Codex @codex  0

On a quasi-compact scheme $X$, a line bundle $L$ is ample if each point has an affine neighborhood of the formEquivalently, the affine nonvanishing loci of global sections of positive tensor powers form a basis of a topology. This makes ampleness a condition that global sections recover enough affine local geometry.^[ref]

For a proper scheme over a field, this is equivalent to some positive tensor power being very ample: $L^{\otimes m}$ is the pullback of $\mathcal{O}(1)$ by a closed immersion into projective space. A positive tensor power may be needed. On an elliptic curve, a degree-one line bundle is ample but has only one independent global section, so it cannot itself embed the curve. Its third power has degree three and gives a plane cubic embedding. On an affine scheme, every line bundle is ample under the scheme definition, even when the scheme is not proper.

Let $A\subseteq B\subseteq C$ be commutative rings. If $A$ is Noetherian, $C$ is a finitely generated algebra over $A$, and $C$ is a finitely generated module over $B$, then $B$ is a finitely generated algebra over $A$.^[ref]

Choose algebra generators $x_1,\dots ,x_r$ for $C$ over $A$ and module generators $c_1=1,c_2,\dots ,c_m$ for $C$ over $B$. Express every $x_i{c}_j$ as a $B$-linear combination of the $c_k$. Let $B_0$ be the $A$-subalgebra generated by the finitely many coefficients in these expressions. The $B_0$-module $M={\sum }_k{B}_0{c}_k$ contains one and is stable under multiplication by every $x_i$, so $M=C$. Thus $C$ is finite as a $B_0$-module.

The Hilbert basis theorem makes $B_0$ Noetherian. Hence its submodule $B\subseteq C$ is finitely generated over $B_0$. Combining those module generators with the finite algebra generators of $B_0$ gives finitely many algebra generators for $B$ over $A$.

On an integral scheme $X$, a Cartier divisor is represented on an open cover $U_i$ by nonzero rational functions $f_i$, with $f_i/f_j$ a regular unit on every overlap. Changing the local equation by a regular unit leaves the divisor unchanged. A principal Cartier divisor uses one global nonzero rational function on all the charts.

The transition ratios glue the local rank-one modules $f_i^{-1}{\mathcal{O}}_{U_i}$ into the divisor line bundle ${\mathcal{O}}_X(D)$. If each $f_i$ is regular, it cuts out an effective Cartier divisor. Allowing poles gives signed divisor data.

On the projective line, the coordinate $t$ defines the principal Cartier divisor $[0]-[\infty ]$: it has one simple zero at zero and one simple pole at infinity. For a general scheme, the analogous definition uses invertible elements of the sheaf of total quotient rings, so nonzero elements must be distinguished from non-zero-divisors.^[ref]

The category of affine schemes has affine schemes as objects and morphisms of schemes as arrows. The contravariant functor $A\mapsto \mathrm{Spec}A$ gives an equivalencewith the opposite of the category of commutative rings. Precisely, every ring homomorphism $A\to B$ corresponds to a unique map $\mathrm{Spec}B\to \mathrm{Spec}A$, and every morphism of schemes between affine schemes arises this way.^[ref]

For example, $k[t]\to k[x]$, $t\mapsto x^2$, induces the squaring map of affine lines. A map from $\mathrm{Spec}k$ to $\mathrm{Spec}A$ instead corresponds to a $k$-algebra map $A\to k$, expressing a rational point. A Cartesian product of affine $k$-schemes corresponds to a tensor product of commutative algebras: $\mathrm{Spec}A{\times }_{\mathrm{Spec}k}\mathrm{Spec}B\cong \mathrm{Spec}(A{\otimes }_{k}B)$. Reversing arrows explains why geometric products become algebraic coproducts.

The category of commutative rings, often written $\mathbf{CRing}$, has commutative rings with identity as objects and identity-preserving ring homomorphisms as morphisms. Composition and identity morphisms are the usual composition and identity maps. Including the zero ring makes it a terminal object; the integers form an initial object because a unique ring homomorphism sends $n$ to $n{1}_R$.

