A closed subset of a scheme carries many closed subscheme structures and an open subset carries one open subscheme structure.
Open and closed subschemes, and closed immersions (Tag D-SCHSUB)
An open subscheme of \(X\) is an open \(U \subseteq \abs{X}\) with \(\OO_U \da \ro{\OO_X}{U}\); every open subset of a scheme is a scheme in exactly one way.
A closed immersion \(i: Z \to X\) is a morphism which is a homeomorphism onto a closed subset and for which \(i^\sharp: \OO_X \to i_* \OO_Z\) is surjective. A closed subscheme is an equivalence class of closed immersions into \(X\).
Closed subschemes of \(\Spec A\) correspond exactly to ideals \(\mfa \normal A\), via \(Z = \Spec(A/\mfa)\). In general, closed subschemes of \(X\) correspond to quasicoherent ideal sheaves \(\mci \subseteq \OO_X\), with \(\OO_Z = \OO_X/\mci\).
(Closed subscheme sequences)
For a closed immersion \(i: Z \to X\) with ideal sheaf \(\mci\) there is an exact sequence of \(\OO_X\)-modules \[ 0 \to \mci \to \OO_X \to i_* \OO_Z \to 0 , \] and tensoring with a locally free \(\OO_X\)-module \(\mce\) keeps it exact: \(0 \to \mci \tensor \mce \to \mce \to i_*(i^* \mce) \to 0\). If \(Z = D\) is an effective Cartier divisor, then \(\mci = \OO_X(-D) = \OO_X(D)^\vee\), giving \[ 0 \to \OO_X(-D) \to \OO_X \to \OO_D \to 0 , \qquad 0 \to \mcl(-D) \to \mcl \to \ro{\mcl}{D} \to 0 \] for every invertible sheaf \(\mcl\). [@Har10a, Proposition II.6.18]
An open subset carries a unique induced scheme structure, while a closed subset can carry many closed-subscheme structures: \(V(x)\) and \(V(x^2)\) in \(\AA^1\) have the same support and different ideal sheaves.
For a closed immersion \(i\), the map on global sections \(\OO_X(X) \to \OO_Z(Z)\) need not be surjective: for \(Z=\ts{0,\infty}\subseteq\PP^1_k\) with its reduced structure, it is the diagonal \(k\to k\times k\).
A closed subscheme of an open subscheme is a locally closed subscheme; an immersion is a morphism that is an isomorphism onto a locally closed subscheme (D-MORIMM).
A closed subscheme is determined by its quasicoherent ideal sheaf: \(V(x)\) and \(V(x^2)\) in \(\AA^1\) have the same underlying point and are different closed subschemes.
The reduced induced structure on a closed subset (Tag D-SCHRED)
Let \(Z \subseteq \abs{X}\) be closed. For \(X = \Spec A\) take \(\mfa \da \Intersect_{\mfp \in Z} \mfp\), a radical ideal, and give \(Z\) the structure \(\Spec(A/\mfa)\). These agree on overlaps, so they glue to a closed subscheme structure on \(Z\) for any \(X\): the reduced induced structure \(Z^\red\).
\(Z^\red\) is the unique reduced closed subscheme of \(X\) with underlying space \(Z\), and it is the smallest closed subscheme with that space: every closed subscheme supported on \(Z\) contains \(Z^\red\) as a closed subscheme. Every morphism \(T\to X\) from a reduced scheme \(T\) whose image lies in \(Z\) factors uniquely through \(Z^\red\).
Taking \(Z = \abs{X}\) gives the reduction \(X^\red \to X\), a homeomorphism that is an isomorphism if and only if \(X\) is reduced. For an ideal \(\mfa\subseteq A\), \(\bigcap_{\mfp\supseteq\mfa}\mfp=\sqrt{\mfa}\), so the reduced induced structure on \(V(\mfa)\subseteq\Spec A\) is \(\Spec(A/\sqrt{\mfa})\).
The scheme-theoretic image of a morphism (Tag D-SCHIMG)
For \(f: X \to Y\), the image of \(f\) lies in a closed subscheme \(Z \subseteq Y\) with ideal sheaf \(\mci_Z\) if the composite \(\mci_Z \to \OO_Y \to f_* \OO_X\) is zero, equivalently if \(f\) factors through \(Z\). The scheme-theoretic image of \(f\) is the smallest closed subscheme of \(Y\) in which the image of \(f\) lies, the intersection of all such \(Z\). When \(\mci \da \ker(\OO_Y \to f_* \OO_X)\) is quasicoherent — for instance when \(f\) is quasicompact and quasiseparated — this ideal sheaf defines it.
