Locally ringed space

A locally ringed space is a ringed space whose stalk ${\mathcal{O}}_{X,x}$ is a local ring at every point. Morphisms of locally ringed spaces induce local homomorphisms on stalks.

A scheme is a locally ringed space covered by open subsets isomorphic to spectra of commutative rings. It retains both the points defined by prime ideals and the local algebra of functions around them.

A scheme is nonreduced when its structure sheaf contains a nonzero nilpotent element. For an affine scheme $\mathrm{Spec}(A/I)$, this is equivalent to the ideal $I$ not being radical.

The spectrum $\mathrm{Spec}A$ of a commutative ring $A$ is the set of its prime ideals, with closed sets $V(I)=\{\mathfrak{p}:I\subseteq \mathfrak{p}\}$ and a structure sheaf whose sections locally look like fractions.

An affine scheme is a scheme isomorphic to $\mathrm{Spec}A$ for some commutative ring $A$. Homomorphisms $A\to B$ correspond contravariantly to morphisms $\mathrm{Spec}B\to \mathrm{Spec}A$.

For $f\in A$, the principal open subset $D(f)=\{\mathfrak{p}:f\notin \mathfrak{p}\}$ of $\mathrm{Spec}A$ is the affine scheme $\mathrm{Spec}{A}_f$.

The affine plane over a field $k$ is the affine scheme $\mathrm{Spec}k[x,y]$.

The punctured affine plane is obtained by deleting the closed point $(x,y)$ from $\mathrm{Spec}k[x,y]$. Its regular functions still form $k[x,y]$, whilewhich has basis represented by $x^{-a}{y}^{-b}$ for $a,b\ge 1$.

A Noetherian scheme has a finite cover by affine schemes $\mathrm{Spec}{A}_i$ with each $A_i$ a Noetherian ring. Equivalently, it is quasi-compact and locally Noetherian.

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.

The affine plane with doubled origin is formed by gluing two copies of ${\mathbb{A}}_k^2$ by the identity away from the origin. The overlap is the punctured affine plane, so this scheme is nonseparated and the two-open affine cover is not acyclic for the structure sheaf.

A scheme is semi-separated when the intersection of any two affine open subsets is affine, equivalently when its diagonal is affine. This condition makes affine covers acyclic for quasi-coherent sheaves.