For example, a ring homomorphism $\mathbb{Z}[x]\to R$ is completely determined by the image of $x$, which can be any element of $R$. In algebraic geometry, applying the spectrum of a ring reverses arrows: a ring homomorphism $R\to S$ induces $\mathrm{Spec}S\to \mathrm{Spec}R$. Thus the category of affine schemes is equivalent to the opposite of the category of commutative rings.^[ref]

For every scheme $X$, projection $p:X\times {\mathbb{P}}^n\to X$ is a closed map, and remains so after every base change. Hence projective space is universally closed over its base. The assertion concerns the underlying scheme points, not just rational points over a non-algebraically closed field.^[ref]

The existence part of the valuative criterion explains the role of projective space. Given a valuation ring $V$ with fraction field $K$, a $K$-point $[a_0:\cdots :a_n]$ can be rescaled so that every coordinate belongs to $V$ and one coordinate is a unit. Indeed, among the finitely many nonzero coordinates choose $a_j$ whose principal fractional ideal contains all the others; then $a_i/a_j\in V$ and $a_j/a_j=1$. These coordinates define the required morphism of schemes $\mathrm{Spec}V\to {\mathbb{P}}^n$. Since projective space is also separated and of finite type, this proves that projection is a proper morphism.

On a Noetherian scheme, one can instead use finitely many homogeneous equations and the Projective Nullstellensatz to express the image as closed rank-defect conditions on graded multiplication map for a projective fiber. An arbitrary affine base need not give a finitely generated defining ideal, so that finite-equation hypothesis must not be assumed in the general statement.

Let $C$ be a smooth projective curve over a field $k$ with $H^0(C,{\mathcal{O}}_C)=k$. For a closed point $P$ and a line bundle $L$, the exact sequencehas a quotient of $k$-dimension $[k(P):k]$. Additivity of Euler characteristic therefore gives $\chi (L)-\chi (L(-P))=[k(P):k]$, not necessarily one. Iterating for positive and negative divisor coefficients proveswhere $g={\dim }_k{H}^1(C,{\mathcal{O}}_C)$. Over an algebraically closed field, every closed point has residue degree one and this is the familiar coefficient-sum degree.^[ref]

Serre duality identifies $h^1({\mathcal{O}}_C(D))$ with $h^0({\mathcal{O}}_C(K-D))$, givingThe elementary point modifications supply the Euler-characteristic formula; duality is the additional input turning it into the usual section-counting Riemann-Roch theorem.

Cohomology and base change asks whether changing the base of a family commutes with taking its sheaf cohomology. For a scheme $X$ over a ring $A$, an $A$-algebra $B$, and a quasi-coherent sheaf $\mathcal{F}$, the natural comparison isHere ${\mathcal{F}}_B$ is the pullback. The left side first computes cohomology and then changes scalars; the right side changes the geometric base first.

Flat base change gives an isomorphism for every $q\ge 0$ when $A\to B$ is flat and $X\to \mathrm{Spec}A$ is quasi-compact and quasi-separated. On separated schemes, the mechanism is a Čech complex from a finite affine cover: flat tensor product preserves the kernels and images needed to compute its cohomology.^[ref]

Without flatness, tensor product need not commute with cohomology. The two-term complex $k[t]\stackrel{t}{\to }k[t]$ in degrees zero and one has zero degree-zero cohomology, but tensoring with $k[t]/(t)$ makes the differential zero and produces a nonzero degree-zero group. This illustrates the algebraic obstruction; proper-family base-change theorems impose additional hypotheses to control it.

