Immersions
Open, closed, and locally closed immersions (Tag D-MORIMM)
\(f : X \to Y\) is an open immersion if it induces an isomorphism of \(X\) onto an open subscheme \(U \subseteq Y\), meaning \(\abs{U} \subseteq \abs{Y}\) is open and \(\OO_U = \ro{\OO_Y}{U}\).
(Closed immersion)
\(f : X \to Y\) is a closed immersion if \(\abs{f}\) is a homeomorphism onto a closed subset of \(\abs{Y}\) and \(f^\sharp : \OO_Y \to f_* \OO_X\) is surjective as a map of sheaves. A closed subscheme is an equivalence class of closed immersions, where two are identified when they fit into a commuting triangle. An immersion is a morphism making \(X\) isomorphic to an open subscheme of a closed subscheme of \(Y\).
(Locally closed immersion)
A morphism \(f \colon X \to Y\) is a locally closed immersion if it factors as \(X \to U \to Y\) with \(X \to U\) a closed immersion and \(U \to Y\) an open immersion.
Every immersion is a locally closed immersion, and a quasicompact locally closed immersion is an immersion; in particular the two notions agree when \(Y\) is locally Noetherian.
A homeomorphism onto a closed subset need not be a closed immersion: the structure morphism \(\Spec k[\varepsilon]/(\varepsilon^2)\to\Spec k\) is a homeomorphism of one-point spaces, and \(k\to k[\varepsilon]/(\varepsilon^2)\) is not surjective. Closed subschemes of \(\Spec A\) correspond bijectively to ideals of \(A\) [@Har10a, Corollary II.5.10]; so \(V(x)\) and \(V(x^2)\) are distinct closed subschemes of \(\AA^1\) with the same underlying point.
The inclusion \(\ts{xy = 0} \sm \ts{0}\to\AA^2\) is a locally closed immersion that is neither an open nor a closed immersion.
A homeomorphism onto a closed subset need not be a closed immersion: the normalization \(\AA^1_k\to V(y^2-x^3)\), \(t\mapsto(t^2,t^3)\), is a homeomorphism, and \(k[t^2,t^3]\to k[t]\) is not surjective.
Affine, finite, and finite type
Affine morphisms (Tag D-MORAFF)
\(f : X \to Y\) is affine if \(f^{-1}(V)\) is affine for every affine open \(V \subseteq Y\). Equivalently, it is enough that this holds for the opens of one affine cover of \(Y\).
The condition may be checked on one affine cover by the affine-communication argument. Every affine morphism is separated and quasicompact.
Closed immersions are affine, any morphism of affine schemes is affine, and finite morphisms are affine. An open immersion need not be affine: \(\AA^2 \sm \ts{0} \to \AA^2\) has non-affine source.
The functors \((f\colon X\to Y)\mapsto f_*\OO_X\) and \(\mathcal A\mapsto\Spec_Y\mathcal A\) are inverse anti-equivalences between affine morphisms to \(Y\) and quasicoherent sheaves of \(\OO_Y\)-algebras; in particular \(X \cong \Spec_Y f_* \OO_X\) for \(f\) affine. Relative spectra occur in the constructions of Stein factorisation and normalisation.
Locally of finite type, finite type, and finite presentation (Tag D-MORFT)
\(f : X \to Y\) is locally of finite type if \(Y\) has an affine cover by \(\Spec B_i\) such that \(f^{-1}(\Spec B_i)\) has an affine cover by \(\Spec A_{ij}\) with each \(A_{ij}\) a finitely generated \(B_i\)-algebra. It is of finite type if in addition each \(f^{-1}(\Spec B_i)\) is covered by finitely many of the \(\Spec A_{ij}\), that is, locally of finite type and quasicompact.
(Finite presentation)
\(f\) is locally of finite presentation if each \(B_i \to A_{ij}\) is of finite presentation: \(B_i[x_1, \dots, x_n] \surjects A_{ij}\) with finitely generated kernel. It is of finite presentation if it is also quasicompact and quasi-separated.