(Constructible set)
In a Noetherian topological space \(X\), the constructible subsets form the smallest family of subsets containing every open set and closed under finite intersections and complements. A subset of \(X\) is constructible if and only if it is a finite disjoint union of locally closed subsets. [@Har10a, Exercise II.3.18]
(Scheme-theoretic closure)
The scheme-theoretic closure of a locally closed immersion \(i \colon X \to Y\) is the scheme-theoretic image of \(i\).
If \(X\) is reduced, the scheme-theoretic image is \(\closure{f(X)}\) with its reduced induced structure. For \(f: \Spec B \to \Spec A\) induced by \(\varphi: A \to B\), the image is \(\Spec(A/\ker \varphi)\).
The set-theoretic image of a morphism need not be closed, open, or locally closed: \(\AA^2 \to \AA^2\), \((x,y) \mapsto (x, xy)\), has image the plane minus the \(y\)-axis, together with the origin. By Chevalley’s theorem, the image of a constructible set under a finite-type morphism of Noetherian schemes is constructible.
The scheme-theoretic image of the morphism \(\Spec k[\eps]/\eps^2 \to \AA^1\) given by \(t\mapsto\eps\) is the double point \(V(t^2)\), because the kernel of \(k[t] \to k[\eps]/\eps^2\) is \((t^2)\). The closure of its set-theoretic image, with the reduced induced structure, is the reduced point \(V(t)\).
The scheme-theoretic image is the smallest closed subscheme through which the morphism factors. For \(\Spec k[\varepsilon]/(\varepsilon^2)\to\AA^1_k\), \(x\mapsto\varepsilon\), it is \(V(x^2)\). By Chevalley’s theorem, the set-theoretic image of a morphism of finite type of noetherian schemes is constructible; the image of \(\AA^2_k\to\AA^2_k\), \((x,y)\mapsto(x,xy)\), is \(\{x\neq0\}\cup\{(0,0)\}\), which is neither open nor closed.
Scheme-theoretic intersection as a fibre product (Tag PR-SCHINT)
For closed subschemes \(X_1, X_2 \subseteq Y\), their intersection is \(\fiberprod{X_1}{Y}{X_2}\), again a closed subscheme of \(Y\). For \(Y = \Spec R\) with \(X_i = \Spec(R/J_i)\) this is \[ \Spec\qty{ R/J_1 \tensor_R R/J_2 } \cong \Spec R/(J_1 + J_2) , \] so intersecting subschemes adds ideals.
On affine schemes, scheme-theoretic intersection is computed by adding the defining ideals.
Intersecting \(y = x^2\) with \(y = 0\) in \(\AA^2\) gives \(k[x,y]/(y - x^2, y) \cong k[x]/(x^2)\), the double point: tangency shows up as a length, and Bézout’s theorem counts lengths rather than points. Intersecting \(y = x^2\) with \(y = 1\) gives \(k[x]/(x^2-1)\), two reduced points, and the total length is \(2\) either way.
Note that the ideal sum need not be radical even when both \(J_i\) are, which is exactly why the reduced induced structure of the intersection is the wrong object for counting.
Points, and dimension
Closed, rational and generic points; specialisation (Tag D-SCHPTS)
The residue field at \(x \in X\) is \(\kappa(x) \da \OO_{X,x}/\mfm_x\); for \(x = \mfp \in \Spec A\) this is \(\Frac(A/\mfp) = A_\mfp/\mfp A_\mfp\). To evaluate \(f \in \OO_X(U)\) at \(x\) is to take its image under \(\OO_X(U) \to \OO_{X,x} \to \kappa(x)\).
A closed point is one whose closure is itself, corresponding to a maximal ideal in an affine chart. A generic point of an irreducible closed \(Z\) is \(\eta\) with \(\closure{\ts{\eta}} = Z\). For \(X\) over \(k\), a rational point is an \(x\) with \(\kappa(x) = k\).