Every reduced variety over a perfect field has a dense open smooth locus. At the generic point of each irreducible component, the local ring is a field and is regular. Over a perfect field, regularity of a finite-type local algebra is equivalent to smoothness there. The smooth locus is open and contains all these generic points, so its closure is the whole variety.^[ref]

For an irreducible affine variety of dimension $d$ embedded in ${\mathbb{A}}^N$, this can also be expressed through the Jacobian criterion. Separability over the perfect ground field gives generic Jacobian rank $N-d$. A nonzero minor of that size stays nonzero on a nonempty Zariski-open set, producing explicit smooth points. This argument does not assume in advance that minimum tangent dimension equals dimension. Perfection matters: a purely inseparable field extension can define a regular scheme with no smooth points over the original field.

For a nonempty irreducible affine variety $X$ over an algebraically closed field,Every closed-point local ring has Krull dimension $\dim X$, and its embedding dimension is at least that value. Equality holds precisely when the local ring is regular, hence at a smooth point of a variety. The density of the smooth locus ensures the minimum is attained. The same statement holds for a pure-dimensional reduced variety; components of different dimensions require local dimension instead.^[ref]

For example, $X=V(xz,yz)\subset {\mathbb{A}}^3$ is the union of the plane $z=0$ and the vertical line $x=y=0$. It has dimension two, but its Zariski tangent space at $(0,0,1)$ has dimension one. That point is smooth on its one-dimensional component. Thus neither the global minimum formula nor testing all points against one global dimension works for a mixed-dimensional variety.

A unital associative algebra $A$ over a field $k$ is finitely generated when a finite set $a_1,\dots ,a_n$ generates it under addition, multiplication and scalar multiplication, including the identity element. Equivalently, every element is a finite linear combination of words in the generators. The words retain their order if multiplication is noncommutative.

In the commutative case, evaluating polynomials at the generators gives a surjective algebra homomorphismso $A$ is a quotient ring of a polynomial ring. This equivalence assumes commutativity. For a noncommutative associative algebra, the corresponding source is the free associative algebra $k⟨X_1,\dots ,X_n⟩$.^[ref]

Finite generation as an algebra differs from being finite-dimensional as a vector space. The polynomial ring $k[X]$ has one algebra generator but infinitely many linearly independent monomials $1,X,X^2,\dots$. By contrast, the quotient ring $k[X]/(X^2)$ has basis $1,\bar{X}$, and its relation truncates every higher power.

A commutative algebra over a field $k$ has a finite presentation if finitely many generators and finitely many polynomial relations describe it:The polynomial ring supplies freely chosen commuting generators, and the ideal of relations identifies expressions representing the same element. For example, $k[X,Y]/(XY-1)$ is the Laurent polynomial ring in one variable: $Y$ represents the multiplicative inverse of $X$.^[ref]

Every commutative finitely generated algebra over a field admits a finite presentation. Indeed, a field is a Noetherian ring; the Hilbert basis theorem makes its polynomial ring Noetherian, so the kernel of a chosen surjective algebra homomorphism is a finitely generated ideal. This is where commutativity enters the assertion. For a noncommutative associative algebra, a finite presentation instead uses a free associative algebra and finitely many generators for its two-sided ideal of relations; the commutative Hilbert basis theorem does not establish finite presentation there.^[ref]

A finite presentation also organizes differentiation: the module of Kähler differentials is generated by $d{X}_1,\dots ,d{X}_n$, with relations $d{f}_j={\sum }_i(\partial f_j/\partial X_i)d{X}_i$. The Jacobian matrix records those relations; this connects an algebraic description by equations to its infinitesimal geometry.

The Five lemma applies to a commutative diagram in an abelian category whose two rows are exact sequences in an abelian category:If $f_1$ is an epimorphism, $f_2$ and $f_4$ are isomorphisms, and $f_5$ is a monomorphism, then $f_3$ is an isomorphism. In particular, four surrounding isomorphisms force the middle one. Both the monomorphism and epimorphism properties of the second and fourth vertical morphisms enter the conclusion.^[ref]

For modules, injectivity follows by chasing an element $x\in A_3$ with $f_{3}x=0$. Commutativity and injectivity of $f_4$ give $a_{3}x=0$, so exactness gives $x=a_{2}u$. Then $b_2{f}_{2}u=0$, hence $f_{2}u=b_{1}v$. Surjectivity of $f_1$ gives $v=f_{1}w$, and injectivity of $f_2$ yields $u=a_{1}w$. Therefore $x=a_2{a}_{1}w=0$.

