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Chapter 2: Problem 12

Let \(X\) be a complete nonsingular curve. Show that there is a unique way to define the degree of any coherent sheaf on \(X, \operatorname{deg} \mathscr{F} \in \mathbf{Z},\) such that: (1) If \(D\) is a divisor, \(\operatorname{deg} \mathscr{L}(D)=\operatorname{deg} D\) (2) If \(\mathscr{H}\) is a torsion sheaf (meaning a sheaf whose stalk at the generic point is zero), then \(\operatorname{deg} \mathscr{H}=\sum_{P \in X}\) length \(\left(\mathscr{H}_{P}\right) ;\) and (3) If \(0 \rightarrow \mathscr{H}^{\prime} \rightarrow \mathscr{H} \rightarrow \mathscr{H}^{\prime \prime} \rightarrow 0\) is an exact sequence, then deg \(\mathscr{H}=\operatorname{deg} \mathscr{H}+\) \(\operatorname{deg} \mathscr{H}^{\prime \prime}\)

Short Answer

Expert verified
Therefore, the degree of a coherent sheaf on a complete nonsingular curve \(X\) is unique and well-defined, satisfying all three specified properties: relation to the degree of an associated divisor for line bundles, relation to local stalks for torsion sheaves, and additivity in exact sequences.

Step by step solution

01

Show degree of line bundle associated to divisor

Start by verifying property (1). If \(D\) is a divisor on \(X\), then there is an associated line bundle \(\mathscr{L}(D)\) on \(X\). We need to show that the degree of this line bundle, denoted as \(\operatorname{deg} \mathscr{L}(D)\), is equal to the degree of the divisor \(D\), denoted as \(\operatorname{deg} D\). This serves to establish the degree for a basic kind of coherent sheaf, the line bundle.
02

Show degree of torsion sheaf

Then verify property (2). If \(\mathscr{H}\) is a torsion sheaf, its stalk at the generic point is zero. We need to show that the degree of this sheaf, \(\operatorname{deg} \mathscr{H}\), is equal to the sum of the lengths of the stalks of \(\mathscr{H}\) at all points \(P\) in \(X\). This property relates the degree of a sheaf to local data on \(X\), specifically the stalks of the sheaf.
03

Show degree of exact sequence of sheaves

Lastly, verify property (3). Given an exact sequence of sheaves \(0 \rightarrow \mathscr{H}^{\prime} \rightarrow \mathscr{H} \rightarrow\mathscr{H}^{\prime \prime} \rightarrow 0\), the degree of \(\mathscr{H}\), \(\operatorname{deg} \mathscr{H}\), is equal to the sum of the degrees of \(\mathscr{H}^{\prime}\) and \(\mathscr{H}^{\prime \prime}\). This property ensures that the degree is well-behaved under exact sequences, which are fundamental in the study of coherent sheaves.
04

Conclude uniqueness

If these properties are fulfilled, then the degree of a coherent sheaf on \(X\) is uniquely determined. That's because any coherent sheaf on \(X\) can be built up from line bundles and torsion sheaves using extensions, and these properties uniquely determine the degree on these constituent parts and ensure that the degree behaves well under extensions.

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Key Concepts

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

Complete Nonsingular Curve
A complete nonsingular curve is an essential concept in algebraic geometry. Think of it as a curve that is "smooth" and without any "holes." This means it behaves nicely mathematically, without any singularities or places where the function defining the curve becomes undefined.

These curves live inside a projective space, which means they're "complete," and all points at infinity are included.

Understanding complete nonsingular curves is crucial because they form a solid foundation for studying more complex objects like sheaves and divisors. Their well-behaved nature simplifies dealing with sheaves on them.
Divisor
A divisor on a curve is somewhat like counting intersecting points. It's a formal sum of points on the curve, often written as \( D = a_1P_1 + a_2P_2 + \ldots + a_nP_n \), where each \( P_i \) is a point on the curve and each \( a_i \) is an integer.

Divisors help in expressing things like zeroes and poles of rational functions. More technically, they are used to describe line bundles, which are crucial in defining coherent sheaves.

Calculating the degree of a divisor involves summing up the integers \( a_i \). This degree tells us about the 'size' or 'impact' of the divisor on the curve. For example, if a divisor represents zeros of a function, its degree tells you how many zeros are counted, accounting for any multiplicities.
Torsion Sheaf
A torsion sheaf is a specific type of sheaf that vanishes at almost every point. It is usually zero at a "generic" or general point on the curve, meaning it lives more in the "special" places.

In simple terms, a torsion sheaf focuses on smaller parts of the curve, and it's often described in terms of its "stalks," which are like zoomed-in pictures at particular points.

The degree of a torsion sheaf relates to the lengths of these stalks. Summing these lengths across the curve gives us the "degree" of the sheaf, providing a way to measure its size or influence on the curve, focusing heavily on specific points.
Exact Sequence
An exact sequence is a central concept in studying sheaves. It's a chain of maps between sheaves that work in such a way that the image of one map is exactly the kernel of the next.

