Morphism of schemes

A morphism of schemes is a morphism of locally ringed spaces. On affine schemes it is contravariantly equivalent to a homomorphism of their coordinate rings.

For a ring map $A\to B$, the module of Kähler differentials represents $A$-derivations: ${\mathrm{Hom}}_B({\Omega }_{B/A},M)\cong {\mathrm{Der}}_A(B,M)$.

For $A\to B\to C=B/I$, there is a right-exact sequencewhere $[i]\mapsto di\otimes 1$ and $db\otimes c\mapsto cd\bar{b}$.

A morphism $f:X\to Y$ is of finite type when every point of $Y$ has an affine neighborhood $V=\mathrm{Spec}A$ for which $f^{-1}V$ has a finite affine cover $\mathrm{Spec}{B}_i$ with each $B_i$ a finitely generated $A$-algebra.

A morphism $f:X\to Y$ is separated when its diagonal $X\to X{\times }_{Y}X$ is a closed immersion. This is the scheme-theoretic analogue of the Hausdorff property.

For a finite-type morphism of Noetherian schemes, separatedness is equivalent to uniqueness in every lifting problem $\mathrm{Spec}K\to X$ over $\mathrm{Spec}R\to Y$, where $R$ is a valuation ring with fraction field $K$.

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

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 with a coherent sheaf, 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 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 $\mathrm{Spec}R$.

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.

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

A morphism $i:Z\to X$ is a closed immersion when it identifies $Z$ homeomorphically with a closed subset of $X$ and the morphism ${\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_Z$ is surjective. Affine-locally it has the form $\mathrm{Spec}(A/I)\to \mathrm{Spec}A$.

A closed subscheme of $X$ is a scheme $Z$ together with a closed immersion $Z\hookrightarrow X$, usually identified with its image and its quotient structure sheaf.

Every closed subset $Z\subseteq X$ has a canonical reduced closed-subscheme structure defined affine-locally by $\mathrm{Spec}(A/\sqrt{I})$ when $Z=V(I)\subseteq \mathrm{Spec}A$. It is the smallest closed subscheme with underlying set $Z$.

The scheme-theoretic image of $f:X\to Y$ is the smallest closed subscheme of $Y$ through which $f$ factors. For an affine morphism $\mathrm{Spec}B\to \mathrm{Spec}A$ induced by $\phi :A\to B$, it is $\mathrm{Spec}(A/\ker \phi )$, whose underlying set is the closure of the set-theoretic image.