Morphism of schemes

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

For 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.

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.

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 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.

Let $W$ be an integral effective Cartier divisor in an integral variety $V$ over a perfect field, and suppose $W$ is not contained in the singular locus of $V$. On a dense open subset both varieties are smooth. The Zariski tangent space of $W$ there has codimension one in that of $V$, so the differential of the local defining equation is nonzero. Thus the conormal sheaf map is injective generically. Its kernel is a subsheaf of a line bundle on the integral variety $W$ and is therefore a torsion-free sheaf; generic vanishing implies zero everywhere. Right exactness is the Conormal exact sequence for Kähler differentials.

For ring maps $A\to B\to C$, the first or transitivity exact sequence isThe maps send $db\otimes c$ to $cdb$ and then kill differentials of elements coming from $B$.

For a morphism of schemes $X\to Y$, the sheaf of relative Kähler differentials is obtained by sheafifying the modules ${\Omega }_{B/A}$ on affine charts $\mathrm{Spec}B\to \mathrm{Spec}A$.

For $n\ge 1$, the cotangent form of the Euler sequence is $0\to {\Omega }_{{\mathbb{P}}^n/k}^1\to \mathcal{O}(-1{)}^{\oplus (n+1)}\to \mathcal{O}\to 0$. The cohomology of twisting sheaves on projective space and the long exact sequence in sheaf cohomology give $H^1({\mathbb{P}}_k^n,{\Omega }^1)\cong k$, with every other cohomology group zero. Therefore the Euler characteristic of a coherent sheaf is $-1$. The nonzero class is the connecting image of the constant section $1$.

For a variety over $k$, the Module of Kähler differentials ${\Omega }_{A/k}$ on each affine chart gives a quasi-coherent sheaf. Compatibility with localization glues these into ${\Omega }_{X/k}^1$. A finite presentation of an algebra gives a finite presentation of a module for its differentials, so this is a coherent sheaf on a variety.

An invertible maximal-rank minor in the Jacobian matrix eliminates generators and gives a local surjection from ${\mathcal{O}}_X^n$ to the Kähler differential sheaf. For an integral smooth variety the map is an isomorphism at the generic point. Its kernel is a submodule of a free module over an integral domain, so it is a torsion-free module; vanishing generically forces it to be zero. This gives local free frames for differentials.

For a smooth $n$-dimensional variety $X$ over a field, its canonical line bundle is ${\omega }_X={\bigwedge }^n{\Omega }_{X/k}^1$. It is the bundle of top-degree algebraic differential forms. A nonzero rational section defines a canonical divisor of a smooth variety. This algebraic definition works in arbitrary characteristic and is distinct from restricting the analytic canonical-bundle definition to complex manifolds.

A canonical divisor $K_X$ on a smooth integral variety is the Cartier divisor of a nonzero rational top-degree differential form; ${\mathcal{O}}_X(K_X)\cong {\omega }_X$. Its linear-equivalence class is independent of the chosen form. The plurigenus is the dimension of the space of sections of a positive tensor power of this bundle.

For a smooth projective variety, its $m$th plurigenus is $P_m(X)=h^0(X,{\omega }_X^{\otimes m})$, for $m\ge 1$. These dimensions describe pluricanonical linear systems. Blowup invariance of plurigenera proves their invariance under a point blowup on a smooth surface.

For a point blowup of a smooth algebraic surface $\psi :\tilde{X}\to X$, where $X$ is a smooth projective surface over an algebraically closed field, multiplication by the exceptional section to power $m$ identifies $H^0(\tilde{X},m{\psi }^{\ast }{K}_X)$ with $H^0(\tilde{X},m{K}_{\tilde{X}})$. Every effective pluricanonical representative must contain $mE$, since after subtracting $j<m$ copies its intersection with $E$ is $-m+j<0$. The projection formula for sheaves and ${\psi }_{\ast }\mathcal{O}=\mathcal{O}$ identify the remaining sections with those on $X$. No effectivity of $K_X$ itself is assumed.

