A scheme \((X, \OO_X)\) is reduced if for every open set \(U \subseteq X\) the ring \(\OO_X(U)\) has no nilpotent elements.
Show that \((X, \OO_X)\) is reduced if and only if for every \(P \in X\) the local ring \(\OO_{X, P}\) has no nilpotent elements.
Let \((X, \OO_X)\) be a scheme. Let \((\OO_X)_{\mathrm{red}}\) be the sheaf associated to the presheaf \(U \mapsto \OO_X(U)_{\mathrm{red}}\), where for any ring \(A\) we denote by \(A_{\mathrm{red}}\) the quotient of \(A\) by its ideal of nilpotent elements. Show that \(\qty{X, (\OO_X)_{\mathrm{red}}}\) is a scheme. We call it the reduced scheme associated to \(X\) and denote it by \(X_{\mathrm{red}}\). Show that there is a morphism of schemes \(X_{\mathrm{red}} \to X\) which is a homeomorphism on the underlying topological spaces.
Let \(f: X \to Y\) be a morphism of schemes and assume that \(X\) is reduced. Show that there is a unique morphism \(g: X \to Y_{\mathrm{red}}\) such that \(f\) is obtained by composing \(g\) with the natural map \(Y_{\mathrm{red}} \to Y\).
Solution
1. If every ring of sections \(\mathcal O_X(U)\) is reduced, then every local ring \(\mathcal O_{X,P}\) is reduced.
Let \[ \xi\in\mathcal O_{X,P} \] be nilpotent. Choose an open neighborhood \[ P\in U \] and a section \[ s\in\mathcal O_X(U) \] representing \(\xi\).
If \[ \xi^n=0, \] then the germ of \(s^n\) at \(P\) is zero. By the definition of equality in a stalk, after shrinking to some open \[ P\in V\subseteq U \] one has \[ (s|_V)^n=0 \] in \(\mathcal O_X(V)\). This ring is reduced by hypothesis, so \[ s|_V=0. \] Hence \[ \xi=0. \] Thus the local ring has no nonzero nilpotents.
2. Conversely, if every local ring \(\mathcal O_{X,P}\) is reduced, then every ring \(\mathcal O_X(U)\) is reduced.
Let \[ s\in\mathcal O_X(U) \] satisfy \[ s^n=0. \] For every \(P\in U\), the germ \(s_P\) satisfies \[ (s_P)^n=0 \] in the reduced local ring \(\mathcal O_{X,P}\). Hence \[ s_P=0 \] for every \(P\).
A section of a sheaf whose germ is zero at every point is itself zero: each point has a neighborhood on which the section vanishes, and the sheaf uniqueness axiom glues those local zeroes. Therefore \[ s=0. \] So \(\mathcal O_X(U)\) is reduced.
3. Therefore\[ \boxed{ X\text{ is reduced} \iff \mathcal O_{X,P}\text{ is reduced for every }P\in X. } \]
Combine 1 and 2.
4. Reduction commutes with localization.For a ring \(A\) and \(f\in A\),\[ \boxed{ (A_f)_{\mathrm{red}} \cong (A_{\mathrm{red}})_{\bar f}, } \]where\[ A_{\mathrm{red}}=A/\sqrt{(0)}. \]
The nilradical localizes: \[ \sqrt{(0)}_{A_f} = (\sqrt{(0)}_A)_f. \] Indeed, if \[ \frac a{f^m} \] is nilpotent in \(A_f\), then for some \(n\), \[ \frac{a^n}{f^{mn}}=0. \] Thus some power \(f^r a^n=0\) in \(A\), so \[ (f^ra)^n=0, \] and the fraction lies in the localization of the nilradical. The reverse inclusion is immediate.
Therefore \[ (A_f)_{\mathrm{red}} = A_f/(\sqrt0)_f \cong (A/\sqrt0)_{\bar f}. \]
5. Let\[ V=\operatorname{Spec}A\subseteq X \]be affine.Then the restriction of \((\mathcal O_X)_{\mathrm{red}}\) to \(V\) is the usual structure sheaf of\[ \operatorname{Spec}A_{\mathrm{red}}. \]
On the distinguished basis \[ D(f)\subseteq V, \] the presheaf whose sheafification defines \((\mathcal O_X)_{\mathrm{red}}\) has values \[ \mathcal O_X(D(f))_{\mathrm{red}} =(A_f)_{\mathrm{red}}. \] By 4 this is \[ (A_{\mathrm{red}})_{\bar f}, \] which is exactly \[ \mathcal O_{\operatorname{Spec}A_{\mathrm{red}}}(D(\bar f)). \] These identifications commute with restriction maps, so after sheafification the two structure sheaves agree on the affine chart.
6. The quotient map\[ A\longrightarrow A_{\mathrm{red}} \]induces a homeomorphism\[ \boxed{ \operatorname{Spec}A_{\mathrm{red}} \xrightarrow{\sim} \operatorname{Spec}A. } \]
Every prime ideal of \(A\) contains the nilradical \[ \sqrt{(0)}. \] Hence extension and contraction along \[ A\to A/\sqrt0 \] give a bijection \[ \operatorname{Spec}(A/\sqrt0) \longleftrightarrow \operatorname{Spec}A. \]
Closed sets correspond because an ideal \(I\subseteq A\) and its image \[ (I+\sqrt0)/\sqrt0 \] define corresponding prime sets. Thus the bijection is a homeomorphism.
