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 morphism is a continuous map $f:X\to Y$ together with a homomorphism ${\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$ of sheaves of rings whose induced map on every stalk is a local homomorphism. Applied to schemes, this is exactly a morphism of schemes.

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.

The functor of points describes a scheme through its morphisms from test schemes. On commutative rings, it is covariant because a ring homomorphism reverses the corresponding affine-scheme map. A functor on rings is represented by $X$ in the algebraic-geometric sense when it is naturally isomorphic to $h_X$. The representing scheme need not be affine; this is different from demanding a representable functor in the category of rings itself.

// Target: module-theory.bigb

Given schemes $X_i$ and isomorphisms between designated open subschemes satisfying inverse and cocycle conditions, identify the underlying open spaces and glue their structure sheaves. The resulting locally ringed space is locally one of the $X_i$, hence a scheme. The same construction works for formal schemes and topological structure sheaves. Compatible morphisms out of the charts glue uniquely.

// Target: ringed-space.bigb

A formal scheme is a locally ringed space with a sheaf of topological rings that is locally a formal spectrum of a complete adic ring. An ordinary scheme is the special case of the discrete topology. A formal completion of a scheme retains all infinitesimal neighbourhoods of a closed subscheme, rather than just that subscheme.

// Target: ringed-space.bigb

For a closed subscheme with ideal sheaf $\mathcal{I}$ in a locally Noetherian scheme, the completion is the formal scheme on $|Y|$ with topological structure sheaf ${\underleftarrow{\lim }}_n({\mathcal{O}}_X/{\mathcal{I}}^{n+1}){|}_{|Y|}$. On a Noetherian affine scheme $\mathrm{Spec}A$, it is $\mathrm{Spf}({\underleftarrow{\lim }}_{n}A/I^{n+1})$. Its adic topology remembers every infinitesimal neighbourhood.

// Target: commutative-algebra.bigb

Restricting the topological structure sheaf of a formal completion of a scheme to an open subscheme commutes with its inverse limit: both sheaves are ${\underleftarrow{\lim }}_n({\mathcal{O}}_U/(\mathcal{I}{|}_U{)}^{n+1})$ on $|U\cap Y|$. Equivalently, the identical infinitesimal neighbourhoods have identical restricted structure sheaves. On formal affine charts this uses completed localization of an adic ring.

// Target: toric-geometry.bigb

For a complete ring with an adic topology defined by $I$, the formal spectrum has the prime ideals containing $I$ as its points and structure sheaf ${\underleftarrow{\lim }}_n{\mathcal{O}}_{\mathrm{Spec}(A/I^{n+1})}$. On a principal open subset $D(f)$ its sections form the completed localization of an adic ring, not generally the ordinary localization $A_f$.

// Target: ringed-space.bigb

A morphism of schemes is smooth when it is locally of finite presentation, flat and has geometrically regular fibres. Smooth fibres alone do not imply that a morphism is smooth: flatness and the dimension behaviour matter as well. The Abel map of an algebraic curve has smooth projective space fibres whose dimensions can jump.

A morphism of schemes is étale when it is flat, unramified and locally of finite presentation. For a morphism between smooth varieties of the same dimension, an invertible differential gives this property locally. A finite étale morphism over an algebraically closed field has reduced finite fibres, and their cardinalities equal its degree when the source and target are connected.

A finite étale morphism is both a finite morphism and an étale morphism. Affine-locally it is represented by a finite locally free algebra with zero module of Kähler differentials. Its fibres are finite products of finite separable field extensions. For an elliptic curve with good reduction at a prime not dividing $m$, multiplication by $m$ on its smooth proper group model is finite étale of degree $m^2$: its differential is multiplication by the unit $m$. A division torsor of a section consequently gives an unramified local extension, the good-prime input to the Weak Mordell-Weil theorem.

A scheme is quasi-compact when its underlying topological space is quasi-compact: every open cover has a finite subcover. Since affine open subsets form a base, this is equivalent to possessing a finite affine open cover. Every affine scheme is quasi-compact.

