Algebraic geometry

For a projective variety over the complex numbers, analytification gives an equivalence between algebraic and analytic coherent sheaves, preserves their cohomology, and is fully faithful on their morphisms. In particular, a holomorphic isomorphism between algebraic line bundles is algebraic. This allows a complex-torus proof of a line-bundle identity to establish the corresponding algebraic identity on a projective elliptic curve.

A property holds at a very general point of an irreducible complex algebraic parameter space if it holds outside a countable union of proper Zariski-closed subsets. This permits countably many exceptional conditions. Over $\mathbb{C}$ that complement is nonempty and dense, for example by the Baire category theorem on a local smooth analytic chart.

For a projective scheme $X$ and a fixed Hilbert polynomial $P$, the Hilbert scheme parametrizes flat families of closed subschemes of $X$ with that polynomial. This turns a family of embedded algebraic curves into an algebraic parameter space.

For a projective scheme $X$ over a field $k$, the Hilbert functor assigns to a $k$-scheme $T$ the set of closed subschemes $W\subseteq X{\times }_{k}T$ which are of finite presentation, proper and flat over $T$. Pullback of families makes it a contravariant functor. After choosing a very ample line bundle, requiring a fixed fibre Hilbert polynomial $P$ gives the subfunctor represented by ${\mathrm{Hilb}}^P(X)$. The unrestricted functor is represented by the disjoint union of these Hilbert schemes.

A first-order embedded deformation of a closed subscheme $Z\subseteq X$ is a closed subscheme $Z_{\epsilon }\subseteq X{\times }_k\mathrm{Spec}(k[\epsilon ]/({\epsilon }^2))$ flat over the dual numbers and with special fibre equal to the given embedded $Z$. It is a tangent vector of the Hilbert scheme, not a deformation modulo arbitrary automorphisms of the ambient scheme. These deformations are naturally the global sections of the normal sheaf.

For a closed subscheme $Z\subseteq X$ with ideal sheaf $\mathcal{I}$, one has $T_{[Z]}\mathrm{Hilb}(X)\cong {\mathrm{Hom}}_{{\mathcal{O}}_X}(\mathcal{I},i_{\ast }{\mathcal{O}}_Z)\cong H^0(Z,{\mathcal{N}}_{Z/X})$. Locally a flat deformation ideal $J\subseteq A[\epsilon ]/({\epsilon }^2)$ satisfies $J\cap \epsilon A=\epsilon I$. Lifting $f\in I$ to $f+\epsilon g\in J$ gives a well-defined map $f\mapsto g\mathrm{mod}I$ killing $I^2$. Conversely the graph condition $g\mathrm{mod}I=\varphi (f)$ defines an ideal with flat quotient by the flatness criterion over dual numbers. These operations commute with restriction and are inverse, giving the global identification without a smoothness assumption.

The closed subscheme of the Grassmannian parametrizing projective $k$-planes contained in an embedded projective variety $X$.

The Fano scheme for projective lines. For $X\subseteq {\mathbb{P}}^3$, it is a closed subscheme of $\mathrm{Gr}(2,4)$ and hence is projective. Requiring the defining homogeneous polynomials of $X$ to vanish along a line gives its closed defining conditions.

Cubic forms in four variables form ${\mathbb{P}}^{19}$. Lines form the four-dimensional Grassmannian $G=\mathrm{Gr}(2,4)$. The incidence pairs $(\ell ,[F])$ with $F{|}_{\ell }=0$ form a projective-space bundle with fibre ${\mathbb{P}}^{15}$ over $G$, since restriction to a line imposes four independent linear conditions. Hence the incidence variety is irreducible of dimension 19. For $F=x_0^2{x}_2+x_1^2{x}_3$, a nearby line written $x_2=a_0{x}_0+a_1{x}_1$, $x_3=b_0{x}_0+b_1{x}_1$ lies on $F$ precisely when all four parameters vanish. This is an isolated point of that fibre. The fiber dimension theorem forces the image to have dimension 19, and projectivity makes the image closed. It is therefore all of ${\mathbb{P}}^{19}$, proving geometric existence over an algebraically closed field.