7. The locally ringed space\[ X_{\mathrm{red}} := \left(X,(\mathcal O_X)_{\mathrm{red}}\right) \]is a scheme.
Choose an affine open cover \[ X=\bigcup_iV_i, \qquad V_i\cong\operatorname{Spec}A_i. \] By <1>5–<1>6, on the same underlying open subset \(V_i\) the reduced structure is isomorphic to \[ \operatorname{Spec}(A_i)_{\mathrm{red}}. \] Thus \(X_{\mathrm{red}}\) is covered by affine schemes and hence is a scheme.
8. The identity map of the underlying topological space, together with the quotient on structure sheaves, defines a morphism\[ \boxed{ \iota:X_{\mathrm{red}}\longrightarrow X } \]which is a homeomorphism on underlying spaces.
Sectionwise quotient gives a morphism of presheaves \[ \mathcal O_X \longrightarrow \left(U\mapsto\mathcal O_X(U)_{\mathrm{red}}\right), \] and hence, after sheafification, a morphism \[ \mathcal O_X \longrightarrow (\mathcal O_X)_{\mathrm{red}}. \] Together with the identity continuous map \[ |X|\to|X|, \] this has the correct contravariant direction for a morphism \[ X_{\mathrm{red}}\to X. \]
Affinely it is the morphism \[ \operatorname{Spec}A_{\mathrm{red}} \longrightarrow \operatorname{Spec}A \] from 6, so it is a homeomorphism locally and hence globally.
On a stalk it is the quotient \[ \mathcal O_{X,P} \longrightarrow (\mathcal O_{X,P})_{\mathrm{red}}. \] The inverse image of the maximal ideal of the quotient is the maximal ideal of the original local ring, so this map is local. Thus \(\iota\) is a morphism of schemes.
9. Let\[ f:X\longrightarrow Y \]be a morphism with \(X\) reduced.The sheaf map\[ f^\sharp:\mathcal O_Y \longrightarrow f_*\mathcal O_X \]annihilates every nilpotent local section of \(\mathcal O_Y\).
Let \(V\subseteq Y\) be open and suppose \[ s\in\mathcal O_Y(V) \] is nilpotent, say \[ s^n=0. \] Then \[ f^\sharp(s)^n =f^\sharp(s^n) =0 \] in \[ \mathcal O_X(f^{-1}V). \] Since \(X\) is reduced, the ring of sections on every open set is reduced by definition. Hence \[ f^\sharp(s)=0. \]
10. The morphism \(f^\sharp\) factors uniquely through the reduced structure sheaf:\[ \boxed{ \mathcal O_Y \longrightarrow (\mathcal O_Y)_{\mathrm{red}} \xrightarrow{g^\sharp} f_*\mathcal O_X. } \]
By 9, on each open set \(V\) the ring map \[ \mathcal O_Y(V) \longrightarrow \mathcal O_X(f^{-1}V) \] kills the nilradical. Hence it factors uniquely through \[ \mathcal O_Y(V)_{\mathrm{red}}. \] These factorizations commute with restrictions because the original maps do. Thus they define a morphism from the presheaf \[ V\longmapsto\mathcal O_Y(V)_{\mathrm{red}} \] to the sheaf \(f_*\mathcal O_X\). By the universal property of sheafification, it factors uniquely through \[ (\mathcal O_Y)_{\mathrm{red}}. \]
11. The continuous map underlying \(g\) is the same as the map underlying \(f\), and the morphism\[ g:X\longrightarrow Y_{\mathrm{red}} \]defined by 10 is a morphism of schemes satisfying\[ f=\iota_Y\circ g. \]
The reduction morphism \[ \iota_Y:Y_{\mathrm{red}}\to Y \] is the identity on the underlying topological space by 8, so any factorization of \(f\) through it must use the same continuous map \[ |X|\to|Y|=|Y_{\mathrm{red}}|. \]
It remains to check that the induced maps on stalks are local. At \(P\in X\), put \[ Q=f(P). \] The stalk map for \(f\) factors as \[ \mathcal O_{Y,Q} \longrightarrow (\mathcal O_{Y,Q})_{\mathrm{red}} \xrightarrow{g_P^\sharp} \mathcal O_{X,P}. \] Since \(f_P^\sharp\) is local, \[ (f_P^\sharp)^{-1}(\mathfrak m_{X,P}) =\mathfrak m_{Y,Q}. \] The maximal ideal of the reduced local ring is \[ \mathfrak m_{Y,Q}/\sqrt0, \] so \[ (g_P^\sharp)^{-1}(\mathfrak m_{X,P}) =\mathfrak m_{Y,Q}/\sqrt0. \] Thus \(g_P^\sharp\) is local, and \(g\) is a morphism of schemes.
The equality \[ f=\iota_Y\circ g \] holds both on underlying spaces and on structure sheaves by the factorization in 10.
12. The factorization \(g\) is unique.
Any such factorization must have the same underlying continuous map, because \[ |Y_{\mathrm{red}}|\to|Y| \] is the identity. On sheaves, its map \[ (\mathcal O_Y)_{\mathrm{red}} \longrightarrow f_*\mathcal O_X \] must compose with \[ \mathcal O_Y\to(\mathcal O_Y)_{\mathrm{red}} \] to give \(f^\sharp\). The uniqueness in 10 therefore forces the sheaf map to be \(g^\sharp\). Hence the scheme morphism is unique.
13. Thus reduction has the universal property\[ \boxed{ \operatorname{Hom}(X,Y_{\mathrm{red}}) \xrightarrow{\sim} \operatorname{Hom}(X,Y) } \]for every reduced scheme \(X\).
Existence and uniqueness are exactly 11 and 12.
14. Q.E.D.
Steps <1>1–<1>3 prove part (a), <1>4–<1>8 prove part (b), and <1>9–<1>13 prove part (c).