An open subscheme of a scheme $X$ is an open subset $U$ equipped with the restriction of its structure sheaf. Its inclusion is an open immersion. Every scheme is covered by affine open subschemes. Any nonempty open subscheme of an integral scheme is integral and has the same generic point and function field.

An irreducible scheme is one whose underlying topological space is a nonempty irreducible topological space. It can have nilpotent elements in its structure sheaf; irreducibility concerns only the topology. A reduced scheme is an integral scheme precisely when it is irreducible.

The dimension of a scheme is the supremum of the lengths of strict chains of irreducible closed subsets of its underlying topological space. On an affine scheme it is the Krull dimension of its coordinate ring. Taking the reduced subscheme leaves the dimension unchanged.

The structure morphism of a scheme $X$ is its unique morphism $X\to \mathrm{Spec}\mathbb{Z}$. More generally, an $S$-scheme comes equipped with a specified morphism to the base scheme $S$.

A scheme $X$ has characteristic $p$ when $p\cdot 1=0$ in its structure sheaf of a scheme. Equivalently, its structure morphism $X\to \mathrm{Spec}\mathbb{Z}$ factors through $\mathrm{Spec}{\mathbb{F}}_p$.

The absolute Frobenius morphism of a scheme of characteristic p is the identity on the underlying topological space and raises every local function to its $p$th power. On an affine chart $\mathrm{Spec}A$ it is induced contravariantly by the Frobenius endomorphism $A\to A$, $a\mapsto a^p$. It need not be an isomorphism of schemes, even when it acts invertibly on global sections.

The algebraic Frobenius endomorphism induces this scheme map contravariantly on affine charts.

For a finite surjective map of normal integral schemes with function fields $K\subseteq L$ in characteristic $p$, the condition $L^p\subseteq K$ lets the Absolute Frobenius morphism of the source factor through the map. On an affine chart $A\subseteq B$, send $b$ to $b^p\in A$: it lies in $K$, is integral over $A$, and $A$ is integrally closed. The condition $K\subseteq L^p$ instead lets the map factor after the source Frobenius. Send $a$ to its unique $p$th root in $L$; that root lies in the normal ring $B$ because it satisfies a monic polynomial over $B$. These rules localize and glue; injectivity of Frobenius in the function field supplies uniqueness.

The structure sheaf ${\mathcal{O}}_X$ records the regular functions on every open subset of a scheme $X$. Its stalk ${\mathcal{O}}_{X,x}$ is the local ring of functions defined near $x$.

A global idempotent on a scheme is zero or one in every local ring. Its one-locus is both open and closed. Conversely a clopen subset determines a global idempotent by gluing one on it and zero on its complement. Under this correspondence, decompositions into nonzero orthogonal idempotents correspond to disjoint clopen decompositions. When all connected components are open, the primitive idempotents correspond exactly to those components. Without orthogonality the assertion fails in characteristic two: $(1,0)=(1,1)+(0,1)$ in ${\mathbb{F}}_2\times {\mathbb{F}}_2$.

The units in the structure sheaf of a scheme form a multiplicative sheaf of abelian groups. On a smooth algebraic curve, a nonzero rational function is a unit at a point precisely when its discrete valuation there is zero. Its inclusion in the sheaf of nonzero rational functions on an irreducible variety relates divisor on an algebraic curve to the Picard group through the long exact sequence in sheaf cohomology.

A regular function on an open subset $U$ of a scheme $X$ is a section of ${\mathcal{O}}_X(U)$. On an affine variety it is locally a quotient of polynomial functions with a denominator that does not vanish.

On a variety, a regular function is a regular map to the affine line; the arbitrary-target regular-map concept is broader.

A global regular function is a global section of the structure sheaf of a scheme, defined on the entire scheme. These functions form its ring $H^0(X,{\mathcal{O}}_X)$. On an affine scheme this recovers the defining ring; on a positive-dimensional projective complete intersection it consists only of constants over the base field.