A projective bundle with fibre ${\mathbb{P}}^{15}$ over the four-dimensional Grassmannian $\mathrm{Gr}(2,4)$. The condition of containing a fixed line imposes four independent linear conditions on the twenty cubic coefficients. The projection to ${\mathbb{P}}^{19}$ is proper. The line $\{x_2=x_3=0\}$ is an isolated point over $x_0^2{x}_2+x_1^2{x}_3$; isolated fibre point forces dominance in equal dimensions then proves that every cubic surface contains a line.

For a line $L$ on a degree-$d$ smooth surface in ${\mathbb{P}}^3$, the adjunction formula gives $L^2=2-d$. If $d\ge 3$, all lines are negative curves, hence there are only countably many by countability of negative curves. Their projective Fano scheme of lines cannot have a positive-dimensional complex component, since such a component has uncountably many points. A zero-dimensional projective scheme of finite type has finitely many points, proving the assertion.

Étale cohomology is sheaf cohomology on the site whose objects are étale morphisms $Y\to X$ and whose coverings are jointly surjective families of such morphisms. It provides cohomological invariants for algebraic varieties over fields of positive characteristic. In Deligne-Lusztig theory, compactly supported groups with $\ell$-adic coefficients, for $\ell$ different from the field characteristic, give finite-dimensional representations of the finite groups acting on the varieties.

An algebraic set is a zero locus of polynomial equations, allowing reducible sets. An affine algebraic set uses ordinary polynomial equations; a projective algebraic set uses homogeneous equations in projective coordinates. On affine or projective space, closed sets in the Zariski topology are algebraic sets, and an irreducible algebraic set is an algebraic variety in the classical convention.

An abelian variety of dimension $g$ has complex multiplication when its rational endomorphism ring contains a commutative semisimple $\mathbb{Q}$-algebra of dimension $2g$. For an elliptic curve in characteristic zero, this means its geometric endomorphism ring of an elliptic curve is an order in an imaginary quadratic field. The resulting rank-one Tate module representations connect torsion points to class field theory.

A complex CM period compares an algebraic invariant differential on a CM elliptic curve with its analytic lattice coordinate. At an ordinary prime, a $p$-adic CM period compares the same differential with the invariant differential of the multiplicative formal group: an isomorphism $\eta$ is normalized by ${\log }_{\hat{E}}\eta (T)={\Omega }_p\log (1+T)$. Matched periods make normalized critical Hecke L-function values algebraic and give precise p-adic L-function interpolation.

For a complex CM torus $\mathbb{C}/\mathfrak{a}$ and an idele $s$ of its CM field, the automorphism corresponding to $s$ by Artin reciprocity takes its lattice to $s^{-1}\mathfrak{a}$ and acts on torsion through the adelic scalar $s^{-1}$. When the curve is defined over a larger field, use the idelic norm to the CM field. Comparison with the original curve gives a scalar in the CM field, producing the Hecke character of a CM elliptic curve.

An elliptic curve with complex multiplication is called a CM elliptic curve. Distinguish geometric endomorphisms from endomorphisms defined over the base field. In characteristic zero, adjoining the imaginary quadratic field selected by the tangent action makes all geometric endomorphisms defined over the base. Good split primes have ordinary reduction; good inert primes have supersingular reduction.

The projective morphism ${\mathrm{Proj}}_X{⨁}_{j\ge 0}{\mathcal{I}}^j\to X$ associated to the powers of the ideal sheaf of a closed centre. Off that centre it is an isomorphism. Blowing up a smooth point of an $n$-dimensional smooth variety replaces the point by a projective space ${\mathbb{P}}^{n-1}$, the algebraic exceptional divisor.

For a blowup of an algebraic variety $\pi :\tilde{X}\to X$ with centre $Z$, the strict transform of an algebraic subvariety of a subvariety $Y$ not contained in $Z$ is the closure of ${\pi }^{-1}(Y\setminus Z)$. It excludes components supported entirely in the algebraic exceptional divisor.

