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Chapter 2: Problem 13

Let \(S\) be a graded ring, generated by \(S_{1}\) as an \(S_{0}\) -algebra. For any integer \(d>0\) let \(S^{\text {(i) }}\) be the graded ring \(\oplus_{n \geqslant 0} S_{n}^{\text {(d) }}\) where \(S_{n}^{\text {d) }}=S_{n d}\). Let \(X=\) Proj \(S\). Show that Proj \(S^{(d)} \cong X,\) and that the sheaf \(O(1)\) on Proj \(S^{(d)}\) corresponds via this isomorphism to \(\mathcal{O}_{\boldsymbol{x}}(d)\) This construction is related to the \(d\) -uple embedding \((\mathrm{I}, \mathrm{Ex} .2 .12)\) in the fol lowing way. If \(x_{0}, \ldots, x_{r}\) is a set of generators for \(S_{1},\) corresponding to an embedding \(X\) \& \(P_{A}^{\prime},\) then the set of monomials of degree \(d\) in the \(x_{i}\) is a set of generators for \(S_{1}^{(t)}=S_{d} .\) These define a projective embedding of Proj \(S^{i d}\) which is none other than the image of \(X\) under the \(d\) -uple embedding of \(\mathbf{P}_{1}^{\prime}\).

Short Answer

Expert verified
The Proj of the degree-d extension of the graded ring \(S\) is isomorphic to the Proj of \(S\), with this isomorphism also maintaining a specific correspondence between the sheaf structures on these spaces, namely, \(\mathcal{O}(1)\) on Proj \(S^{(d)}\) corresponds to \(\mathcal{O}_{X}(d)\) via this isomorphism.

Step by step solution

01

Understanding Graded Rings and Projective Space

Firstly, acknowledge that a graded ring is a ring that is decomposed into a direct sum of additive groups, with the property that the product of any two elements of a fixed degree belongs to the group of that degree. The graded ring \(S\) defined here is generated by \(S_1\) as an \(S_0\) -algebra, meaning every element of the ring is expressible as a polynomial in elements of \(S_1\), with coefficients in \(S_0\). The Projective space denoted \(X\) is the set of all lines through the origin in the vector space \(S_0\). The Proj operator used for \(S\) and \(S^{(d)}\) assigns to these graded rings some object in projective space.
02

Defining Degree-d Extension and Understanding Embeddings

Next, clarify the degree-d version of \(S\), denoted \(S^{(d)}\), which is generated by \(S_{nd}\) for \(n>=0\). This is equivalent to taking the \(d^{th}\) finite Veronese subring of \(S\). The addition of \(d\) -uple embeddings, which is a map from projective space, \(X\), to itself, allows one to introduce a new graded ring \(S^{id}\), whose degree \(d\) monomials in \(x_{i}\) form a set of generators, hence providing a projective embedding of Proj \(S^{id}\) into \(X\).
03

Proving Isomorphism and Understanding Related Sheaf Structures

Finally, prove the isomorphism by showing that there is a well-defined, bijective map between Proj \(S\) and Proj \(S^{(d)}\), which respects the graded ring structures. Subsequently, draw a connection between the sheaf structures on these spaces, specifically that the structure sheaf \(O(1)\) on Proj \(S^{(d)}\) corresponds through the isomorphism to \(\mathcal{O}_{X}(d)\) on \(X\). One can verify this by examining what \(\mathcal{O}(1)\) does to the homogeneous elements of \(S^{(d)}\) and observing that it performs the same action as \(\mathcal{O}_{X}(d)\) does to the homogeneous elements of \(S\). Evidence of this correspondence can be traced back to the invariance of the multiplicative structures of \(S\) and \(S^{(d)}\) under the degree \(d\) Veronese map.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graded Rings
Graded rings are an important concept in algebra that can be visualized as a ring being divided into different layers, or 'grades.' Each element of a graded ring belongs to a specific layer depending on its degree. The fascinating aspect of graded rings is their structure.
  • They are decomposed into a direct sum of additive groups, typically denoted as \( S = S_0 \oplus S_1 \oplus S_2 \oplus \ldots \).
  • They display the property that the multiplication of two elements from these groups results in another element whose degree is the sum of the original degrees.
  • In many scenarios, every element of the ring can be expressed as a polynomial in generators of a particular degree, often \( S_1 \), with coefficients in \( S_0 \).