Over an algebraically closed field, every regular function on a nonempty irreducible projective variety is constant. Regard the function as a morphism to ${\mathbb{P}}^1$ avoiding infinity. Its graph is closed; the closedness of projection from projective space makes its image closed in ${\mathbb{P}}^1$. A proper closed subset there is finite, and an irreducible finite image is a singleton. Irreducibility is the usual variety convention; on a disconnected projective algebraic set different components may have different constants.

For $W={\mathbb{A}}_k^2\setminus \{0\}$, restriction to $D(x)$ and $D(y)$ givesThe intersection is taken in $k(x,y)$ and the equality follows from unique factorization: a reduced denominator dividing powers of both $x$ and $y$ is a unit. The inclusion $W\hookrightarrow {\mathbb{A}}^2$ induces an isomorphism on global functions but is not an isomorphism, so $W$ is not an affine variety.

A scheme is reduced when every local ring ${\mathcal{O}}_{X,x}$ is a reduced ring, equivalently when its structure sheaf has no nonzero nilpotent element. The affine scheme $\mathrm{Spec}A$ is reduced exactly when $A$ is reduced.

A Noetherian scheme with no embedded associated primes is reduced if it is reduced at all its generic points. Locally, a nonzero nilpotent ideal would have an associated prime, necessarily a minimal prime. Its localization there would be nonzero, contradicting generic reducedness. This is useful with the unmixedness of a complete intersection.

An integral scheme is a nonempty scheme that is both reduced and irreducible. An affine scheme $\mathrm{Spec}A$ is integral exactly when $A$ is an integral domain.

On an integral scheme $X$, every nonempty open $U$ contains its generic point $\eta$. Taking a germ gives an injective ring homomorphism from its regular functions to the function field ${\mathcal{O}}_{X,\eta }$. Injectivity follows on each nonempty affine open subscheme from the injection of an integral domain into its field of fractions, and then from the sheaf gluing axiom. This lets regular functions on different open subsets be compared inside one field.

A group scheme over a base scheme $S$ is an $S$-scheme $G$ with multiplication $m:G{\times }_{S}G\to G$, an identity section $e:S\to G$, and inversion $i:G\to G$ satisfying the group axioms as equalities of morphisms.

A homomorphism of group schemes $f:H\to G$ is a morphism over the base that commutes with multiplication: $f\circ m_H=m_G\circ (f\times f)$. It consequently preserves the identity and inversion.

For a commutative group scheme $G$, the multiplication-by-$n$ morphism $[n{]}_G:G\to G$ adds a point to itself $n$ times. On an abelian variety, it is surjective for every nonzero integer $n$.

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 of the dual number ring $k[ϵ]/({ϵ}^2)$ is a nonreduced scheme of length two supported at one point. Its nonzero nilpotent $ϵ$ retains first-order information that the underlying singleton topological space forgets. It differs from the disjoint union of two reduced $k$-points.

A scheme over an algebraically closed field is punctual when its underlying topological space consists of one closed rational point. It may have nontrivial nilpotent structure, such as $\mathrm{Spec}k[\epsilon ]/({\epsilon }^2)$.

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.

Contraction sends the zero prime ideal to zero. A complex closed point $(x-z)$ contracts to $(x-z)$ when $z$ is real, and to $((x-\mathrm{Re}z{)}^2+(\mathrm{Im}z{)}^2)$ otherwise. Thus the map on the spectrum of a commutative ring is surjective and identifies each nonreal conjugate pair. Both spectra have only finite proper Zariski-closed sets, so this is also the topological quotient by complex conjugation.