Finite type is a hypothesis of the valuative criterion of properness and part of the definition of a proper morphism. Over a locally Noetherian base \(Y\), locally of finite type and locally of finite presentation coincide, because every ideal of \(B[x_1, \dots, x_n]\) is finitely generated for \(B\) Noetherian; for \(Y\) Noetherian, finite type and finite presentation coincide. Over the non-Noetherian ring \(B=k[x_1,x_2,\ldots]\), the quotient \(B\to B/(x_1,x_2,\ldots)\cong k\) is of finite type and not of finite presentation. Finite presentation is the finiteness condition in the definition of smooth and étale morphisms and in limit arguments.
Finite morphisms, and finite against finite type (Tag D-MORFIN)
\(f : X \to Y\) is finite if \(Y\) has an affine cover by \(\Spec B_i\) with \(f^{-1}(\Spec B_i) = \Spec A_i\) affine and \(A_i\) a \(B_i\)-module of finite type (D-DEFFGMOD).
(Maps of proper curves)
Let \(C\) and \(C'\) be irreducible curves proper over a field \(k\). Every \(k\)-morphism \(f: C \to C'\) is proper, so \(f(C)\) is an irreducible closed subset of \(C'\): \(f\) is either constant with value a closed point or surjective. If \(C\) is nonsingular, \(C'\) is integral and \(f\) is surjective, then \(f\) is finite and \(k(C')\subseteq k(C)\) is a finite extension. [@Har10a, Corollary II.4.8, Proposition II.6.8]
A module generated by \(a_1,\ldots,a_n\) is generated as an algebra by the same elements, so finite morphisms are of finite type. The converse fails: \(k[x]\) has one algebra generator over \(k\) and infinite dimension as a \(k\)-vector space, so \(\AA^1_k \to \Spec k\) is of finite type and not finite.
Finite morphisms are affine and proper, and their fibres are finite sets. Closedness of a finite morphism follows from lying over and going up for the integral ring map \(B_i\to A_i\). If \(f\) is finite and flat and \(Y\) is Noetherian, then \(f_*\OO_X\) is locally free, and its rank at \(y\) is \(\dim_{\kappa(y)}\Gamma(X_y,\OO_{X_y})\). A nonconstant morphism of nonsingular projective curves over a field is finite, and its degree is the degree of the extension of function fields.
Finite means finitely generated as a module, while finite type means finitely generated as an algebra; \(\AA^1_k \to \Spec k\) separates the two notions.
Quasi-finite morphisms (Tag D-MORQF)
\(f : X \to Y\) is locally quasi-finite if it is locally of finite type with discrete fibres: every point of every fibre \(X_y = \fiberprod{X}{Y}{\Spec \kappa(y)}\) is isolated, so \(X_y\) is a zero-dimensional scheme. It is quasi-finite if it is in addition quasicompact, equivalently of finite type with finite fibres.
Among finite-type morphisms, finite fibres characterise quasi-finiteness, which is weaker than finiteness. For example, the projection \(V(xy-1)\to\AA^1_k\) onto the \(x\)-coordinate has finite fibres but is not finite: its image \(D(x)\) is not closed.
The hyperbola projection is quasi-finite and not finite (Tag FE-MORHYP)
Let \(X = V(xy - 1) \subseteq \AA^2_k\) and let \(f : X \to \AA^1_k\) be the projection to \(x\). Then \(X \cong \GG_m\), every fibre is a single reduced point or empty, so \(f\) is quasi-finite, and the image is \(\AA^1 \sm \ts{0}\), which is not closed. A finite morphism is closed, so \(f\) is not finite, and one sees it on rings: \(k[x] \to k[x, x^{-1}]\) is of finite type and not module-finite.
The same hyperbola also witnesses failure of universal closedness for \(\AA^1\): it is closed in \(\AA^1\times\AA^1\) while its projection is not closed.
The example has two consequences. Zariski’s main theorem repairs it: a separated quasi-finite morphism is an open immersion followed by a finite one, and here the finite morphism is the identity of \(\AA^1\) and the open immersion is \(\GG_m \injects \AA^1\). Over a locally Noetherian base, finite is exactly proper plus quasi-finite, and the failure above is the failure of properness.
Finite equals proper plus quasi-finite (Tag PR-MORFINCHAR)
Let \(\pi \colon X \to Y\) be a morphism of schemes.
