Closed Subschemes of Projective Space

We now examine closed subschemes of \(\mathbb{P}^n=\mathbb{P}_\mathbb{K}^n\) as examples of closed subschemes. Although \(\mathbb{P}^n\) is slightly more complicated than affine schemes, it is still easier to handle than a general scheme, because by §Projective Space and the Proj Construction, ⁋Definition 4, any closed subset of \(\mathbb{P}^n\) can always be written as the zero set of homogeneous polynomials in \(\mathbb{K}[\x_0,\ldots, \x_n]\). In other words, although these homogeneous polynomials are not functions defined on \(\mathbb{P}^n\), at least when representing closed subsets we can use a method almost analogous to affine schemes.

In this post, we lift this correspondence to the level of schemes. That is, we see that a homogeneous ideal defines a closed subscheme of \(\mathbb{P}^n\), and conversely that any closed subscheme of \(\mathbb{P}^n\) is obtained in this way. Throughout this post, \(A_\bullet=\mathbb{K}[\x_0,\ldots,\x_n]\) is the graded ring equipped with the standard grading, and \(\mathbb{P}^n=\Proj A_\bullet\).

Construction of \(V_+(\mathfrak{a})\)

We already know that \(\Proj\) is not a functor. This was because for a graded ring homomorphism \(\phi_\bullet:A_\bullet \rightarrow B_\bullet\) and, with \(B_+\) not contained in a homogeneous prime ideal \(\mathfrak{q}\), its inverse image \(\phi^{-1}(\mathfrak{q})\) may contain \(A_+\), so that even if \(\mathfrak{q}\) is a point of \(\Proj B_\bullet\), \(\phi^{-1}(\mathfrak{q})\) need not be a point of \(\Proj A_\bullet\). However, if \(\phi_\bullet\) is a surjection, the situation changes. In this case, since \(\phi(A_+)=B_+\), \(\phi^{-1}(\mathfrak{q})\) containing \(A_+\) means that \(\mathfrak{q}\) contains \(B_+\), and thus \(\mathfrak{q}\) would not have been a point of \(\Proj B_\bullet\) to begin with, allowing us to avoid the problem. On the other hand, in affine schemes a surjective ring homomorphism corresponded precisely to a closed subscheme (the discussion after §Closed Subschemes, ⁋Proposition 3), and the same holds for projective space.

Proposition 1 Let a homogeneous ideal \(\mathfrak{a}\subseteq A_\bullet\) and the canonical projection \(\pi:A_\bullet \rightarrow A_\bullet/\mathfrak{a}\) be given. Then \(\pi\) induces a closed embedding

\[\iota:\Proj (A_\bullet/\mathfrak{a}) \rightarrow \Proj A_\bullet=\mathbb{P}^n\]

whose image is \(Z_+(\mathfrak{a})\).

Proof

As we saw in [Algebraic Structures] §Graded Rings, \(A_\bullet/\mathfrak{a}\) is a graded ring, and the ideal correspondence for quotient rings sends homogeneous ideals to homogeneous ideals. In addition, since \(\pi\) is surjective, \(\pi(A_+)=(A_\bullet/\mathfrak{a})_+\), and thus \(\mathfrak{q}\) not containing \((A_\bullet/\mathfrak{a})_+\) is equivalent to \(\pi^{-1}(\mathfrak{q})\) not containing \(A_+\). In particular, \(\mathfrak{q}\mapsto \pi^{-1}(\mathfrak{q})\) is a bijection from the points of \(\Proj(A_\bullet/\mathfrak{a})\), i.e., homogeneous prime ideals not containing \((A_\bullet/\mathfrak{a})_+\), to the homogeneous prime ideals containing \(\mathfrak{a}\) but not \(A_+\) in \(A_\bullet\), and the latter is precisely \(Z_+(\mathfrak{a})\). (§Projective Space and the Proj Construction, ⁋Definition 2) Also, under this correspondence, a closed set of the form \(Z_+(\bar{\mathfrak{b}})\) corresponds to \(Z_+(\pi^{-1}(\bar{\mathfrak{b}}))\cap Z_+(\mathfrak{a})\), so \(\iota\) is a homeomorphism onto its image \(Z_+(\mathfrak{a})\).