These features are particularly useful in various mathematical fields, such as algebraic geometry, where they facilitate the construction of projective varieties. By understanding how these elements interact, one can interpret and manipulate spaces in projective geometries.
Sheaf Theory
Sheaf theory provides a systematic way to track local data attached to the open sets of a topological space and find a consistent global representation. It's like piecing together a complex puzzle from smaller parts.
  • A sheaf assigns data, often in the form of functions, to every open set in a topological space while ensuring consistency across overlaps.
  • Sheaves can be considered as a generalization of functions because they allow for 'functions' that are defined only locally, but when pieced together, give a global picture.
  • In algebraic geometry, sheaves are pivotal because they enable the tracking of polynomial functions within projective varieties and help relate algebraic structures with geometric intuition.

In the context of proj construction, we frequently encounter structure sheaves \( \mathcal{O} \), which hold information about the functions on projective varieties. The related sheaf structures involved in the isomorphism between \( \text{Proj } S \) and \( \text{Proj } S^{(d)} \) allow for underlying algebraic relationships to be revealed and studied in depth.
Projective Embeddings
Projective embeddings are a way to represent algebraic varieties within projective space. This concept is central when dealing with proj constructions as it provides an intuitive embedding process.
  • Projective space is essentially a set of lines through the origin of a vector space, and a projective embedding maps an algebraic variety into this space.
  • An important aspect of projective embeddings is the degree of the embedding, which involves considering all combinations of functions up to a certain degree, allowing varieties to be viewed more simplistically in higher dimensions.
  • Through these embeddings, complex algebraic properties can be visualized, manipulated, and understood as geometric objects.

In practice, projective embeddings, such as the \( d \)-uple embedding involved in the proj construction, illustrate how algebraic structures can be systematically represented within projective space. This correspondence between algebraic objects and geometric representations provides not only visualization but insights into the interactions between different mathematical entities.

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Most popular questions from this chapter

The Grothendieck Group of a Nonsingular Curce. Let \(X\) be a nonsingular curve over an algebraically closed field \(k .\) We will show that \(K(X) \cong \operatorname{Pic} X \oplus \mathbf{Z},\) in several steps. (a) For any divisor \(D=\sum n_{i} P_{i}\) on \(X\), let \(\psi(D)=\sum n_{i \hat{i}}\) ' \(\left(k\left(P_{i}\right)\right) \in K(X),\) where \(k\left(P_{i}\right)\) is the skyscraper sheaf \(k\) at \(P_{t}\) and 0 elsewhere. If \(D\) is an effective divisor, let \(\mathrm{C}_{D}\) be the structure sheaf of the associated subscheme of codimension \(1,\) and show that \(\psi(D)=\dot{\gamma}\left(C_{D}\right) .\) Then use (6.18) to show that for any \(D, \psi(D)\) depends only on the linear equivalence class of \(D,\) so \(\psi\) defines a homomorphism \(\psi: \mathrm{Cl} X \rightarrow K(X)\) (b) For any coherent sheaf \(\mathscr{F}\) on \(X\), show that there exist locally free sheaves \(\delta_{0}\) and \(\mathscr{E}_{1}\) and an exact sequence \(0 \rightarrow \mathscr{E}_{1} \rightarrow \mathscr{B}_{0} \rightarrow \mathscr{H} \rightarrow 0 .\) Let \(r_{0}=\operatorname{rank} \delta_{0}\) \(r_{1}=\operatorname{rank} \delta_{1},\) and define det \(\tilde{\mathscr{H}}=\left(\bigwedge^{r_{0}} \mathscr{E}_{0}\right) \otimes\left(\bigwedge^{r_{1}} \delta_{1}\right)^{-1} \in \operatorname{Pic} X .\) Here \(\wedge \mathrm{de}\) notes the exterior power (Ex. 5.16). Show that det \(\mathscr{H}\) is independent of the resolution chosen, and that it gives a homomorphism det: \(K(X) \rightarrow\) Pic \(X\) Finally show that if \(D\) is a divisor, then \(\operatorname{det}(\psi(D))=\mathscr{L}(D)\) (c) If \(\mathscr{F}\) is any coherent sheaf of rank \(r,\) show that there is a divisor \(D\) on \(X\) and an exact sequence \(0 \rightarrow \mathscr{P}(D)^{\oplus r} \rightarrow \mathscr{H} \rightarrow \mathscr{I} \rightarrow 0\), where \(\mathscr{J}\) is a torsion sheaf. Con- clude that if \(\mathscr{F}\) is a sheaf of rank \(r,\) then \(\gamma(\mathscr{F})-r \gamma\left(\mathcal{O}_{X}\right) \in \operatorname{Im} \psi\) (d) Using the maps \(\psi,\) det, rank,and \(1 \mapsto \gamma\left(C_{X}\right)\) from \(\mathbf{Z} \rightarrow K(X),\) show that \(K(X) \cong\) Pic \(X \oplus \mathbf{Z}\)

