sheaf assigns to \\(D(f)\\) the localization \\(A\_f\\). This single formula explains why localization is restriction to an open set and why an affine scheme is a locally ringed space.

The structure sheaf becomes natural only after localization and the basic open sets \(D(f)\) are understood. The formula for regular functions on \(D(f)\) is not an isolated definition; it is the geometric meaning of \(A_f\).

The defining formula on basic opens

Let \(X=\operatorname{Spec}A\). For every \(f\in A\), define

\[\mathcal O_X(D(f))=A_f.\]

If \(D(g)\subseteq D(f)\), then \(V(f)\subseteq V(g)\). Using the standard identity

\[\sqrt{(f)}=\bigcap_{\mathfrak p\supseteq(f)}\mathfrak p,\]

this is equivalent to \(g\in\sqrt{(f)}\). Thus some power of \(g\) lies in \((f)\), so \(f\) becomes invertible in \(A_g\). The restriction from \(D(f)\) to \(D(g)\) is therefore induced by the universal property of localization:

\[A_f\to A_g.\]

Explicitly, if \(g^N=hf\), then in \(A_g\) the element \(g^N\) is a unit, and

\[f\cdot hg^{-N}=1.\]

Hence \(f\) is a unit in \(A_g\), so any fraction with powers of \(f\) in the denominator restricts to a valid element of \(A_g\).

In practice, restricting functions means allowing more denominators. The identity

\[D(f)\cap D(g)=D(fg)\]

means overlaps are controlled by the localization \(A_{fg}\).

Sections over arbitrary opens

For an arbitrary open set \(U\subseteq X\), a section \(s\in\mathcal O_X(U)\) can be described as a function assigning to each \(\mathfrak p\in U\) an element

\[s(\mathfrak p)\in A_{\mathfrak p}\]

such that locally near every \(\mathfrak p\), the section is represented by a single fraction \(a/f\). More explicitly, for every \(\mathfrak p\in U\), there is a basic open \(D(f)\subseteq U\) containing \(\mathfrak p\) and elements \(a\in A\), \(n\ge0\), such that

\[s(\mathfrak q)=\frac a{f^n}\in A_{\mathfrak q}\]

for all \(\mathfrak q\in D(f)\).

This local-fraction description makes the sheaf axiom conceptually clear.

Values of sections at points

If \(\mathfrak p\in U\) and \(s\in\mathcal O_X(U)\), the germ of \(s\) at \(\mathfrak p\) lies in the local ring

\[\mathcal O_{X,\mathfrak p}\cong A_{\mathfrak p}.\]

Composing with the quotient map to the residue field gives a value

\[s(\mathfrak p)\in\kappa(\mathfrak p).\]

For a global section \(a\in A=\Gamma(X,\mathcal O_X)\), this value is exactly the image of \(a\) in \(\kappa(\mathfrak p)\). Hence the residue-field value map

\[\mathfrak p\mapsto a(\mathfrak p)\in\kappa(\mathfrak p)\]

is the pointwise shadow of the global regular function \(a\).

The word “shadow” matters. The structure sheaf is not merely the assignment of residue values to points. It remembers that regular functions may be represented locally by fractions, that those fractions restrict to smaller opens, and that compatible local fractions glue. On a nonreduced scheme, residue values can even miss nonzero nilpotents: in \(k[\epsilon]/(\epsilon^2)\), the element \(\epsilon\) has zero value in the unique residue field, but it is not the zero global section.

Why the structure presheaf glues

Suppose \(D(f)=\bigcup_i D(f_i)\), and suppose sections

\[s_i\in A_{f_i}\]

agree on overlaps \(D(f_i f_j)\). The claim is that they glue uniquely to a section on \(D(f)\), hence to an element of \(A_f\).

The algebra behind this statement is localization plus a finite-subcover principle for basic opens in affine spectra: if \(D(f)\subseteq\bigcup_i D(f_i)\), then \(f\) lies in the radical of the ideal generated by finitely many \(f_i\). This permits denominators to be cleared after replacing equalities by sufficiently high powers.