Now let us verify the scheme morphism structure and the surjectivity of the sheaf morphism on the standard affine cover. For each \(i\), letting \(\bar{\x}_i=\pi(\x_i)\), we have \(\iota^{-1}(D_+(\x_i))=D_+(\bar{\x}_i)\) by the above correspondence. Under the identification of the scheme structures given to \(\Proj\) by §Projective Space and the Proj Construction, ⁋Theorem 10, namely \(D_+(\x_i)\cong\Spec A_{(\x_i)}\) and \(D_+(\bar{\x}_i)\cong \Spec (A_\bullet/\mathfrak{a})_{(\bar{\x}_i)}\), the restriction of \(\iota\) is the morphism between affine schemes induced by the ring homomorphism

\[A_{(\x_i)} \rightarrow (A_\bullet/\mathfrak{a})_{(\bar{\x}_i)};\qquad \frac{f}{\x_i^d}\mapsto \frac{\pi(f)}{\bar{\x}_i^d}\]

Since \(\pi\) is surjective, this ring homomorphism is also surjective, and by the exactness of localization, its kernel is

\[\mathfrak{a}_{(\x_i)}=\left\{\frac{a}{\x_i^d}\middle\vert\text{$a\in\mathfrak{a}$ homogeneous of degree $d$}\right\}\]

That is, on each chart, \(\iota\) is a closed embedding of the form \(\Spec\bigl(A_{(\x_i)}/\mathfrak{a}_{(\x_i)}\bigr) \rightarrow \Spec A_{(\x_i)}\), and these are compatible on the \(D_+(\x_i\x_j)\). Indeed, under the identification of §Projective Space and the Proj Construction, ⁋Lemma 9, \(D_+(\x_i\x_j)\) is \(\Spec A_{(\x_i\x_j)}\), and localizing the above ring homomorphism further to \(D_+(\x_i\x_j)\), the homomorphism \(A_{(\x_i\x_j)} \rightarrow (A_\bullet/\mathfrak{a})_{(\bar{\x}_i\bar{\x}_j)}\) is given by \(f/(\x_i\x_j)^d\mapsto \pi(f)/(\bar{\x}_i\bar{\x}_j)^d\), which is symmetric with respect to the roles of \(i\) and \(j\). Hence they glue to a single scheme morphism \(\iota\). On the other hand, as the proof of §Closed Subschemes, ⁋Proposition 3 shows, it suffices to check being a closed embedding on a single affine open cover, so \(\iota\) is a closed embedding.

We denote the closed subscheme thus obtained by \(V_+(\mathfrak{a})\). That is, \(V_+(\mathfrak{a})\) is \(Z_+(\mathfrak{a})\) as a topological space, and \(\Proj(A_\bullet/\mathfrak{a})\) as a scheme. As the notation suggests, this is the projective version of the correspondence \(\mathfrak{a}\mapsto \Spec(B/\mathfrak{a})\) in affine schemes.

Example 2 For a nonzero homogeneous polynomial \(f\) of positive degree and the ideal \((f)\) generated by it, we call \(V_+(f)=\Proj\bigl(\mathbb{K}[\x_0,\ldots,\x_n]/(f)\bigr)\) a hypersurface of degree \(\deg f\). If \(f=0\) then \(V_+(0)=\mathbb{P}^n\), and if \(f\) is a nonzero constant then \(V_+(f)=\emptyset\), so this condition is necessary. For example, in \(\mathbb{P}^2\), \(V_+(\x_0\x_2-\x_1^2)\) is a conic.

On the other hand, comparing the two closed subschemes of \(\mathbb{P}^2\), \(V_+(\x_0)\) and \(V_+(\x_0^2)\), their underlying spaces are the same as \(Z_+(\x_0)=Z_+(\x_0^2)\), but their scheme structures differ. Indeed, on the chart \(D_+(\x_2)\cong\Spec\mathbb{K}[\x_0/\x_2,\x_1/\x_2]\), the former is given by the ideal \((\x_0/\x_2)\) and the latter by \((\x_0^2/\x_2^2)\), and the coordinate ring of the latter has nilpotent elements. This is the same kind of non-reduced thickening as the double point examined in §Closed Subschemes, ⁋Example 1. However, since this thickening is supported not on a point but on the line \(V_+(\x_0)\cong \mathbb{P}^1\), this is not a double point but a double line.