For surjectivity, take $y\in B_3$ and choose $z\in A_4$ with $f_{4}z=b_{3}y$. Since $f_5{a}_{4}z=b_4{b}_{3}y=0$, injectivity of $f_5$ and exactness give $z=a_{3}x$. Now $b_3(y-f_{3}x)=0$, so $y-f_{3}x=b_{2}u$ for some $u\in B_2$. Write $u=f_{2}v$. Commutativity gives $y=f_3(x+a_{2}v)$.

The weaker one-sided hypotheses $f_1,f_4$ epic and $f_2,f_5$ monic do not suffice. Over a nonzero field $k$, use the exact rows $0\to 0\to k\stackrel{1}{\to }k\to 0$ and $0\to k\stackrel{1}{\to }k\to 0\to 0$, with every vertical map zero. The first and fourth are epimorphisms, and the second and fifth are monomorphisms, but the middle zero map $k\to k$ is not an isomorphism.

Let $C$ be a smooth projective curve over an algebraically closed field, let $p_0\in C$, and let $E$ be a vector bundle of finite positive rank. Then $E\otimes {\mathcal{O}}_C(m{p}_0)$ is globally generated for all sufficiently large integers $m$. This is the curve version of generation by sufficiently positive twists of an ample line bundle.^[ref]

For a uniform bound, take an upper degree bound for line subbundles on a smooth curve $B$ for $E^{\ast }\otimes K_C$. If $m-1>B$, a nonzero map ${\mathcal{O}}_C(m{p}_0-p)\to E^{\ast }\otimes K_C$ would have saturated image of degree at least $m-1$, a contradiction. Serre duality gives $H^1(E(m{p}_0-p))=0$ for every point $p$. The evaluation exact sequence then makes $H^0(E(m{p}_0))\to E(m{p}_0){|}_p$ surjective at every point.

On the projective line, $E=\mathcal{O}(a)$ gives the concrete bound $m+a\ge 0$: the degree-$m+a$ homogeneous polynomials generate the twisted line bundle. The chosen field hypothesis makes every closed point have degree one, as used in the bound $m-1$.

For an embedded projective variety $X\subseteq {\mathbb{P}}_k^n$, let $I(X)$ be its homogeneous vanishing ideal. Its homogeneous coordinate ring is the graded ringThe grading records polynomial degree, and $\mathrm{Proj}S(X)$ recovers the embedded projective variety. The affine scheme $\mathrm{Spec}S(X)$ is its affine cone.

The chosen homogeneous ideal matters: different graded quotients can have the same Proj construction. For the usual degree-one polynomial grading, saturating an ideal by the irrelevant ideal recovers the ideal determined by the associated closed closed subscheme. Thus an arbitrary presentation $X=V_{+}(I)$ should not silently identify $I$ with $I(X)$.^[ref]

The ring also depends on the projective embedding. A projective line in its standard embedding has homogeneous coordinate ring $k[s,t]$, while its degree-two Veronese map image has homogeneous coordinate ring $k[X,Y,Z]/(XZ-Y^2)$.

For two plane curves $C=V(f)$ and $D=V(g)$ meeting only at an isolated point $p$ locally on a smooth surface over an algebraically closed field $k$, their intersection multiplicity isIn local plane coordinates centered at $p$, this is also ${\dim }_{k}k[[x,y]]/(f,g)$. The quotient is finite-dimensional because the local intersection is isolated. This length counts contact that the underlying point set alone cannot detect.^[ref]

The coordinate axes have quotient $k[[x,y]]/(x,y)\cong k$ and multiplicity one. The curves $y=0$ and $y=x^m$, $m\ge 1$, instead givewith basis $1,x,\dots ,x^{m-1}$, hence multiplicity $m$. For $m\ge 2$ the higher value records tangency. If the curves share a local component, this quotient has infinite length and the isolated-intersection formula does not apply. Summing the local multiplicities for projective plane curves without a common component gives the product of degrees in Bézout's theorem.

