Proper morphism

A morphism is proper when it is separated, of finite type, and universally closed. Properness is stable under base change and composition.

A variety over a field $k$ is complete when its map to $\mathrm{Spec}k$ is universally closed: after any base change, projection sends closed subsets to closed subsets. For a separated finite-type variety, this is equivalent to being proper. In the classical algebraically closed setting one may test projections $X\times Y\to Y$ for arbitrary varieties. Images of rational points over a non-algebraically-closed field must not replace geometric images.

For a finite-type affine morphism of Noetherian schemes, apply its valuative lifting property to each domain quotient $B/\mathfrak{q}$ for a minimal prime and all valuation rings of its fraction field containing the image of $A$. The lift forces $B/\mathfrak{q}$ into every such valuation ring, hence makes it integral over that image. For each $b\in B$, multiply the finitely many resulting monic equations. Its value at $b$ lies in the nilradical. The nilradical of a Noetherian ring is nilpotent, so a power of this monic polynomial annihilates $b$. Thus $B$ is integral over $A$. Finitely many algebra generators, each satisfying a monic equation, have their powers reduced to finitely many monomials, proving that $B$ is a finite $A$-module. Nilpotents cannot be ignored in this proof.

Suppose $gf$ is proper, $f$ is surjective, and $g$ is separated and of finite type. Surjectivity of scheme morphisms is preserved by base change: a nonempty fiber remains nonempty after a residue-field extension. After any base change to $Z$, for a closed subset $C$ of the new target of $f$, its inverse image is closed and has image $g(C)$ under the proper composite. Thus $g(C)$ is closed. This proves universal closedness of $g$, completing the three conditions for properness.

If $X\stackrel{f}{\to }Y\stackrel{g}{\to }Z$ has proper composite and $g$ is separated, the graph $X\to X{\times }_{Z}Y$ is a closed immersion, being a base change of the diagonal of $g$. The projection $X{\times }_{Z}Y\to Y$ is a base change of the proper composite. Their composition is $f$, so $f$ is proper.

If $f:X\to Y$ is a morphism of schemes and $(U_i)$ is an open cover of $Y$, then $f$ is proper exactly when every restriction $f^{-1}(U_i)\to U_i$ is proper. The three defining properties—finite type, separatedness, and universal closedness—can each be checked on an open cover of the target.

A finite-type morphism can have a proper fiber over every closed point without being proper. Properness controls compatible specialization in families, as expressed by the valuative criterion for properness, rather than only the individual closed fibers.

For a proper morphism $X\to \mathrm{Spec}A$ with $A$ Noetherian and an $A$-flat coherent sheaf $\mathcal{F}$, locally on the base there is a bounded complex of finite free modules $K^{\bullet }$ such thatnaturally for every $A$-module $M$.

In a proper flat family of algebraic curves, if the first sheaf cohomology of a line bundle vanishes on every fibre of an open set, its direct image there is a vector bundle of rank given by the Riemann-Roch theorem and commutes with arbitrary base change. Locally a finite free complex computes both fibre sheaf cohomology groups. Vanishing of the last cokernel makes its last map surjective; splitting that map leaves a finite locally free kernel in degree zero. This is the mechanism behind charts of the Picard scheme of a curve and evaluation descriptions of the theta divisor.

For a proper morphism of Noetherian schemes and a coherent sheaf flat over the base, the fiber dimension $s\mapsto {\dim }_{\kappa (s)}{H}^p(X_s,{\mathcal{F}}_s)$ is upper semicontinuous, and the fiberwise Euler characteristic is locally constant.

For a line bundle $M$ on $C\times T$, with $C$ a smooth projective curve, this function is locally constant. Twist both $M$ and $M^{-1}$ by a large power of an ample line bundle on $C$. Near a chosen fiber their first sheaf cohomology groups vanish. The Riemann-Roch theorem and upper semicontinuity of their zeroth sheaf cohomology groups then bound the degree from above and below by the degree on that fiber. Connectedness of $T$ is necessary for a single constant across all fibers.

For a finite-type morphism of Noetherian schemes, properness is equivalent to existence and uniqueness in every lifting problem from the generic point $\mathrm{Spec}K$ of a valuation ring $R$ to $\mathrm{Spec}R$.

This is the existence-and-uniqueness part of the valuative criterion.

A morphism $X\to S$ is projective when it factors as a closed immersion $X\hookrightarrow {\mathbb{P}}_S^n$ followed by the projection to $S$. Every projective morphism is proper.

Projection $X\times {\mathbb{P}}^n\to X$ sends closed subsets to closed subsets, with geometric points or schemes understood. On an affine base, choose finitely many homogeneous equations for a closed subset. Its fiber is nonempty exactly when every graded multiplication map for a projective fiber fails to be surjective, by the Projective Nullstellensatz. The image is thus an intersection of closed rank-defect loci. An affine open cover proves the general case. This is the universally closed part of the proper morphism property of projective space.

