Closed subscheme

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.

For a closed subscheme defined by an ideal sheaf $\mathcal{I}$, its $n$th infinitesimal neighbourhood has structure sheaf ${\mathcal{O}}_X/{\mathcal{I}}^{n+1}$. All these neighbourhoods have the same underlying topological space because the radicals of $\mathcal{I}$ and its positive powers agree. Their inverse limit of structure sheaves is the formal completion of a scheme.

// Target: ringed-space.bigb

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.