A morphism $f:X\to Y$ is affine when the inverse image of every affine open subscheme of $Y$ is affine. For a quasi-coherent sheaf $\mathcal{F}$ on $X$, one has $R^q{f}_{\ast }\mathcal{F}=0$ for $q>0$.

For an affine morphism $f:X\to Y$ and a quasi-coherent sheaf $\mathcal{F}$ on $X$, the direct image sheaf has the same sheaf cohomology on $Y$ as $\mathcal{F}$ on $X$. To prove this, take a flasque resolution $\mathcal{F}\to {\mathcal{I}}^{\bullet }$. Its direct images are flasque sheaves. On each affine open $V\subset Y$, the complex of sections of the direct images is $\Gamma (f^{-1}V,{\mathcal{I}}^{\bullet })$. By vanishing of quasi-coherent cohomology on an affine scheme, this augmented complex is exact. Checking on the affine basis proves that $f_{\ast }{\mathcal{I}}^{\bullet }$ resolves $f_{\ast }\mathcal{F}$. Its global sections are identical to those of the original resolution.

A morphism $f:X\to Y$ is flat at $x\in X$ when the local ring ${\mathcal{O}}_{X,x}$ is a flat module over ${\mathcal{O}}_{Y,f(x)}$. It is flat when it is flat at every point.

A finite-type morphism of schemes is quasi-finite if its fibres are finite discrete schemes. A proper quasi-finite morphism is a finite morphism. In particular, a proper finite-type morphism of projective varieties with no positive-dimensional fibres is finite; a positive-dimensional projective fibre would contain an integral curve.

A morphism is universally closed when every base change is a closed map on underlying topological spaces.

Glue two copies of ${\mathbb{P}}_k^1$ by the identity away from one chosen point. The resulting finite-type scheme is universally closed over $k$: after any base change, a closed subset meets each projective-line chart in a closed subset whose image is closed, and the total image is their finite union. The two copies of the chosen point cannot be separated, so the structure morphism is not separated and therefore is not proper.

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.

A morphism of schemes is separated when its diagonal is a closed immersion. This is a property of the morphism, distinct from the broader morphism or diagonal constructions.

A scheme over a base $S$ is separated when its structure morphism to $S$ is a separated morphism. An absolute scheme is usually called separated when it is separated over $\mathrm{Spec}\mathbb{Z}$.

An algebraic variety that is separated over its ground field. Two affine open charts have an affine intersection: their intersection is the inverse image of the closed diagonal inside the product of the two affine charts. This ensures a finite affine cover is acyclic for quasi-coherent sheaves.

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 $K$.

This is the uniqueness part of the valuative criterion.

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

If $f:X\to Y$ is a morphism of schemes and $(U_i)$ is an open cover of $Y$, then $f$ is proper exactly when every restriction $f^{-1}(U_i)\to U_i$ is proper. The three defining properties—finite type, separatedness, and universal closedness—can each be checked on an open cover of the target.

A finite-type morphism can have a proper fiber over every closed point without being proper. Properness controls compatible specialization in families, as expressed by the valuative criterion for properness, rather than only the individual closed fibers.

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 $M$.

For a proper morphism of Noetherian schemes and a coherent sheaf flat over the base, 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 line bundle $M$ on $C\times T$, with $C$ a smooth projective curve, this function is locally constant. Twist both $M$ and $M^{-1}$ by a large power of an ample line bundle on $C$. Near a chosen fiber their first sheaf cohomology groups vanish. The Riemann-Roch theorem and upper semicontinuity of their zeroth sheaf cohomology groups then bound the degree from above and below by the degree on that fiber. Connectedness of $T$ is necessary for a single constant across all fibers.

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 $\mathrm{Spec}R$.

This is the existence-and-uniqueness part of the valuative criterion.

A morphism $X\to S$ is projective when it factors as a closed immersion $X\hookrightarrow {\mathbb{P}}_S^n$ followed by the projection to $S$. Every projective morphism is proper.

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.

Projectivity is a property of the scheme over its specified base, expressed by its structural projective morphism.