The divisor contracted by a blowup of an algebraic variety to its centre. For the blowup of a smooth point on an $n$-dimensional smooth variety it is ${\mathbb{P}}^{n-1}$. Being a closed projective subvariety of positive dimension, it can obstruct affineness of a fibre containing it.

The Zariski topology on an affine scheme $\mathrm{Spec}A$ has closed sets $V(I)=\{\mathfrak{p}:I\subseteq \mathfrak{p}\}$ for ideals $I\subseteq A$. Its basic open sets are the principal open subschemes $D(f)$.

A Zariski cover of a scheme is a family of open immersions whose images cover its underlying topological space. Such families define the Zariski topology as a Grothendieck topology on the category of schemes; a cover need not be finite.

A subset of an algebraic variety is Zariski dense when its closure in the Zariski topology is the whole variety. For an affine variety, this is equivalent to saying that every regular function vanishing on the subset vanishes on the entire variety.

A presheaf of sets on a topological space $X$ is a contravariant functor from the inclusion-ordered category $\mathcal{O}(X)$ of open subsets to $\mathbf{Set}$. Its maps $F(U)\to F(V)$ for $V\subseteq U$ are restriction maps.

Every restriction is the identity. Its stalk at each point is $T$, and its étale space of a presheaf is $X\times T$ with $T$ discrete. It is generally not a sheaf; in particular its value on the empty open set is $T$. Its sheafification is the constant sheaf of sets.

A natural transformation between presheaves assigns maps $F(U)\to G(U)$ commuting with every restriction. For presheaves of abelian groups or modules, those maps preserve the specified algebraic structure.

A sheaf of sets is a presheaf in which compatible sections on any open cover glue to one unique section on the union.

A sheaf may additionally carry abelian group or module structure on its sections, with restriction and gluing preserving that structure. The underlying sheaf of sets retains the same local-to-global axiom.

For inclusion of a point $i:\{x\}\to X$, this is the direct-image sheaf of a set. It is right adjoint to the stalk functor. It uses the singleton terminal set on opens omitting $x$, rather than the zero group used for abelian-group skyscraper sheaves.

The space of germs of a presheaf has basic opens given by the germs of one section over an open set. Projection to its base point is a local homeomorphism. Continuous sections of this projection give sheafification. For the constant presheaf of sets, the basic opens are $U\times \{t\}$, giving the ordinary product with a discrete fibre.

The sections are locally constant functions into a discrete set $T$, with ordinary restriction. It is the sheafification of the constant presheaf of sets. Its value on an empty open set is a singleton, even if $T$ is empty. For a topological space, this construction is left adjoint to the global sections functor: a map from it to a sheaf is a family of global sections indexed by $T$.

A morphism of sheaves assigns a map $\mathcal{F}(U)\to \mathcal{G}(U)$ on every open set, commuting with all restriction maps. For sheaves of abelian groups, sheaves of rings or sheaves of modules, the maps must also preserve the relevant algebraic structure. It induces maps on every stalk, and equality of sheaf morphisms can be checked there.

For a morphism of sheaves $\varphi :\mathcal{F}\to \mathcal{G}$ of abelian groups or modules, the cokernel sheaf is the sheafification of the presheaf $U\mapsto \mathcal{G}(U)/\varphi (\mathcal{F}(U))$. Its stalk at $x$ is $\mathrm{coker}({\mathcal{F}}_x\to {\mathcal{G}}_x)$. The presheaf quotient itself need not satisfy the sheaf gluing axiom, which explains why a surjective morphism of sheaves need not be surjective on global sections.

For a continuous map $f:X\to Y$, an f-morphism from a sheaf on $Y$ to a sheaf on $X$ is a family of the displayed maps commuting with restrictions. It is equivalently a morphism of sheaves $\mathcal{G}\to f_{\ast }\mathcal{F}$. For each $x\in X$ it induces a map ${\mathcal{G}}_{f(x)}\to {\mathcal{F}}_x$ on stalks, by pulling back the neighbourhood representing a germ.