For a reduced ring with a nonempty disconnected spectrum of a commutative ring, choose a separation $V(I)\bigsqcup V(J)$. Disjointness gives $I+J=R$; covering the spectrum gives $IJ\subseteq \sqrt{(0)}=0$. For comaximal ideals, $I\cap J=IJ$. The Chinese remainder theorem then supplies a product of two nonzero quotient rings. Equivalently, if $i+j=1$ with $i\in I$, $j\in J$, then $i,j$ are nontrivial orthogonal idempotents. This connects topology of the spectrum to ring product decomposition by an idempotent without a Noetherian assumption.

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

Suppose a scheme is the scheme-theoretic union of affine closed subschemes $Y=\mathrm{Spec}A$ and $Z=\mathrm{Spec}B$, meaning their ideal sheaves have zero intersection. The intersection $W$ is closed in both, hence affine with coordinate ring $C$ and surjections $A\to C$, $B\to C$. Then the whole scheme is $\mathrm{Spec}(A{\times }_{C}B)$. Both projections of this fiber product are surjective; their kernels have zero intersection and product, and their quotient by the sum is $C$. Thus its spectrum has the same closed pieces, intersection and topology. The structure sheaf is the kernel of $i_{\ast }{\mathcal{O}}_Y\oplus j_{\ast }{\mathcal{O}}_Z\to k_{\ast }{\mathcal{O}}_W$ on both spaces, identifying their locally ringed structures. A reduced Noetherian scheme has finitely many irreducible components, so induction proves it is affine if and only if its reduced irreducible components are affine. The intersection $W$ need not be reduced.

A Noetherian scheme is affine exactly when every coherent ideal sheaf has zero first sheaf cohomology. One direction is vanishing of quasi-coherent cohomology on an affine scheme. For the converse, ideal-sheaf vanishing produces affine principal neighbourhoods from ideal-sheaf vanishing. A finite such cover yields a unit-ideal certificate from a principal affine cover, and those affine charts glue to the spectrum of a commutative ring of global sections. The criterion, including the finite-type ideal version for quasi-compact quasi-separated schemes, is recorded in Stacks Project, Section 30.3.

If finitely many $X_{f_i}$ cover $X$, the morphism ${\mathcal{O}}_X^r\to {\mathcal{O}}_X$ with coefficients $f_i$ is onto. If its kernel has vanishing first sheaf cohomology, the long exact sequence in sheaf cohomology lifts the global section $1$ to coefficients $g_i$. This turns a geometric cover into a unit ideal in the ring of global sections.

Let $P$ have an affine open neighbourhood $U$, and let $Y=X\setminus U$. Evaluation gives ${\mathcal{I}}_Y\twoheadrightarrow k_P$ with kernel ${\mathcal{I}}_Y\cap {\mathcal{I}}_P$. If first sheaf cohomology of every coherent ideal sheaf vanishes, a global section $f$ of ${\mathcal{I}}_Y$ can be chosen with $f(P)=1$. Then $X_f\subseteq U$ is a principal open subset of an affine variety and is affine.

If $H^1(X,\mathcal{I})=0$ for every coherent ideal sheaf, projection to the last coordinate makes any coherent $\mathcal{F}\subseteq {\mathcal{O}}_X^r$ an extension of a coherent submodule of ${\mathcal{O}}_X^{r-1}$ by an ideal sheaf. The long exact sequence in sheaf cohomology proves the assertion by mathematical induction on $r$.

For an affine scheme $X=\mathrm{Spec}A$, its ring of global sections is $A$. A morphism of schemes between affine schemes is determined contravariantly by its map on these rings. Consequently a morphism between affine schemes is an isomorphism of schemes exactly when that ring map is an isomorphism. The affine hypothesis is essential, as the punctured affine plane shows.

An affine open subscheme is an open subset of a scheme with its restricted structure sheaf, which is itself an affine scheme. Every scheme has an open cover by such subschemes. On an affine scheme, the principal open subschemes give an open basis.

An open subset of an algebraic variety equipped with an identification with an affine variety. Such charts make quasi-coherent sheaves and regular maps accessible through their local coordinate rings.