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

If a nonnegatively graded ring $S$ is generated by $S_1$ as an $S_0$-algebra, the standard affine opens of Proj $D_{+}(f)$ with $f\in S_1$ cover $\mathrm{Proj}S$. Indeed a homogeneous prime containing every degree-one element would contain the irrelevant ideal of a graded ring. This condition makes all twisting sheaves on Proj invertible and makes graded sheafification compatible with tensor products of sheaves.

A graded module $M$ over a graded ring $S$ determines a quasi-coherent sheaf on the Proj construction $\mathrm{Proj}S$ by taking the degree-zero part of its localization on each standard affine open of Proj. Localization identifies these module sheaves on overlaps. The construction is exact, by exactness of localization and exactness of taking a fixed graded component.

Under the degree-one generation condition for Proj, cover $X=\mathrm{Proj}S$ by degree-one charts. The degree-one localization of a graded module identifies the two local modules in the displayed formula, using localization commutes with tensor products. The maps are natural and agree on overlaps, so they glue to an isomorphism of sheaves of modules. The analogous assertion can fail for general positively graded rings.

With the graded shift convention $S(n{)}_d=S_{n+d}$, this is the sheaf associated with a graded module $S(n)$. Under the degree-one generation condition for Proj, it is an invertible sheaf: on $D_{+}(f)$ with $\mathrm{deg}f=1$, multiplication by $f^n$ freely generates its local module for every integer $n$. Without that hypothesis, it need not be invertible. The twisting sheaf on projective space is the standard special case. Stacks Project, Section 27.10 records the hypotheses and local trivializations.

For a nonnegatively graded ring $S={⨁}_{d\ge 0}{S}_d$, the irrelevant ideal isThe points of $\mathrm{Proj}S$ are precisely the homogeneous prime ideals that do not contain $S_{+}$.

For a positive-degree homogeneous element $f\in S$, the standard openis the affine scheme $\mathrm{Spec}(S_f{)}_0$.

A graded homomorphism $S\to T$ induces a morphism to $\mathrm{Proj}S$ on the open subset of $\mathrm{Proj}T$ where the inverse image of a prime does not contain $S_{+}$. On $D_{+}(\phi (f))$ it is induced by $(S_f{)}_0\to (T_{\phi (f)}{)}_0$.

If a graded homomorphism $S\to T$ is an isomorphism in every sufficiently large degree, then it induces an isomorphismMultiplying degree-zero localized fractions by a sufficiently large power of the chart denominator reduces the claim to the assumed high-degree isomorphisms.

For a quasi-coherent graded ${\mathcal{O}}_X$-algebra $\mathcal{A}$, the relative Proj is obtained by gluing the schemes $\mathrm{Proj}\mathcal{A}(U)$ over affine open subsets $U\subseteq X$. If $\mathcal{A}$ is of finite type and generated in degree one, a surjection $\mathrm{Sym}(\mathcal{E})\twoheadrightarrow \mathcal{A}$ realizes ${\mathrm{Proj}}_X\mathcal{A}$ as a closed subscheme of the projective bundle ${\mathbb{P}}_X(\mathcal{E})$.

For a nonnegatively graded ring $S={⨁}_{m\ge 0}{S}_m$ and $d\ge 1$, its $d$th Veronese subring isgraded by $(S^{(d)}{)}_m=S_{dm}$. Passing from $S$ to $S^{(d)}$ does not change its projective scheme: the standard affine charts give a canonical isomorphism $\mathrm{Proj}S\cong \mathrm{Proj}{S}^{(d)}$.

The degree-$d$ Veronese embedding sends a projective point to all monomials of degree $d$ in its homogeneous coordinates. For example,is a closed immersion with image the conic $uw-v^2=0$.

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 nilpotent thickening is a closed immersion defined by a nilpotent ideal sheaf of a closed subscheme. Its underlying topological map is a homeomorphism, because every prime ideal contains every nilpotent element. For example, quotienting the dual numbers by $(\epsilon )$ changes the structure sheaf without changing the single point of the spectrum.

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.