The Ricmunn-Rocili Problem. Let \(\lambda\) be a nonsingular projective variety over an algebracally closed field, and let \(D\) be a divisor on \(X\). For any \(n>0\) we consider the complete linear system \(|n D| .\) Then the Riemann-Roch problem is to determine \(\operatorname{dim}|n D|\) as a function of \(n,\) and. in particular, its behavior for large \(n\). If \(\mathscr{L}\) is the corresponding invertible sheaf, then \(\operatorname{dim}|n D|=\operatorname{dim} \Gamma\left(X, \mathscr{L}^{n}\right)-1,\) so an equivalent problem is to determine \(\operatorname{dim} \Gamma\left(X, \mathscr{L}^{n}\right)\) as a function of \(n\) (a) Show that if \(D\) is very ample, and if \(X \subseteq P_{k}^{n}\) is the corresponding embedding in projective space, then for all \(n\) sufficiently large, \(\operatorname{dim}|n D|=P_{X}(n)-1\) polynomial function of \(n,\) for \(n\) large. (b) If \(D\) corresponds to a torsion element of Pic \(X\), of order \(r,\) then \(\operatorname{dim}|n D|=0\) if \(r | n,-1\) otherwise. In this case the function is periodic of period \(r\) It follows from the general Riemann- Roch theorem that \(\operatorname{dim}|n D|\) is a polynomial function for \(n\) large, whenever \(D\) is an ample divisor. See \((I V, 1.3 .2),(V, 1.6)\) and Appendix A. In the case of algebraic surfaces, Zariski [7] has shown for any effective divisor \(D\), that there is a finite set of polynomials \(P_{1}, \ldots, P_{r},\) such that for all \(n\) sufficiently large, \(\operatorname{dim}|n D|=P_{t(n)}(n),\) where \(i(n) \in\\{1,2, \ldots, r\\}\) is a function of \(n\)

Let \(\left(X, C_{X}\right)\) be a locally ringed space, and let \(f: \mathscr{L} \rightarrow . ll\) be a surjective map of invertible sheaves on \(X\). Show that \(f\) is an isomorphism. [Hint: Reduce to a question of modules over a local ring by looking at the stalks.]

A morphism \(f: X \rightarrow Y\) of schemes is quasi-compact if there is a cover of \(Y\) by open affines \(V_{i}\) such that \(f^{-1}\left(V_{i}\right)\) is quasi-compact for each \(i .\) Show that \(f\) is quasicompact if and only if for every open affine subset \(V \subseteq Y, f^{-1}(V)\) is quasi-compact.

(a) Let \(X\) be a scheme over a scheme \(Y\), and let \(\mathscr{L}, \mathscr{M}\) be two very ample invertible sheaves on \(X .\) Show that \(\mathscr{L} \otimes \mathscr{M}\) is also very ample. [Hint: Use a Segre embedding. \(]\) (b) Let \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\) be two morphisms of schemes. Let \(\mathscr{L}\) be a very ample invertible sheaf on \(X\) relative to \(Y\), and let \(\mathscr{M}\) be a very ample invertible sheaf on \(Y\) relative to \(Z\). Show that \(\mathscr{L} \otimes f^{*} . \mathscr{M}\) is a very ample invertible sheaf on \(X\) relative to \(Z\).
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