\(\pi\) is finite if and only if it is integral and locally of finite type.
If \(\pi\) is finite, then it is quasi-finite: every fibre is a finite discrete set [@Har10a, Exercise II.3.5].
If \(\pi\) is finite, then \(X \cong \operatorname{\mathbf{Proj}}_Y \mathcal{S}\) over \(Y\) for a quasicoherent graded \(\OO_Y\)-algebra \(\mathcal{S}\) generated in degree \(1\) by the finite type module \(\mathcal{S}_1\); in particular \(\pi\) is projective in the sense of D-SCHRELSPECPROJ, and proper.
A morphism of locally Noetherian schemes is finite exactly when it is proper and quasi-finite, equivalently proper with finite fibres [@Har10a, Exercise III.11.2]. Without Noetherian hypotheses: finite exactly when proper, locally of finite presentation, and with finite fibres.
This characterization converts the module-theoretic definition of finiteness into geometric conditions. The implications are \[ \text{finite} \implies \text{projective} \implies \text{proper} \implies \text{universally closed}, \] with none of them reversible: \(\PP^1_k \to \Spec k\) is projective and not finite, and complete non-projective threefolds give proper morphisms to a point that are not projective.
The converse direction, proper plus finite fibres giving finite, recognizes normalization maps and proper zero-dimensional-fibre morphisms as finite.
Projective
Projective and quasi-projective morphisms (Tag D-MORPROJ)
\(f : X \to Y\) is projective if it factors as a closed immersion followed by the projection, \[ X \injects \PP^N_Y \to Y, \qquad \PP^N_Y \da \fiberprod{\PP^N_\ZZ}{\Spec \ZZ}{Y} , \] for some \(N\). It is quasi-projective if it factors as an open immersion into a scheme projective over \(Y\).
For \(Y\) Noetherian, projective implies proper. The proof combines properness of closed immersions, stability of properness under composition, and universal closedness of \(\PP^N_\ZZ \to \Spec \ZZ\) via elimination.
For a graded ring \(S\) with \(S_0 = A\) and \(S\) generated in degree one by finitely many elements, \(\Proj S \to \Spec A\) is projective. The blowup of a Noetherian scheme \(X\) along a closed subscheme with ideal sheaf \(\mci\) is the relative \(\Proj\) of the Rees algebra \(\bigoplus_{d\ge0}\mci^d\), and the blowup morphism is projective [@Har10a, Proposition II.7.13]; for \(X\) a projective variety, the blowup is again a projective variety.
A projective variety over an algebraically closed field \(k\) is the same as an integral closed subscheme of some \(\PP^n_k\), equivalently a closed subscheme that is reduced and irreducible. An arbitrary closed subscheme of \(\PP^n_k\) is a projective \(k\)-scheme but need not be a variety: \(V(x_0^2) \subseteq \PP^1_k\) is a nonreduced point, and \(V(x_0 x_1) \subseteq \PP^1_k\) is two points.
The implication chain is \[ \text{closed immersion} \implies \text{finite} \implies \text{projective} \implies \text{proper} \implies \text{universally closed} , \] and the first four classes consist of morphisms of finite type. A universally closed morphism need not be of finite type: \(\Spec\bar\QQ\to\Spec\QQ\) is integral, hence universally closed, and not of finite type. \(\PP^1_k\to\Spec k\) is projective but not finite, and the projection \(V(xy-1)\to\AA^1_k\), \((x,y)\mapsto x\), is quasi-finite but not proper, since its image \(\AA^1_k\setminus\{0\}\) is not closed.
Base change
The fibre of a morphism as a base change (Tag D-MORFIB)
For \(f : X \to Y\) and \(y \in Y\) with residue field \(\kappa(y) = \OO_{Y,y}/\mfm_y\), the fibre is \[ X_y \da \fiberprod{X}{Y}{\Spec \kappa(y)} . \]
(Geometric points and fibres)
A geometric point of a scheme \(Y\) is a morphism \(\bar{y} \colon \Spec \Omega \to Y\) with \(\Omega\) an algebraically closed field. The geometric fibre of \(f \colon X \to Y\) over \(\bar{y}\) is \(X_{\bar{y}} = X \times_Y \Spec \Omega\). For a property \(P\) of schemes over a field such as connected, irreducible, reduced or integral, \(f\) has geometrically \(P\) fibres if every geometric fibre has \(P\), and a \(k\)-scheme \(X\) is geometrically \(P\) if \(X \times_k \Spec \bar{k}\) has \(P\).