A projective complete intersection is a closed subscheme of a projective space defined by a homogeneous regular sequence of positive degrees. A length-$r$ sequence in $k[t_0,\dots ,t_N]$ gives dimension $N-r$ when $r\le N$. Sheafifying its successive quotient sequences gives exact restriction sequences of twisting sheaves on projective space, allowing computation of its sheaf cohomology. It need not be smooth or reduced.

A projective complete intersection defined by a regular sequence in a polynomial ring has pure codimension equal to the sequence length and has no embedded associated primes. Thus its components account for the whole degree, with their generic multiplicities. This fact excludes hidden embedded points when a contained curve already has the full intersection degree.

For a length-$r$ homogeneous regular sequence of positive degrees in ${\mathbb{P}}_k^N$, with $r<N$, the global regular functions on its projective complete intersection are exactly $k$. In the successive restriction sequences, intermediate cohomology vanishing for a projective complete intersection makes the first-cohomology group on the preceding stage zero. Inductively all negative twists have zero global sections, so at twist zero the restriction of constants is an isomorphism. The positive-dimension assumption is essential: a zero-dimensional complete intersection can have more global functions, including nilpotents.

For a length-$r$ projective complete intersection $X_r\subseteq {\mathbb{P}}_k^N$, the groups $H^q(X_r,{\mathcal{O}}_{X_r}(m))$ vanish for every integer $m$ and $0<q<N-r$. Starting with cohomology of twisting sheaves on projective space, induct through $0\to {\mathcal{O}}_{X_{j-1}}(m-d_j)\to {\mathcal{O}}_{X_{j-1}}(m)\to {\mathcal{O}}_{X_j}(m)\to 0$. Both adjacent groups in the long exact sequence in sheaf cohomology vanish precisely in the indicated range.

A quasi-coherent sheaf of ideals $\mathcal{I}\subseteq {\mathcal{O}}_X$ gives a closed subscheme by gluing $\mathrm{Spec}(A/J)$ on affine open subschemes where $\mathcal{I}=\tilde{J}$. Compatibility follows from exactness of localization. On the closed set $Z=\{x:{\mathcal{I}}_x\ne {\mathcal{O}}_{X,x}\}$, the structure sheaf is $i^{-1}({\mathcal{O}}_X/\mathcal{I})$, where $i:Z\hookrightarrow X$ is the inclusion. Its pushforward is ${\mathcal{O}}_X/\mathcal{I}$.

For a closed immersion $i:Z\hookrightarrow X$, the ideal sheaf is the kernel ${\mathcal{I}}_{Z/X}=\ker ({\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_Z)$. It therefore fits into the short exact sequence of sheaves

For a closed point $P$ with residue field $k(P)$, evaluation gives a morphism from the structure sheaf onto the skyscraper sheaf $k_P$. Its kernel is the ideal sheaf of $P$. On a Noetherian scheme it is a coherent ideal sheaf.

For distinct closed points over an algebraically closed field, evaluation gives the short exact sequence $0\to {\mathcal{I}}_{\{P,Q\}}\to {\mathcal{O}}_X\to k_P\oplus k_Q\to 0$. If all global regular functions are constant, the long exact sequence in sheaf cohomology embeds $k^2/k(1,1)$ into $H^1(X,{\mathcal{I}}_{\{P,Q\}})$. Thus this ideal detects a concrete obstruction to affineness.

A finite locally free sheaf of ideals on a reduced scheme has rank at most one at every point. On a nonempty affine open subscheme where its rank is $r$, localize its inclusion into the structure sheaf at a minimal prime ideal. The resulting local ring is a field $K$, giving an injection $K^r\hookrightarrow K$, hence $r\le 1$. This works without a Noetherian assumption. The empty scheme is a vacuous exception to claims phrased as nonexistence of a sheaf of a prescribed rank.

If a homogeneous polynomial $f_d$ of degree $d$ cuts out a projective hypersurface $i:X\hookrightarrow {\mathbb{P}}^n$, multiplication by $f_d$ gives the short exact sequence

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 construction retains the scheme structure on the closed subscheme, not merely the set of points in an image.