A morphism of sheaves of modules is surjective when the induced map on every stalk is onto. Equivalently every target section can locally be lifted. It need not be onto on global sections; the long exact sequence in sheaf cohomology describes that obstruction.

For a morphism of sheaves of abelian groups or modules, the kernel sheaf has sections $\ker (\mathcal{F}(U)\to \mathcal{G}(U))$. These already satisfy the sheaf gluing axiom. Its stalk at $x$ is the kernel of ${\mathcal{F}}_x\to {\mathcal{G}}_x$, since filtered colimits of modules preserve exactness. A sheaf morphism is injective exactly when its kernel sheaf is zero.

A sheaf of abelian groups is a sheaf of sets whose sections on each open set form an abelian group and whose restriction maps are group homomorphisms. Its stalk of a sheaf is the corresponding group of germs. Kernels are computed on sections, whereas cokernels are obtained by sheafification of the sectionwise cokernel. An exact sequence can therefore be checked on stalks.

The direct sum of sheaves of abelian groups is the sheafification of the presheaf $U\mapsto {⨁}_i{\mathcal{F}}_i(U)$. It need not equal that presheaf: sections are locally represented using finitely many summands, while globally infinitely many can occur. For point-supported skyscraper sheaves at closed points, allowable point supports are a locally finite family of subsets.

A sheaf of abelian groups is injective if every morphism to it from a subsheaf extends to the containing sheaf. Every such sheaf embeds in an injective sheaf. Injective sheaves are flasque: the inclusion of extension by zero constant-integer sheaves associated to $U\subseteq V$ turns the extension property into surjectivity of restriction from $V$ to $U$.

An exact sequence of sheaves of abelian groups is a sequence of sheaf homomorphisms for which the image of each map equals the kernel of the next. Exactness can be checked on every stalk of a sheaf. Exactness of the associated sequences of sections on every open set is a stronger condition: a surjective sheaf homomorphism need only admit lifts locally.

A sequence $0\to {\mathcal{F}}^{\prime }\to \mathcal{F}\to {\mathcal{F}}^{\prime \prime }\to 0$ of sheaves is exact exactly when the induced sequence on every stalk is exact.

For an abelian group $A$, the constant sheaf has sections over an open set given by its locally constant $A$-valued functions. It is the sheafification of the constant presheaf, and its sections on a disconnected open set need not have a single common value.

Every nonempty open subset of an irreducible space is connected. A locally constant function into a fixed abelian group $D$ is therefore constant on that open set. Restrictions between nonempty opens identify with the identity of $D$, and restrictions to the empty set are surjective onto zero. Thus the constant sheaf is flasque, with global sections $D$ and zero higher sheaf cohomology on a nonempty irreducible space. For an integral scheme its constant function-field sheaf has its natural structure-sheaf action by regular functions viewed in the function field.

The sheaf gluing axiom says that sections $s_i$ on an open cover $(U_i)$ that agree on every overlap $U_i\cap U_j$ glue to a unique section on ${\bigcup }_i{U}_i$.

A ringed space $(X,{\mathcal{O}}_X)$ is a topological space $X$ equipped with a sheaf of rings ${\mathcal{O}}_X$. A morphism consists of a continuous map $f:X\to Y$ and a compatible morphism $f^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$.

A morphism of ringed spaces $f:(X,{\mathcal{O}}_X)\to (Y,{\mathcal{O}}_Y)$ consists of a continuous map $f:X\to Y$ and a morphism of sheaves of rings $f^{-1}{\mathcal{O}}_Y\to {\mathcal{O}}_X$.

A sheaf of rings assigns a ring $\mathcal{O}(U)$ to every open set, restriction homomorphisms to inclusions, and satisfies local identity and gluing. It supplies the local functions on a ringed space.

This is a ring-valued Sheaf (mathematics), with ring homomorphisms as restriction maps.

A sheaf of topological rings takes values in topological rings, with continuous restriction maps. For an open cover, the sheaf condition is an equalizer of the product of sections on the cover and the product on pairwise overlaps, in the category of topological rings. The structure sheaf of a formal scheme is an inverse limit of discrete structure sheaves with this topology.