Over the real numbers, the closed equation $y_0^2=t{y}_1^2$ in ${\mathbb{A}}^1\times {\mathbb{P}}^1$ projects on real points to $\{t\ge 0\}$, which is not Zariski closed. Projective properness concerns scheme or geometric images, not the image of rational points over an arbitrary field. Similarly, $y_0^2+y_1^2$ has no real projective zero, but its homogeneous ideal contains no power of the irrelevant ideal of projective space, since it has a complex projective zero.

A projective scheme over a base $S$ is an $S$-scheme admitting a closed immersion into some projective space ${\mathbb{P}}_S^n$. Every projective morphism is proper.

Projectivity is a property of the scheme over its specified base, expressed by its structural projective morphism.

For a graded ring $S$, $\mathrm{Proj}S$ consists of homogeneous prime ideals not containing the irrelevant ideal $S_{+}$. Its standard affine opens satisfy $D_{+}(f)\cong \mathrm{Spec}(S_f{)}_0$.

If a nonnegatively graded ring $S$ is generated by $S_1$ as an $S_0$-algebra, the standard affine opens of Proj $D_{+}(f)$ with $f\in S_1$ cover $\mathrm{Proj}S$. Indeed a homogeneous prime containing every degree-one element would contain the irrelevant ideal of a graded ring. This condition makes all twisting sheaves on Proj invertible and makes graded sheafification compatible with tensor products of sheaves.

A graded module $M$ over a graded ring $S$ determines a quasi-coherent sheaf on the Proj construction $\mathrm{Proj}S$ by taking the degree-zero part of its localization on each standard affine open of Proj. Localization identifies these module sheaves on overlaps. The construction is exact, by exactness of localization and exactness of taking a fixed graded component.

Under the degree-one generation condition for Proj, cover $X=\mathrm{Proj}S$ by degree-one charts. The degree-one localization of a graded module identifies the two local modules in the displayed formula, using localization commutes with tensor products. The maps are natural and agree on overlaps, so they glue to an isomorphism of sheaves of modules. The analogous assertion can fail for general positively graded rings.

With the graded shift convention $S(n{)}_d=S_{n+d}$, this is the sheaf associated with a graded module $S(n)$. Under the degree-one generation condition for Proj, it is an invertible sheaf: on $D_{+}(f)$ with $\mathrm{deg}f=1$, multiplication by $f^n$ freely generates its local module for every integer $n$. Without that hypothesis, it need not be invertible. The twisting sheaf on projective space is the standard special case. Stacks Project, Section 27.10 records the hypotheses and local trivializations.

For a nonnegatively graded ring $S={⨁}_{d\ge 0}{S}_d$, the irrelevant ideal isThe points of $\mathrm{Proj}S$ are precisely the homogeneous prime ideals that do not contain $S_{+}$.

For a positive-degree homogeneous element $f\in S$, the standard openis the affine scheme $\mathrm{Spec}(S_f{)}_0$.

A graded homomorphism $S\to T$ induces a morphism to $\mathrm{Proj}S$ on the open subset of $\mathrm{Proj}T$ where the inverse image of a prime does not contain $S_{+}$. On $D_{+}(\phi (f))$ it is induced by $(S_f{)}_0\to (T_{\phi (f)}{)}_0$.

If a graded homomorphism $S\to T$ is an isomorphism in every sufficiently large degree, then it induces an isomorphismMultiplying degree-zero localized fractions by a sufficiently large power of the chart denominator reduces the claim to the assumed high-degree isomorphisms.

For a quasi-coherent graded ${\mathcal{O}}_X$-algebra $\mathcal{A}$, the relative Proj is obtained by gluing the schemes $\mathrm{Proj}\mathcal{A}(U)$ over affine open subsets $U\subseteq X$. If $\mathcal{A}$ is of finite type and generated in degree one, a surjection $\mathrm{Sym}(\mathcal{E})\twoheadrightarrow \mathcal{A}$ realizes ${\mathrm{Proj}}_X\mathcal{A}$ as a closed subscheme of the projective bundle ${\mathbb{P}}_X(\mathcal{E})$.

For a nonnegatively graded ring $S={⨁}_{m\ge 0}{S}_m$ and $d\ge 1$, its $d$th Veronese subring isgraded by $(S^{(d)}{)}_m=S_{dm}$. Passing from $S$ to $S^{(d)}$ does not change its projective scheme: the standard affine charts give a canonical isomorphism $\mathrm{Proj}S\cong \mathrm{Proj}{S}^{(d)}$.

The degree-$d$ Veronese embedding sends a projective point to all monomials of degree $d$ in its homogeneous coordinates. For example,is a closed immersion with image the conic $uw-v^2=0$.