\(X = \Spec \QQ(i) \to \Spec \QQ\) has a one-point fibre, which is integral, but its geometric fibre \(\Spec \QQ(i) \otimes_\QQ \overline{\QQ} \cong \Spec \overline{\QQ} \sqcup \Spec \overline{\QQ}\) is not connected. Over \(k = \FF_p(t)\), \(\Spec k[x]/(x^p - t)\) is integral, and after base change to \(\bar{k}\) it becomes \(\Spec \bar{k}[x]/((x - t^{1/p})^p)\), which is not reduced.
The fibre \(X_y\) is a scheme over \(\kappa(y)\); for \(f\) proper and \(\mcf\) coherent on \(X\), the groups \(H^i(X_y,\mcf|_{X_y})\) are finite-dimensional \(\kappa(y)\)-vector spaces. Base change to \(\Spec\OO_{Y,y}\) instead retains the part of the family over the generalisations of \(y\).
The underlying space of \(X_y\) is homeomorphic to \(f^{-1}(y)\), while the scheme structure records multiplicity and residue-field information. For \(\Spec \ZZ[i] \to \Spec \ZZ\), the fibre over \((p)\) is \(\Spec\FF_p[x]/(x^2+1)\): two points, one point with residue field \(\FF_{p^2}\), or one nonreduced point, according as \(p\) splits, is inert, or ramifies. Since \(\ZZ[i]\) is a free \(\ZZ\)-module of rank \(2\), each fibre has dimension \(2\) over \(\FF_p\), as does the generic fibre \(\Spec\QQ(i)\) over \(\QQ\); for a finite flat morphism \(f\) of Noetherian schemes, \(y\mapsto\dim_{\kappa(y)}\Gamma(X_y,\OO_{X_y})\) is locally constant on \(Y\).
Which properties are stable under base change and composition (Tag PR-MORBC)
The following are stable under arbitrary base change and under composition: open and closed immersions, affine, finite, integral, quasicompact, locally of finite type, of finite type, quasi-finite, flat, surjective, separated, proper, projective, smooth, unramified, étale.
The length of the list reflects common stability mechanisms under tensor product and fibre product. A property defined by a condition on the ring maps in an affine cover is stable under base change whenever the corresponding module-theoretic condition survives \(\wait \tensor_B B'\), and finite generation, flatness, and surjectivity all do.
The instructive entries are the ones that are not on it. Being an isomorphism onto the image, being a closed map, and being dominant are not stable, and universal closedness is defined by forcing the issue: it is exactly “closed after every base change”, which is why the definition of proper contains the word universally. \(\AA^1_k \to \Spec k\) is closed and not universally closed, and the witness is again the hyperbola in \(\fiberprod{\AA^1}{k}{\AA^1}\).
Stability under base change is what makes a property a statement about fibres, and that is its real use: a proper morphism has proper fibres, a flat morphism has flat base changes, and a smooth morphism stays smooth after any extension of the base field.
If a property \(P\) is stable under base change and \(f\colon X\to Y\) has \(P\), then every fibre \(X_y\to\Spec\kappa(y)\) has \(P\). Closedness is not stable under base change: \(\AA^1_k\to\Spec k\) is closed, but its base change \(\AA^2_k\to\AA^1_k\), \((x,y)\mapsto x\), sends the closed set \(V(xy-1)\) onto \(\AA^1_k\setminus\{0\}\). Properness therefore requires universal closedness.
Locality of properties of morphisms
Classes of morphisms stable under base change and local on the source or target (Tag D-MORLOCAL)
Let \(P\) be a class of morphisms of schemes.
\(P\) is stable under base change if for every \(f \colon X \to Y\) in \(P\) and every morphism \(g \colon Y' \to Y\), the base change \(X \times_Y Y' \to Y'\) is in \(P\).