Proof outline

Agreement of two fractions after localization means equality after multiplying by some power of the relevant denominator. On overlaps \(D(f_i f_j)\), the compatibility condition says the two local representatives become equal in \(A_{f_i f_j}\). Since \(D(f)\) is quasi-compact, a cover by distinguished opens has a finite subcover; algebraically this says that for some finite set of indices, \(f\in\sqrt{(f_{i_1},\ldots,f_{i_r})}\). Clearing denominators with this finite relation produces one fraction in \(A_f\) whose restrictions are the \(s_i\).

This is the standard basis-sheaf construction: giving \(\mathcal O_X(D(f))=A_f\) with the localization restriction maps on the basis \({D(f)}\) determines a unique sheaf on \(X\), because the basis is closed under finite intersections and the displayed gluing condition holds on basis covers.

Stalks of the structure sheaf

Let \(\mathfrak p\in\operatorname{Spec}A\). The basic open neighborhoods of \(\mathfrak p\) are exactly \(D(f)\) with \(f\notin\mathfrak p\). Therefore

\[\mathcal O_{X,\mathfrak p} = \varinjlim_{\mathfrak p\in D(f)}A_f \cong (A\setminus\mathfrak p)^{-1}A =A_{\mathfrak p}.\]

The isomorphism sends the germ represented by \(a/f^n\in A_f\), with \(f\notin\mathfrak p\), to the same fraction in \(A_{\mathfrak p}\). It is well-defined because passing to a smaller basic neighborhood means further localization. It is surjective since every fraction \(a/s\in A_{\mathfrak p}\) is represented on \(D(s)\). It is injective because if two representatives \(a/f^n\in A_f\) and \(b/g^m\in A_g\) have the same image in \(A_{\mathfrak p}\), then equality holds after multiplying by some \(t\notin\mathfrak p\), hence after restricting both representatives to the common smaller neighborhood \(D(fgt)\).

Because \(A_{\mathfrak p}\) is local, every affine spectrum with its structure sheaf is a locally ringed space.

Definition of affine scheme

The affine scheme associated to \(A\) is

\[\operatorname{Spec}A = \left(|\operatorname{Spec}A|,\mathcal O_{\operatorname{Spec}A}\right),\]

where the underlying topological space has prime ideals as points and Zariski topology, and the sheaf is the structure sheaf above.

Affine scheme

An affine scheme is a locally ringed space isomorphic to \(\operatorname{Spec}A\) for some commutative ring \(A\).

Global sections

The global sections of the structure sheaf are

\[\Gamma(\operatorname{Spec}A,\mathcal O_{\operatorname{Spec}A}) = \mathcal O_{\operatorname{Spec}A}(\operatorname{Spec}A) =A.\]

Since \(\operatorname{Spec}A=D(1)\), this is \(\mathcal O(D(1))=A_1\cong A\). This is the precise recovery theorem for affine schemes: the ring we started with is recovered as the ring of global regular functions.

Examples

For \(X=\operatorname{Spec}k[x]\), the basic open \(D(x)\) is the affine line with the closed point \((x)\) removed. Its ring of functions is

\[k[x]_x=k[x,x^{-1}].\]

For \(X=\operatorname{Spec}\mathbb Z\), the basic open \(D(p)\) consists of primes not containing \(p\). Its functions are

\[\mathbb Z[1/p].\]

For \(A=k[x,y]/(xy)\), \(D(x)\) removes the component where \(x\) vanishes generically and keeps the part where \(x\) is invertible. Since \(xy=0\) and \(x\) becomes a unit on \(D(x)\), one obtains \(y=0\) in \(A_x\).

Common confusion

The equality \(\mathcal O(D(f))=A_f\) does not say that every section is a globally defined polynomial function. It says regular functions on the open set where \(f\) is nonzero may have powers of \(f\) in the denominator.