Homogeneous ideal of a closed subscheme

We now show conversely that any closed subscheme of \(\mathbb{P}^n\) is of the form \(V_+(\mathfrak{a})\).

Theorem 3 In \(\mathbb{P}^n\), for any closed subscheme \(Z\), there exists a homogeneous ideal \(\mathfrak{a}\subseteq A_\bullet\) such that \(Z=V_+(\mathfrak{a})\).

Proof

For each \(i\), \(Z\cap D_+(\x_i)\) is a closed subscheme of the affine scheme \(D_+(\x_i)\cong\Spec A_{(\x_i)}\), so by the discussion after §Closed Subschemes, ⁋Proposition 3 it corresponds to a unique ideal \(\mathfrak{b}_i\subseteq A_{(\x_i)}\). Now consider the set of homogeneous elements

\[T=\left\{f\in A_\bullet\middle\vert\text{$f$ homogeneous,}\quad \frac{f}{\x_i^{\deg f}}\in \mathfrak{b}_i\text{ for all }i\right\}\]

and let \(\mathfrak{a}\) be the ideal generated by \(T\). Since \(T\) consists of homogeneous elements, \(\mathfrak{a}\) is a homogeneous ideal. Our claim is that for all \(i\), \(\mathfrak{a}_{(\x_i)}=\mathfrak{b}_i\) holds.

First, \(\mathfrak{a}_{(\x_i)}\) is generated by elements of the form \(f/\x_i^{\deg f}\) (\(f\in T\)) as an ideal of \(A_{(\x_i)}\), and by the definition of \(T\) all these generators belong to \(\mathfrak{b}_i\), so \(\mathfrak{a}_{(\x_i)}\subseteq \mathfrak{b}_i\).

Conversely, let \(g=f/\x_i^d\in \mathfrak{b}_i\) be given, where \(f\) is a homogeneous polynomial of degree \(d\). If we show that for sufficiently large \(N\) we have \(\x_i^Nf\in T\), then \(g=(\x_i^Nf)/\x_i^{N+d}\in\mathfrak{a}_{(\x_i)}\) and the proof is complete. To this end, for each \(j\), so that \((\x_i^Nf)/\x_j^{N+d}\in \mathfrak{b}_j\), let us find an \(N\). The case \(j=i\) holds for any \(N\), so let \(j\neq i\).

The key point is that \(\mathfrak{b}_i\) and \(\mathfrak{b}_j\) are compatible on the intersection \(D_+(\x_i\x_j)\). If we consider the ideal sheaf of \(Z\), \(\mathcal{I}_{Z/\mathbb{P}^n}=\ker\iota^\sharp\) (§Closed Subschemes, ⁋Definition 5), the kernel is computed on sections, so \(\mathcal{I}_{Z/\mathbb{P}^n}(D_+(\x_i))=\mathfrak{b}_i\). On the other hand, by §Projective Space and the Proj Construction, ⁋Lemma 9, \(D_+(\x_i\x_j)\) is, in \(\Spec A_{(\x_i)}\), the principal open subset \(D(\x_j/\x_i)\), and since the sections of the structure sheaf and \(\iota_\ast\mathcal{O}_Z\) on an affine scheme are all given by localization, by the exactness of localization we have

\[\mathcal{I}_{Z/\mathbb{P}^n}(D_+(\x_i\x_j))=(\mathfrak{b}_i)_{\x_j/\x_i}=(\mathfrak{b}_j)_{\x_i/\x_j}\]

Here both sides are ideals of \(A_{(\x_i\x_j)}\).