\(P\) is stable under composition if \(f \colon X \to Y\) and \(g \colon Y \to Z\) in \(P\) imply \(g \circ f \in P\).
If \(P\) contains every isomorphism and is stable under composition, the schemes together with the morphisms in \(P\) form a subcategory \(\Sch_P \subseteq \Sch\) with the same objects.
(Locality for the Zariski topology)
Let \(P\) be a class of morphisms of schemes.
\(P\) is local on the target if for every morphism \(f \colon X \to Y\) and every open cover \(Y = \bigcup_i V_i\): \(f \in P\) if and only if every restriction \(f^{-1}(V_i) \to V_i\) is in \(P\).
\(P\) is local on the source if for every morphism \(f \colon X \to Y\) and every open cover \(X = \bigcup_i U_i\): \(f \in P\) if and only if every restriction \(f|_{U_i} \colon U_i \to Y\) is in \(P\).
\(P\) is affine-local on the target if for every morphism \(f \colon X \to Y\) and every cover of \(Y\) by affine opens \(V_i\): \(f \in P\) if and only if every \(f^{-1}(V_i) \to V_i\) is in \(P\); so a single affine cover of \(Y\) decides membership. Affine-local on the source is defined in the same way with affine covers of \(X\).
(Locality for a Grothendieck topology)
Let \(\tau\) be the Zariski, étale, smooth or fppf topology. A class \(P\) of morphisms is \(\tau\)-local on the target if for every \(f \colon X \to Y\) and every \(\tau\)-covering \(\{Y_i \to Y\}\), \(f \in P\) if and only if every base change \(X \times_Y Y_i \to Y_i\) is in \(P\). It is \(\tau\)-local on the source if for every \(f \colon X \to Y\) and every \(\tau\)-covering \(\{X_i \to X\}\), \(f \in P\) if and only if every composite \(X_i \to X \to Y\) is in \(P\). For the Zariski topology these are the notions above.
Affine morphisms are stable under base change and composition and local on the target (D-MORAFF), but not local on the source: for a field \(k\), \(X = \AA^2_k \setminus \{0\}\) is covered by the affine opens \(D(x)\) and \(D(y)\), each affine over \(\Spec k\), while \(X \to \Spec k\) is not affine because \(X\) is not an affine scheme.
Closed immersions are local on the target but not on the source: \(\Spec k \sqcup \Spec k \to \Spec k\) restricts to an isomorphism on each of the two open components, but it is not injective, so it is not a closed immersion.
Flat morphisms and morphisms locally of finite presentation are local on the source and on the target for the fppf topology; smooth and étale morphisms are local on the source for the étale topology and local on the target for the fppf topology. Proper and separated morphisms are local on the target for the fppf topology and are not local on the source: \(\PP^1_k\) is covered by two copies of \(\AA^1_k\), and \(\AA^1_k \to \Spec k\) is not proper.
Cancellation
The cancellation theorem for properties of morphisms (Tag PR-MORCANCEL)
Let \(P\) be a class of morphisms of schemes stable under base change and composition (D-MORLOCAL). Let \(\pi \colon X \to Y\) and \(\rho \colon Y \to Z\) be morphisms and \(\tau = \rho \circ \pi\). If \(\tau \in P\) and the diagonal \(\delta_\rho \colon Y \to Y \times_Z Y\) is in \(P\), then \(\pi \in P\).
Let \(P\) be stable under base change and composition, and \(\rho \circ \pi \in P\).
If \(P\) contains all locally closed immersions, then \(\pi \in P\).
If \(P\) contains all closed immersions and \(\rho\) is separated, then \(\pi \in P\) (P-AGH248PROPSTAB, part (e)).
If \(P\) contains all quasicompact morphisms and \(\rho\) is quasiseparated, then \(\pi \in P\).
Proper morphisms are stable under base change and composition and contain the closed immersions. If \(X\) and \(Y\) are proper over a field \(k\), with structure morphisms \(\tau \colon X \to \Spec k\) and \(\rho \colon Y \to \Spec k\), then every \(k\)-morphism \(\pi \colon X \to Y\) is proper, by part 2 of the corollary, since \(\rho\) is separated.