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

In an affine variety with coordinate ring $A$, every dense open subset $U$ contains a dense principal open subset. Write its closed complement as $V(I)$. Density says that $I$ is contained in none of the finitely many minimal prime ideals of the reduced ring $A$. Prime avoidance supplies $s\in I$ outside all these primes. Thus $D(s)\subseteq U$, and $s$ is a non-zero-divisor. Noetherian Zariski topology also makes $U$ a finite union of principal opens; one can add $D(s)$ to this cover. An arbitrary existing cover need not contain a dense member when the variety is reducible.

The affine line over a field $k$ is ${\mathbb{A}}_k^1=\mathrm{Spec}k[t]$.

The one-dimensional scheme has an affine space of rational points over its base field; its scheme structure also records non-rational points and local rings.

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

The two-dimensional scheme has an affine space of rational points over its base field; its scheme structure also records non-rational points and local rings.

The points of the real affine plane as a scheme are its prime ideals: the zero ideal, the principal ideals of irreducible real polynomials, and its maximal ideals. The latter have residue field $\mathbb{R}$ or $\mathbb{C}$, by the Zariski lemma and the real closed field property. They correspond respectively to real coordinate pairs and conjugate pairs of nonreal complex coordinate pairs. Thus the scheme contains both nonclosed generic points and closed points absent from the real locus. Its topology is the Zariski topology, and its structure sheaf has sections $\mathbb{R}[x,y{]}_h$ on each principal open subscheme $D(h)$.

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 $a,b\ge 1$.

The punctured affine plane $U={\mathbb{A}}_k^2\setminus \{(0,0)\}$ is not an affine scheme. On its cover $D(x)\cup D(y)$, global sections are $k[x,y{]}_x\cap k[x,y{]}_y=k[x,y]$ inside the field of fractions. Coprimality of $x$ and $y$ proves this equality. If $U$ were affine, its inclusion into ${\mathbb{A}}_k^2$ would be an isomorphism by affine scheme reconstruction from global sections, contradicting the omitted point.

Affine three-space is $\mathrm{Spec}k[x_1,x_2,x_3]$.

The three-dimensional scheme has an affine space of rational points over its base field; its scheme structure also records non-rational points and local rings.

For $U={\mathbb{A}}_k^3\setminus \{0\}$,andThe last group has basis represented by $x_1^{-a}{x}_2^{-b}{x}_3^{-c}$ for $a,b,c\ge 1$, and all groups in degrees at least three vanish.

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 locally Noetherian scheme is regular when every local ring ${\mathcal{O}}_{X,x}$ is a regular local ring.

A Noetherian scheme is regular in codimension one when every local ring at a codimension-one point is regular. For an integral scheme these local rings are then discrete valuation rings.

An integral locally Noetherian scheme is normal when every local ring is an integrally closed domain.

More generally a scheme is normal when every local ring is an integrally closed domain; the scheme need not be globally integral, for example a disjoint union of normal integral components.

For an integral scheme, take on each affine chart the integral closure of its coordinate ring in its function field. Integral closure commutes with localization, so these rings glue to a normal scheme with an integral birational morphism to the original scheme. This is the normalization. It need not be finite without additional hypotheses; for an algebraic curve of finite type over a field it is finite. It is an isomorphism exactly when the original scheme is normal.

An algebraic variety whose local rings are integrally closed domains. In particular an irreducible variety that is a normal variety has normal affine coordinate rings, enabling codimension-two extension of regular functions on a normal variety.

A singular point on a normal two-dimensional algebraic variety has an integrally closed local domain which is not a regular local ring. A rational double point is an important isolated example with a crepant minimal resolution.

A rational double point is a normal surface singularity whose minimal resolution has a configuration of smooth rational $(-2)$-curves of type $A$, $D$ or $E$. Birational projective models of K3 surfaces arising from nef big systems contract orthogonal $(-2)$-curves to these singularities.