Now the image of \(g\in \mathfrak{b}_i\) in \(A_{(\x_i\x_j)}\) belongs to \((\mathfrak{b}_j)_{\x_i/\x_j}\). On the other hand, for \(h=f/\x_j^d\in A_{(\x_j)}\), in \(A_{(\x_i\x_j)}\) we have \(h=(\x_i/\x_j)^d g\), so the image of \(h\) also belongs to \((\mathfrak{b}_j)_{\x_i/\x_j}\). That is, there exists \(m_j\geq 0\) such that

\[\left(\frac{\x_i}{\x_j}\right)^{m_j}h=\frac{\x_i^{m_j}f}{\x_j^{m_j+d}}\in \mathfrak{b}_j\]

Then setting \(N=\max_{j\neq i}m_j\), since \(\mathfrak{b}_j\) is an ideal, for all \(j\) we have \((\x_i^Nf)/\x_j^{N+d}=(\x_i/\x_j)^{N-m_j}\cdot(\x_i^{m_j}f)/\x_j^{m_j+d}\in \mathfrak{b}_j\), and therefore \(\x_i^Nf\in T\).

In summary, \(Z\) and \(V_+(\mathfrak{a})\) have the same ideal sheaf as closed subschemes of \(\mathbb{P}^n\), so applying §Closed Subschemes, ⁋Lemma 9 in both directions yields closed embeddings factoring through each other. The proof realizes this embedding on each affine open subset \(U=\Spec A\) as a morphism between the quotients given by the sections of the two ideal sheaves, and since the two ideals are equal in our case, this is the identity morphism; hence the obtained closed embedding is an isomorphism, and therefore \(Z=V_+(\mathfrak{a})\).

Unlike the affine case, this correspondence is not one-to-one. For example, for any homogeneous ideal \(\mathfrak{a}\) and \(N\geq 1\), \(\mathfrak{a}\) and \(\mathfrak{a}A_+^N\) always define the same closed subscheme. Indeed, \(V_+(\mathfrak{a})\) is determined on each chart \(D_+(\x_i)\) by the ideal \(\mathfrak{a}_{(\x_i)}\) computed in the proof of Proposition 1, and

\[\mathfrak{a}A_+^N\subseteq \mathfrak{a}\]

is obvious, so \((\mathfrak{a}A_+^N)_{(\x_i)}\subseteq \mathfrak{a}_{(\x_i)}\). Conversely, an element of \(\mathfrak{a}_{(\x_i)}\) is, for a degree \(d\) homogeneous element \(f\in \mathfrak{a}\), of the form \(f/\x_i^d\), and writing this as \(\x_i^Nf/\x_i^{N+d}\) shows that this element belongs to \((\mathfrak{a}A_+^N)_{(\x_i)}\). That is, for all \(i\), we have \(\mathfrak{a}_{(\x_i)}=(\mathfrak{a}A_+^N)_{(\x_i)}\), and from this we see that the above correspondence is not one-to-one.

Saturation

The problem revealed in the above computation is that when computing the ideal on a chart, powers of \(\x_i\) are absorbed into the denominator, so after multiplying by a sufficiently high power of \(\x_i\), any element entering \(\mathfrak{a}\) gives the same information as an element of \(\mathfrak{a}\) itself. In other words, the data that each element of \(\mathfrak{a}\) gives on a chart is already known from the higher-degree element obtained by multiplying that element by a power of \(\x_i\), so in fact if two homogeneous ideals agree in sufficiently large degree, they define the same closed subscheme.

Read conversely, even if a degree \(d\) homogeneous element \(f\) does not belong to \(\mathfrak{a}\), as long as for some \(N\) we have \(\x_i^Nf\in \mathfrak{a}\), then \(f/\x_i^d=\x_i^Nf/\x_i^{N+d}\) already belongs to \(\mathfrak{a}_{(\x_i)}\). Such an \(f\) is information already contained by \(\mathfrak{a}\) on the chart, so even if for all \(i\) we collect all such \(f\) to enlarge the ideal, each \(\mathfrak{a}_{(\x_i)}\) remains unchanged, and hence the closed subscheme remains unchanged as well.

Definition 4 For a homogeneous ideal \(\mathfrak{a}\subseteq A_\bullet\), the collection of elements such that for each \(i\), \(\x_i^Nf\in \mathfrak{a}\) for some \(N\geq 0\), where \(f\in A_\bullet\), is called the saturation of \(\mathfrak{a}\) and is denoted \(\mathfrak{a}^\sat\). A homogeneous ideal with \(\mathfrak{a}=\mathfrak{a}^\sat\) is called saturated.