// Target: algebra.bigb

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 underlying topological space of every scheme is a Kolmogorov space: any two distinct points are distinguished by an open subset containing just one. Choose an affine open subscheme containing the first point. If it omits the second, it already distinguishes them. Otherwise, the two points correspond to distinct prime ideals, and an element in one but not the other defines a principal open subscheme distinguishing them. Thus distinct points of a scheme have distinct closures.

A scheme is locally Noetherian if it has an affine open cover by spectra of Noetherian rings. It is a Noetherian scheme exactly when it is also a quasi-compact scheme: a finite subcover then supplies the required finite affine open cover. An infinite gluing of spectra of Noetherian rings along open subschemes is still locally Noetherian, but need not be a quasi-compact scheme.

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

Let $A$ be a commutative ring and $I$ an ideal all of whose elements are nilpotent elements. Every ring homomorphism from $A$ to a field kills $I$, because a field has no nonzero nilpotent elements. Thus the closed immersion $\mathrm{Spec}(A/I)\hookrightarrow \mathrm{Spec}A$ induces a bijection on points valued in every field, even when it is not an isomorphism of schemes. The dual numbers provide the smallest example. Testing the functor of points on all affine schemes does detect this information, by affine test schemes determine scheme morphisms.

Restricting the functor of points to affine schemes gives a fully faithful functor from schemes to covariant set-valued functors on commutative rings. Given a natural transformation, apply it to the inclusions of an affine open cover of $X$. Naturality on affine open subschemes of overlaps makes the resulting maps to $Y$ agree, so they glue. For an arbitrary map from an affine scheme to $X$, cover its inverse images of these charts by affine open subschemes and apply naturality again. This proves that the glued map induces the prescribed natural transformation, and uniqueness follows from the original cover. In particular, the restricted functor of points determines the scheme up to an isomorphism of schemes.

A Zariski sheaf $F$ is a functor represented by a scheme if it has a set of open subfunctors represented by affine schemes whose pullbacks cover every test scheme. The overlaps are open subschemes in both charts, and their identifications satisfy the compatibility on triple overlaps. Gluing of schemes along open subschemes produces a representing scheme; the Zariski sheaf condition identifies its functor of points with $F$. Conversely, every affine open cover of a scheme supplies such a presentation. A family of local charts covers in this sense without requiring every global element to belong to a single chart.

A subfunctor $H\subseteq F$ on the opposite of the category of schemes is open if every natural transformation $h_T\to F$ pulls it back to a functor represented by a scheme $U$, with the projection represented by an open immersion $U\hookrightarrow T$. Thus membership in $H$ defines an open subscheme in every family parameterized by $T$. This condition is preserved by further pullbacks.

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 $x$.

The nilradical sheaf of a scheme is the ideal sheaf of locally nilpotent sections in its structure sheaf. On an affine open subscheme $\mathrm{Spec}A$ it is the quasi-coherent sheaf associated to the nilradical of $A$, since the nilradical commutes with localization. Its quotient defines the reduced induced subscheme. On a Noetherian scheme it has a uniform nilpotence exponent: take nilpotence bounds on a finite affine open cover and use their maximum. On a general scheme, a locally nilpotent section need not have a single nilpotence exponent over a non-quasi-compact open subset.

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.

A regular function $f$ on an entire affine variety is a polynomial function, even if the variety is reducible. Put $J=\{g\in A(X):gf\in A(X)\}$, an ideal of its coordinate ring. Near a point $p$, write $f=a/b$ with $b$ nonzero, and choose a principal open subset $D(h)$ containing $p$ inside that neighbourhood. Then $(hb)f=ha$ on all of $X$: on $D(h)$ this is the fraction identity, and off $D(h)$ both sides vanish. Hence $hb\in J$ and $(hb)(p)\ne 0$. No point is a common zero of $J$, so the weak Hilbert Nullstellensatz implies $J=A(X)$. Thus $1\in J$ and $f\in A(X)$. This proof does not require a fraction field for a reducible coordinate ring.

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.