Over an algebraically closed field of characteristic different from two, a nondegenerate quadratic form of rank at least three is irreducible, its cone has only the vertex singular, and its hypersurface ring is Cohen-Macaulay. Thus $(R_1)$ and $(S_2)$ in the Serre criterion for normality hold. In characteristic two, the reduced zero variety of ${\sum }_i{t}_i^2$ is instead the hyperplane ${\sum }_i{t}_i=0$, which is normal. The unreduced double-hyperplane scheme is not normal.

For an integral normal variety that is Noetherian, removing a closed subset of codimension at least two does not create new regular functions. Affine-locally, a normal Noetherian domain is the intersection of its localizations at height-one primes inside its fraction field. The complement still contains every height-one point, so a regular function on it lies in each such localization and therefore in the original ring. On projective space this also follows directly by excluding irreducible polynomial factor of the denominators on each polynomial affine chart.

A Noetherian ring is normal exactly when it satisfies $(R_1)$, regularity in codimension one, and $(S_2)$, depth at least $\min (2,\dim )$ at every localization.

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 morphisms of schemes $f:X\to Y$ and $g:Y\to Z$, the graph is the displayed map. It is the pullback of the diagonal morphism $Y\to Y{\times }_{Z}Y$ along $(f\circ p_X,p_Y)$. Thus a separated morphism $g$ makes ${\Gamma }_f$ a closed immersion. If also $g\circ f$ is a closed immersion, the projection $X{\times }_{Z}Y\to Y$ is a closed immersion by base change of a morphism of schemes, and its composite with ${\Gamma }_f$ proves that $f$ is a closed immersion.

A morphism of schemes into an affine scheme is uniquely determined by its homomorphism on global sections. Conversely, a ring homomorphism $A\to \Gamma (X,{\mathcal{O}}_X)$ restricts on every affine open subscheme of $X$ to a map into $\mathrm{Spec}A$; equality on affine covers of overlaps glues these maps. No quasi-compactness or affineness assumption on $X$ is needed.

A proper morphism $f:X\to Y$ factors as $X\stackrel{g}{\to }Z\stackrel{h}{\to }Y$, with $h$ finite and $g_{\ast }{\mathcal{O}}_X={\mathcal{O}}_Z$; its fibers are connected. When $X$ is normal, $Z$ is normal. For a morphism from a smooth surface onto a curve, $Z$ is therefore a smooth projective curve over an algebraically closed field.

An isomorphism of schemes is a morphism of schemes admitting a two-sided inverse morphism. It is simultaneously a homeomorphism of the underlying spaces and an isomorphism of their structure sheaves.

The fibre product $X{\times }_{S}Y$ represents pairs of morphisms to $X$ and $Y$ with equal composites to $S$. Affine-locally,

For two closed subschemes of a scheme $X$, their intersection is their fiber product of schemes over $X$. On an affine chart, ideals $I,J$ give the quotient by $I+J$. This retains nonreduced structure and intersection multiplicities which a set-theoretic intersection would lose.

For morphisms of schemes $X\to S$, $Y\to S$, a pair of points $x,y$ lifts to a point of $X{\times }_{S}Y$ exactly when their base images agree. For sufficiency use the nonempty spectrum of a commutative ring of $\kappa (x){\otimes }_{\kappa (s)}\kappa (y)$, by tensor product of field extensions is nonzero. This establishes existence, not uniqueness, of a lift.

The fibre of a morphism of schemes over $y\in Y$ is its base change to the residue field $\kappa (y)$. For $\mathrm{Spec}B\to \mathrm{Spec}A$, with $y$ corresponding to $\mathfrak{p}$, its fibre ring is $B{\otimes }_A\kappa (\mathfrak{p})$. Its structure sheaf records nilpotents and multiplicities that the set-theoretic inverse image cannot see.