Then by definition, it is obvious that \(\mathfrak{a}^\sat\) is an ideal containing \(\mathfrak{a}\). Also, since \(\mathfrak{a}\) is homogeneous, \(\x_i^Nf\in \mathfrak{a}\) holds for each homogeneous component of \(f\), and therefore \(\mathfrak{a}^\sat\) is also a homogeneous ideal.

Proposition 5 For homogeneous ideals \(\mathfrak{a},\mathfrak{b}\subseteq A_\bullet\), the following hold.

1. \(V_+(\mathfrak{a})=V_+(\mathfrak{a}^\sat)\).
2. \(V_+(\mathfrak{a})=V_+(\mathfrak{b})\) if and only if \(\mathfrak{a}^\sat=\mathfrak{b}^\sat\).

In particular, closed subschemes of \(\mathbb{P}^n\) are in one-to-one correspondence with saturated homogeneous ideals.

Proof

As we saw in the proof of Proposition 1, \(V_+(\mathfrak{a})\cap D_+(\x_i)\) is determined by the ideal \(\mathfrak{a}_{(\x_i)}\subseteq A_{(\x_i)}\), and since closed subschemes of an affine scheme correspond one-to-one with ideals (the discussion following §Closed Subschemes, ⁋Proposition 3), \(V_+(\mathfrak{a})=V_+(\mathfrak{b})\) is equivalent to having, for all \(i\), \(\mathfrak{a}_{(\x_i)}=\mathfrak{b}_{(\x_i)}\).

1. Since \(\mathfrak{a}\subseteq \mathfrak{a}^\sat\), we have \(\mathfrak{a}_{(\x_i)}\subseteq (\mathfrak{a}^\sat)_{(\x_i)}\). Conversely, if \(f\in \mathfrak{a}^\sat\) is a homogeneous element of degree \(d\) and \(\x_i^Nf\in \mathfrak{a}\), then

   \[\frac{f}{\x_i^d}=\frac{\x_i^Nf}{\x_i^{N+d}}\in \mathfrak{a}_{(\x_i)}\]

   so the reverse inclusion also holds.

2. One direction follows from (1). Conversely, suppose \(V_+(\mathfrak{a})=V_+(\mathfrak{b})\) and let \(f\in \mathfrak{a}^\sat\) be a homogeneous element of degree \(d\). By the computation in (1), \(f/\x_i^d\in \mathfrak{a}_{(\x_i)}=\mathfrak{b}_{(\x_i)}\), so there exists a degree \(e\) homogeneous element \(g\in \mathfrak{b}\) such that in \(A_{(\x_i)}\), \(f/\x_i^d=g/\x_i^e\). Since \(A_\bullet\) is an integral domain, \(A_\bullet \rightarrow A_{\x_i}\) is injective, so this means \(\x_i^ef=\x_i^dg\in \mathfrak{b}\). Since \(i\) was arbitrary, \(f\in \mathfrak{b}^\sat\), and checking homogeneous components yields \(\mathfrak{a}^\sat\subseteq \mathfrak{b}^\sat\). Exchanging the roles of \(\mathfrak{a}\) and \(\mathfrak{b}\) gives the reverse inclusion.

The final claim holds because by Theorem 3, any closed subscheme is of the form \(V_+(\mathfrak{a})\), and applying (2) to \(V_+(\mathfrak{a})=V_+(\mathfrak{a}^\sat)\) given by (1) yields \((\mathfrak{a}^\sat)^\sat=\mathfrak{a}^\sat\), that is, \(\mathfrak{a}^\sat\) is saturated, so this is obtained from a saturated ideal, and (2) also gives the uniqueness of such an ideal.

The second result of Proposition 5 tells us that \(\mathfrak{a}^\sat\) is the largest homogeneous ideal defining \(V_+(\mathfrak{a})\). This is because if \(V_+(\mathfrak{b})=V_+(\mathfrak{a})\), then \(\mathfrak{b}\subseteq \mathfrak{b}^\sat=\mathfrak{a}^\sat\). That is, saturation is the operation of choosing a canonical representative among the ideals defining the same closed subscheme, and since in \(\x_i^Nf\in \mathfrak{a}\), \(N\) can be made arbitrarily large, this representative is determined by the sufficiently large degree part of \(\mathfrak{a}\) alone. Thus, as mentioned at the beginning of this section, if two homogeneous ideals agree in sufficiently large degree, their saturations are equal, and therefore they define the same closed subscheme. On the other hand, from this perspective, the ideal constructed in the proof of Theorem 3 is already saturated, because \(T\) in that proof was the collection of elements such that for all \(i\), \(f/\x_i^{\deg f}\in \mathfrak{b}_i\) for homogeneous elements \(f\), and the above computation shows that these are precisely the homogeneous elements of \(\mathfrak{a}^\sat\).

