Ring Theory 09: Morphisms and Global Sections

Ring theory / 09

Morphisms and global sections

Functions pull back along maps. Therefore a homomorphism \\(A\to B\\) defines a morphism \\(\operatorname{Spec}B\to\operatorname{Spec}A\\), and affine schemes form the opposite category of commutative rings.

The anti-equivalence between commutative rings and affine schemes is the categorical form of the affine dictionary. This article fills in the underlying map on points, the compatibility with structure sheaves, and the global-section recovery.

From a ring map to a map of spectra

Let \(\varphi:A\to B\) be a unital ring homomorphism. Define

\[f:\operatorname{Spec}B\to\operatorname{Spec}A,\qquad f(\mathfrak q)=\varphi^{-1}(\mathfrak q).\]

The inverse image of a prime ideal is prime, so this is well-defined.

It is continuous because

\[f^{-1}(D(a))=D(\varphi(a)).\]

Indeed, \(\mathfrak q\in f^{-1}(D(a))\) precisely when \(a\notin\varphi^{-1}(\mathfrak q)\), equivalently \(\varphi(a)\notin\mathfrak q\).

The sheaf map

A morphism of locally ringed spaces

\[(\operatorname{Spec}B,\mathcal O_{\operatorname{Spec}B}) \to (\operatorname{Spec}A,\mathcal O_{\operatorname{Spec}A})\]

requires more than a continuous map. It also requires a compatible morphism of sheaves of rings.

On a basic open \(D(a)\subseteq\operatorname{Spec}A\), the inverse image is \(D(\varphi(a))\subseteq\operatorname{Spec}B\). The ring map induces

\[A_a\to B_{\varphi(a)}.\]

These maps are compatible with restrictions, so after sheafification they define the structural morphism

\[f^\#:\mathcal O_{\operatorname{Spec}A}\to f_*\mathcal O_{\operatorname{Spec}B},\]

equivalently \(f^{-1}\mathcal O_{\operatorname{Spec}A}\to\mathcal O_{\operatorname{Spec}B}\).

At a point \(\mathfrak q\in\operatorname{Spec}B\), the induced local map is

\[A_{\varphi^{-1}(\mathfrak q)}\to B_{\mathfrak q}.\]

It is local: the inverse image of the maximal ideal \(\mathfrak qB_{\mathfrak q}\) is exactly \(\varphi^{-1}(\mathfrak q)A_{\varphi^{-1}(\mathfrak q)}\).

Global sections

For an affine scheme \(X=\operatorname{Spec}A\),

\[\Gamma(X,\mathcal O_X)=A.\]

Therefore any morphism of affine schemes

\[g:\operatorname{Spec}B\to\operatorname{Spec}A\]

induces a pullback homomorphism on global sections

\[g^\#:\Gamma(\operatorname{Spec}A,\mathcal O)\to \Gamma(\operatorname{Spec}B,\mathcal O),\]

that is,

\[A\to B.\]

For the morphism constructed from a ring map \(A\to B\), this pullback is exactly the original ring map.

Anti-equivalence

The construction

\[A\mapsto\operatorname{Spec}A\]

defines an equivalence between the opposite category of commutative rings and the category of affine schemes with scheme morphisms:

\[\mathbf{CRing}^{\operatorname{op}}\simeq\mathbf{AffSch}.\]

Equivalently, for all commutative rings \(A,B\),

\[\operatorname{Hom}_{\mathbf{AffSch}}(\operatorname{Spec}B,\operatorname{Spec}A) \cong \operatorname{Hom}_{\mathbf{CRing}}(A,B).\]

The order of \(A\) and \(B\) is not a typo. It is the same reversal familiar from functions: a map of spaces \(Y\to X\) pulls functions on \(X\) back to functions on \(Y\).

Proof outline

A ring map \(A\to B\) gives the continuous map on prime spectra and the compatible local maps on structure sheaves described above. Conversely, a scheme morphism \(\operatorname{Spec}B\to\operatorname{Spec}A\) gives a ring map on global sections \(A=\Gamma(\operatorname{Spec}A,\mathcal O)\to\Gamma(\operatorname{Spec}B,\mathcal O)=B\). These constructions are inverse because the inverse image of each basic open \(D(a)\) is \(D(\varphi(a))\), and the structure sheaf on basic opens is determined by localization.

Examples

Closed immersions

The quotient map \(A\to A/I\) gives

\[\operatorname{Spec}(A/I)\hookrightarrow\operatorname{Spec}A.\]

The image is \(V(I)\), and the structure sheaf on the closed subscheme is the quotient of the restricted structure sheaf by the quasi-coherent ideal sheaf generated by \(I\).

Basic open immersions

The localization map \(A\to A_f\) gives

\[\operatorname{Spec}A_f\to\operatorname{Spec}A\]

with image \(D(f)\). This morphism identifies \(\operatorname{Spec}A_f\) with the open subscheme \(D(f)\).

Polynomial maps

A \(k\)-algebra map

\[k[y_1,\ldots,y_m]\to k[x_1,\ldots,x_n]\]

is determined by polynomials \(g_i(x_1,\ldots,x_n)\), the images of the \(y_i\). It corresponds to the polynomial map

\[\mathbb A^n_k\to\mathbb A^m_k,\qquad x\mapsto(g_1(x),\ldots,g_m(x)).\]

This is the classical picture embedded in the affine-scheme formalism.

Pullback of modules

Let \(f:\operatorname{Spec}B\to\operatorname{Spec}A\) be induced by \(A\to B\). For an \(A\)-module \(M\), the inverse-image pullback of \(\widetilde M\) is

\[f^*\widetilde M = f^{-1}\widetilde M\otimes_{f^{-1}\mathcal O_{\operatorname{Spec}A}}\mathcal O_{\operatorname{Spec}B} \cong \widetilde{M\otimes_A B}.\]

On basic opens this is the same localization-and-extension calculation:

\[(M\otimes_A B)_{\varphi(a)} \cong M_a\otimes_{A_a}B_{\varphi(a)}.\]

Thus module-theoretic change of scalars is geometric pullback of quasi-coherent sheaves.

Common confusion

The map on points goes from \(\operatorname{Spec}B\) to \(\operatorname{Spec}A\), but the map on functions goes from \(A\) to \(B\). Keeping this contravariance explicit prevents many sign and direction errors.