Blow up $(0,1,0)$ in ${\mathbb{A}}_{x,y,z}^3$ and compose with $xy:{\mathbb{A}}^3\to {\mathbb{A}}^1$. Nonzero fibres are ${\mathbb{G}}_m\times {\mathbb{A}}^1$. The zero fibre consists of the strict transforms of the two coordinate planes and the exceptional ${\mathbb{P}}^2$, all with multiplicity one. Its plane crossings make it singular; its closed ${\mathbb{P}}^2$ makes it nonaffine. The centre is deliberately on just one component of $xy=0$, keeping the fibre reduced.

The scheme-theoretic fibres over positive real points, negative real points, zero, and nonreal closed points have real coordinate algebras ${\mathbb{R}}^2\times \mathbb{C}$, ${\mathbb{C}}^2$, $\mathbb{R}[x]/(x^4)$, and ${\mathbb{C}}^4$, respectively. Their ordinary irreducible component counts are $3,2,1,4$, so there are four real-scheme isomorphism classes. The last case pulls back an irreducible quadratic to a squarefree degree-eight polynomial, with four conjugate pairs of nonreal roots. The zero fibre retains its nilpotents and is not a reduced point.

A morphism of schemes between integral schemes can have a scheme-theoretic fibre that is neither reduced nor irreducible. For the map of affine lines given by $t=x^2(x-1{)}^2$, the fibre at $t=0$ has ring $k[x]/(x^2(x-1{)}^2)$. The Chinese remainder theorem identifies it with the product of two dual number rings, so it consists of two nonreduced double points. Its nilpotent structure is invisible to the set-theoretic fibre.

For an injective ring homomorphism between nonzero integral domains $A\to B$, the scheme-theoretic fibre over the generic point of $\mathrm{Spec}A$ is $\mathrm{Spec}(S^{-1}B)$ with $S=A\setminus \{0\}$. Localization at these nonzero elements is a nonzero integral domain, so the generic fibre is a nonempty integral scheme. A surjective morphism between these affine schemes necessarily gives an injective ring map: a prime above $(0)$ contains its kernel.

For morphisms $X\to S$ and $Y\to S$, maps $T\to X{\times }_{S}Y$ are naturally equivalent to pairs of maps $T\to X$ and $T\to Y$ whose composites to $S$ agree.

Given $f:X\to Y$ and $g:Y^{\prime }\to Y$, the base change of $f$ along $g$ is the projectionProperties such as being proper and flat are stable under arbitrary base change.

Flatness of a projective family does not by itself force global sections to commute with base change. For a discrete valuation ring with uniformizer $\pi$, glue $\mathrm{Spec}R[\pi t,t^2,t^3]$ to $\mathrm{Spec}R[t^{-1}]$ along $\mathrm{Spec}R[t,t^{-1}]$. In the projective model with coordinates $(1,\pi t,t^2,t^3)$, the global ring is $R$. The special-fibre first chart has an additional square-zero element $a$ annihilated by $t^2,t^3$; it vanishes on the overlap and therefore extends by zero to a global section. The special-fibre global ring is $k[a]/(a^2)$, and base change from $R$ reaches only its constants.

For a scheme over $\mathbb{R}$, its base change of a morphism of schemes to $\mathbb{C}$ has scheme-theoretic fibre over a point $p$ equal to $\mathrm{Spec}(\kappa (p){\otimes }_{\mathbb{R}}\mathbb{C})$. This is the spectrum of $\kappa (p)[s]/(s^2+1)$. It has two points if $-1$ is a square in the residue field, and otherwise one point with a quadratic field extension. In both cases it is reduced, because the quadratic polynomial has no repeated root in characteristic zero. The distinction applies to generic points as well as closed points.

If $X\to Y$ is a surjective morphism of schemes, the projection $X{\times }_Y{Y}^{\prime }\to Y^{\prime }$ is surjective for every $Y^{\prime }\to Y$. For a point $y\prime$, choose a point $x$ over its base image and apply the point-lifting property of a scheme fibre product. No flatness assumption is necessary.