Example 6 In \(\mathbb{P}^1\), consider \(\mathfrak{a}=(\x_0^2,\x_0\x_1)\). On \(D_+(\x_1)\), the ideal \(\mathfrak{a}_{(\x_1)}\) is generated by \(\x_0^2/\x_1^2\) and \(\x_0\x_1/\x_1^2=\x_0/\x_1\), so it is \((\x_0/\x_1)\), while on \(D_+(\x_0)\) it contains \(\x_0^2/\x_0^2=1\) and hence is all of \(A_{(\x_0)}\). Therefore \(V_+(\mathfrak{a})\) is the reduced closed subscheme consisting of the single point \([0:1]\), namely \(V_+(\x_0)\).

Reading this via saturation, since \(\x_0^N\cdot \x_0\in (\x_0^2)\) and \(\x_1^N\cdot \x_0\in (\x_0\x_1)\), we have \(\x_0\in \mathfrak{a}^\sat\). Conversely, if \(f\in \mathfrak{a}^\sat\), then \(\x_1^Nf\in \mathfrak{a}\subseteq (\x_0)\), and since \((\x_0)\) is a prime ideal and \(\x_1\notin (\x_0)\), we have \(f\in (\x_0)\). Thus \(\mathfrak{a}^\sat=(\x_0)\), and \(\mathfrak{a}\) defines this point but is not saturated.

In the language of cones, what \(\mathfrak{a}\) cuts out in \(\mathbb{A}^2\) is the line \(\x_0=0\) with an embedded point at the origin (§Algebraic Structure of Schemes, ⁋Example 11), and saturation erases this component leaving only the line. The origin does not appear in any chart of \(\mathbb{P}^1\), so this difference is invisible in \(V_+(\mathfrak{a})\).

Projective scheme

In [Algebraic Varieties] §Projective Varieties, ⁋Definition 3, we called the closed subsets of \(\mathbb{P}^n\) cut out by homogeneous polynomials projective algebraic sets, and the irreducible ones among them projective varieties. The results of this post allow us to lift this definition to the language of schemes.

Definition 7 Over a field \(\mathbb{K}\), a scheme \(X\) is called a projective scheme if there exist \(n\geq 0\) and a closed embedding \(X \rightarrow \mathbb{P}^n_\mathbb{K}\). (§Closed Subschemes, ⁋Definition 2)

By Theorem 3 and Proposition 5, up to isomorphism, projective schemes over \(\mathbb{K}\) are precisely, for saturated homogeneous ideals \(\mathfrak{a}\subseteq \mathbb{K}[\x_0,\ldots,\x_n]\), the schemes \(\Proj\bigl(\mathbb{K}[\x_0,\ldots,\x_n]/\mathfrak{a}\bigr)\). Conversely, for an algebra generated by elements of degree \(1\), namely a finitely generated graded \(\mathbb{K}\)-algebra, a projective scheme is its \(\Proj\). Indeed, for such an algebra \(B_\bullet\), choosing from its degree \(1\) part \(n+1\) generators yields a surjection \(\mathbb{K}[\x_0,\ldots,\x_n] \rightarrow B_\bullet\), and Proposition 1 carries this to a closed embedding \(\Proj B_\bullet \rightarrow \mathbb{P}^n_\mathbb{K}\). On the other hand, Definition 7 removes irreducibility from the classical definition, and when \(\mathbb{K}\) is algebraically closed, what corresponds to a projective variety is an integral projective scheme. (§Algebraic Structure of Schemes, ⁋Definition 1)

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References

[Har] R. Hartshorne, Algebraic geometry. Graduate texts in mathematics. Springer, 1977.
[Vak] R. Vakil, The rising sea: Foundation of algebraic geometry. Available online.

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