Write \(\tilde x \leadsto x\), “\(\tilde x\) specialises to \(x\)”, when \(x \in \closure{\ts{\tilde x}}\); equivalently \(\mfp_{\tilde x} \subseteq \mfp_x\) in an affine chart.
(General points and general fibres)
Let \(X\) be an irreducible scheme. A property holds at a general point of \(X\) if there is a dense open \(U \subseteq X\) such that it holds at every point of \(U\). For a morphism \(f \colon X \to Y\) with \(Y\) irreducible, a property holds for the general fibre of \(f\) if there is a dense open \(V \subseteq Y\) such that the fibre \(X_y\) has it for every \(y \in V\).
Every nonempty open subset of an irreducible scheme contains its generic point. On \(\AA^1_k = \Spec k[t]\) over an algebraically closed field \(k\), the property “\(\kappa(x) = k\)” holds at every closed point and fails at the generic point \(\eta\), where \(\kappa(\eta) = k(t)\); the set of closed points is dense but not open. For the family \(V(xy - t) \to \AA^1_t\), the general fibre is a smooth conic \(xy = c\) with \(c \neq 0\), while the generic fibre is the curve \(xy = t\) over the field \(k(t)\).
A section of the structure sheaf evaluates in the residue field \(\kappa(x)\), which can vary with \(x\); so a section is not in general a function from \(\abs X\) to one fixed field. On \(\Spec \ZZ\) the element \(5\) evaluates to \(0 \in \FF_5\), to \(5 \in \FF_7\), and to \(5 \in \QQ\) at the generic point. On a nonreduced scheme a nonzero section can vanish at every point: \(\eps \in k[\eps]/\eps^2\). So a section of \(\OO_X\) is not determined by its values at the points of \(X\).
Specialisation is a partial order on the points of a scheme; on \(\Spec A\) it is inclusion of primes. For a variety over an algebraically closed field, the points of the classical variety are the closed points of the associated scheme. For a discrete valuation ring \(R\) with fraction field \(K\), \(\Spec R\) consists of a generic point specialising to a closed point, and the valuative criteria (D-8XX95) test lifts of morphisms \(\Spec K\to X\) to \(\Spec R\).
The value of \(f\in\OO_X(U)\) at \(x\in U\) is its image \(f(x)\in\kappa(x)\). On \(\Spec\ZZ\), \(15\) has value \(0\) in \(\FF_3\) and in \(\FF_5\), value \(1\) in \(\FF_7\), and value \(15\) in \(\QQ\) at the generic point. Specialization, \(x\rightsquigarrow y\) when \(y\in\overline{\{x\}}\), is a partial order on the points of \(X\); for a variety over an algebraically closed field, the closed points are the points of the classical variety. Generic points and their use in “generically” are in properties from the ring.
Dimension and codimension of a scheme (Tag D-SCHDIM)
The dimension of \(X\) is the supremum of \(n\) over chains \[ Z_0 \subsetneq Z_1 \subsetneq \cdots \subsetneq Z_n \] of irreducible closed subsets of \(X\); the length counts links, not sets. The codimension of an irreducible closed \(Z \subseteq X\) is the supremum of lengths of such chains starting at \(Z_0 = Z\); for arbitrary closed \(Y\) take the infimum over irreducible \(Z \subseteq Y\).
\(\dim \Spec A = \krulldim A\), since irreducible closed subsets of \(\Spec A\) correspond to primes, order-reversingly. The codimension of \(V(\mfp)\) is the height of \(\mfp\). More generally, if \(Z \subseteq X\) is an irreducible closed subset with generic point \(\zeta\), then \(\codim_X Z = \dim \OO_{X,\zeta}\), so the irreducible closed subsets of codimension one are exactly the closures of the points whose local ring has dimension one.
Dimension depends only on the underlying topological space, so \(\dim X = \dim X^\red\). For example, \(\Spec k[\eps]/\eps^2\) has dimension \(0\), and its ring of functions has length \(2\), equal to its dimension as a \(k\)-vector space.
For example, \(\dim \Spec \ZZ = 1\) and \(\dim \ZZ[x] = 2\). The equality \(\dim A[x] = \dim A + 1\) holds for Noetherian \(A\) but fails in general. For an integral scheme \(X\) of finite type over a field, \(\dim X = \trdeg_k k(X)\), and \(\codim_X Z + \dim Z = \dim X\) for every irreducible closed subset \(Z\).