A locally closed immersion factors as a closed immersion followed by an open immersion. Equivalently, it identifies its source with a closed subscheme of an open subscheme of the target.

An open immersion is a morphism of schemes that identifies its source with an open subscheme of its target. It is an isomorphism onto that open subscheme as a locally ringed space.

The diagonal morphism sends a point to the pair consisting of that point twice. Every diagonal of schemes is a locally closed immersion, and a morphism is separated exactly when its diagonal is a closed immersion.

For morphisms $f,g:X\to Y$ over a base $S$, their scheme-theoretic coincidence locus is the fibre productIt is the equalizer of $f$ and $g$, hence is universal among subschemes on which they agree. It is locally closed because every diagonal is an immersion, and closed when $Y\to S$ is separated.

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 an algebra $B$ over $A$, the module ${\Omega }_{B/A}$ represents $A$-linear derivations: every derivation $B\to N$ factors uniquely through the universal map $d:B\to {\Omega }_{B/A}$.

For finite separable $L/F$, every $k$-derivation on $F$ into an $L$-module extends uniquely to $L$. Differentiate the minimal polynomial of each algebraic generator and divide by its nonzero derivative. Consequently the Transitivity exact sequence for Kähler differentials identifies ${\Omega }_{L/k}$ with $L{\otimes }_F{\Omega }_{F/k}$, and ${\Omega }_{L/F}=0$.

The Kähler differentials commute with extension of the base ring. A derivation of the base-changed algebra that kills $k^{\prime }$ is determined by its restriction to $A\otimes 1$, and the Leibniz rule provides its unique extension. No flatness hypothesis is required for this isomorphism.

Kähler differentials commute with localization of a ring. The quotient rule $d(a/s)=(sda-ads)/s^2$ uniquely extends a derivation after inverting $s$; its universal property proves the displayed module isomorphism.

The Kähler differentials of a polynomial ring are free on the differentials of its variables. Assigning arbitrary values to these generators uniquely defines a derivation. Applying the Conormal exact sequence for Kähler differentials to a quotient gives its presentation by the formal derivatives of its defining equations.

Composing an $A$-linear map with $d:A\to {\Omega }_{A/k}$ gives a $k$-derivation. Every derivation factors uniquely this way because its linearity and product rule are exactly the defining relations of the module of Kähler differentials.

For a commutative $k$-algebra $A$, let $I$ be the kernel of multiplication $\mu :A{\otimes }_{k}A\to A$. The formula $a\mapsto [1\otimes a-a\otimes 1]$ is a derivation of an algebra into the quotient module $I/I^2$, with $A$ acting through the first factor. It induces an isomorphism from the module of Kähler differentials. An inverse is induced by $q(x\otimes y)=xdy$: the identity $q(rs)=\mu (r)q(s)+\mu (s)q(r)$ shows $q(I^2)=0$. For $z={\sum }_i{x}_i\otimes y_i\in I$, the equality ${\sum }_i{x}_i{y}_i=0$ gives $[z]={\sum }_i{x}_i[1\otimes y_i-y_i\otimes 1]$, proving that the two maps are inverse. For an affine variety, $I$ is the defining ideal of its diagonal morphism.

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

For a closed immersion, this quasi-coherent sheaf measures the first-order equations of the subvariety. Its local module is $I/I^2$, and its map to the restricted Kähler differential sheaf sends the class of $f$ to $df$. This is the definition in Stacks Project, Section 29.32.

For a closed immersion $i:Z\hookrightarrow X$ with ideal sheaf $\mathcal{I}$, the normal sheaf is ${\mathcal{N}}_{Z/X}=\mathcal{H}o{m}_{{\mathcal{O}}_Z}(\mathcal{I}/{\mathcal{I}}^2,{\mathcal{O}}_Z)$. It is the dual of the conormal sheaf. For a regular embedding it is a locally free sheaf; in the smooth case it agrees with the normal bundle. Without regularity it need not be locally free, but its global sections still classify first-order embedded deformations.