Intersection theory assigns algebraic cycles and intersection multiplicities to intersections of algebraic varieties, and organizes these calculations modulo rational equivalence. On a smooth projective variety $X$, the intersection product makes its Chow groups into a graded ring:When representatives meet properly, one counts their common components with the appropriate intersection multiplicities. A moving argument extends this construction to arbitrary classes and makes it independent of the representatives.^[ref]

For projective plane curves of degrees $d$ and $e$ without a common component, Bézout theorem gives total intersection number $de$. The value counts a tangency more than once; it is not just the number of distinct points. The same formalism defines a self-intersection number even though a curve does not meet itself properly. For a line $L$ in ${\mathbb{P}}^2$, $L^2=1$ because another line represents the same Chow class and meets it once. By contrast, the exceptional divisor of the blowup of a smooth algebraic surface has square $-1$, measured by its normal bundle. These calculations connect divisors, Chern classes and birational geometry.

Let $C$ be a smooth integral algebraic curve and let $p$ be a closed point. Its local ring ${\mathcal{O}}_{C,p}$ is a one-dimensional regular local ring, hence a discrete valuation ring. If $\pi$ is a uniformizer, every nonzero rational function has a unique expressionwith $u$ a unit in ${\mathcal{O}}_{C,p}$. The integer $m$ is its order of vanishing at $p$: positive for a zero, negative for a pole, and zero for a local unit.^[ref]

On the affine line at $t=0$, the local ring is $k[t{]}_{(t)}$, the element $t$ is a uniformizer, and $t^3/(1+t)$ has order of vanishing three. At the generic point of $C$, the local ring is the function field $k(C)$, which has dimension zero; the closed-point hypothesis distinguishes these cases.

For a smooth projective curve $C$ of genus $g$ over an algebraically closed field $k$, its Picard scheme represents the sheafified Picard functor of a curve. It is a disjoint union of smooth proper schemes ${\mathrm{Pic}}_{C/k}^d$, indexed by the degree of a line bundle, each of dimension $g$. Its $k$-points are degree-$d$ line bundles up to isomorphism.^[ref]

The degree-zero component is the Jacobian variety. Its group action on every degree component, by tensor product of sheaves, is free and transitive in families: degree-$d$ components are torsors rather than canonically chosen copies of the Jacobian variety. A chosen degree-$d$ line bundle supplies such an identification by tensoring. Over a general field, the same geometric components exist for a smooth projective geometrically connected curve, but they need not have a rational point, so that identification cannot be assumed over the ground field.

The construction has concrete charts: on the locus of nonspecial degree-$g$ line bundles, Riemann-Roch theorem gives one independent global section, whose zero divisor recovers the bundle. Over the algebraically closed field, translates of this open locus cover the Picard scheme.^[ref]

A prime Weil divisor on a Noetherian integral scheme $X$ is an integral closed subscheme of codimension one. Its generic point $\eta$ has a one-dimensional local ring ${\mathcal{O}}_{X,\eta }$.

If $X$ is a normal variety, this local ring is a one-dimensional Noetherian normal domain, hence a discrete valuation ring. More generally it suffices that this local ring be regular. A uniformizer $\pi$ then writes every nonzero rational function as $u{\pi }^m$ with $u$ a unit, giving its integer order along the divisor. This is the valuation used in the principal Weil divisor construction.^[ref]

The codimension-one definition alone does not imply regularity. On the cusp $\mathrm{Spec}k[t^2,t^3]$, the closed point $(t^2,t^3)$ is a prime Weil divisor. Its local ring has a maximal ideal requiring two generators, as its quotient by the square has basis $t^2,t^3$. It is therefore not a discrete valuation ring. Passing to the normalization $k[t]$ supplies a different local ring with uniformizer $t$; normality is a substantive hypothesis, not part of the definition of every prime divisor.