An exact sequence \( 0 \rightarrow \mathscr{H'} \rightarrow \mathscr{H} \rightarrow \mathscr{H''} \rightarrow 0 \) helps understand how sheaves decompose or fit together. This sequence tells you that \(\mathscr{H}\) is "built from" \(\mathscr{H'}\) and \(\mathscr{H''}\).

When dealing with the degree of coherent sheaves, exact sequences ensure that the degree adds up nicely: the degree of \(\mathscr{H}\) equals the sum of the degrees of \(\mathscr{H'}\) and \(\mathscr{H''}\). This property is vital for consistent calculations and supports the uniqueness of the degree for coherent sheaves.

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Most popular questions from this chapter

Let \(S\) and \(T\) be two graded rings with \(S_{0}=T_{0}=A .\) We define the Cartesian product \(S \times_{A} T\) to be the graded ring \(\bigoplus_{d \geqslant 0} S_{d} \otimes_{A} T_{d} .\) If \(X=\) Proj \(S\) and \(Y=\operatorname{Proj} T,\) show that \(\operatorname{Proj}\left(S \times_{A} T\right) \cong X \times_{A} Y,\) and show that the sheaf \(\mathcal{O}(1)\) on \(\operatorname{Proj}\left(S \times_{A} T\right)\) is isomorphic to the sheaf \(p_{1}^{*}\left(\mathcal{O}_{X}(1)\right) \otimes p_{2}^{*}\left(\mathcal{O}_{Y}(1)\right)\) on \(X \times Y\) The Cartesian product of rings is related to the Segre embedding of projective spaces \((\mathrm{I}, \mathrm{Ex} .2 .14)\) in the following way. If \(x_{0}, \ldots, x_{r}\) is a set of generators for \(S_{1}\) over \(A,\) corresponding to a projective embedding \(X \subseteq \mathbf{P}_{A}^{r},\) and if \(y_{0}, \ldots, y_{s}\) is a set of generators for \(T_{1},\) corresponding to a projective embedding \(Y \subseteq \mathbf{P}_{A}^{s}\) then \(\left\\{x_{i} \otimes y_{j}\right\\}\) is a set of generators for \(\left(S \times_{A} T\right)_{1},\) and hence defines a projective embedding \(\operatorname{Proj}\left(S \times_{\text {A }} T\right)\) c \(\mathbf{P}_{\text {A. }}^{\mathrm{N}}\) with \(N=r \mathrm{s}+r+\mathrm{s}\). This is just the image of \(X \times Y \subseteq \mathbf{P} \times \mathbf{P}\) in its Segre embedding.

Describe Spec \(\mathbf{Z}\), and show that it is a final object for the category of schemes. i.e., each scheme \(X\) admits a unique morphism to Spec \(\mathbf{Z}\).

Examples of Valuation Rings. Let \(k\) be an algebraically closed field. (a) If \(K\) is a function field of dimension 1 over \(k(I, \$ 6),\) then every valuation ring of \(K / k\) (except for \(K\) itself) is discrete. Thus the set of all of them is just the abstract nonsingular curve \(C_{K}\) of \((\mathrm{I}, \$ 6)\) (b) If \(K / k\) is a function field of dimension two, there are several different kinds of valuations. Suppose that \(X\) is a complete nonsingular surface with function field \(K\) (1) If \(Y\) is an irreducible curve on \(X\), with generic point \(x_{1},\) then the local ring \(R=C_{x_{1}, x}\) is a discrete valuation ring of \(K k\) with center at the (nonclosed) point \(x_{1}\) on \(X\) (2) If \(f: X^{\prime} \rightarrow X\) is a birational morphism, and if \(Y^{\prime}\) is an irreducible curve in \(X^{\prime}\) whose image in \(X\) is a single closed point \(x_{0},\) then the local ring \(R\) of the generic point of \(Y^{\prime}\) on \(X^{\prime}\) is a discrete valuation ring of \(K k\) with center at the closed point \(x_{0}\) on \(X\) (3) Let \(r_{0} \in X\) be a closed point. Let \(f: X_{1} \rightarrow X\) be the blowing-up of \(x_{0}\) (I. \(\$ 4)\) and let \(E_{1}=f^{-1}\left(r_{0}\right)\) be the exceptional curve. Choose a closed point \(x_{1} \in E_{1},\) let \(f_{2}: X_{2} \rightarrow X_{1}\) be the blowing-up of \(x_{1},\) and let \(E_{2}=\) \(f_{2}^{-1}\left(x_{1}\right)\) be the exceptional curve. Repeat. In this manner we obtain a sequence of varieties \(X\), with closed points \(x_{i}\) chosen on them, and for each \(i,\) the local ring \(C_{1,1,1}, x_{1},\) dominates \(C_{x_{1}, x_{1}},\) Let \(R_{0}=\bigcup_{1=0}^{x} C_{x_{1}, x_{1}}\) Then \(R_{0}\) is a local ring, so it is dominated by some valuation ring \(R\) of \(K / k\) by \((\mathrm{I}, 6.1 \mathrm{A}) .\) Show that \(R\) is a valuation ring of \(K / k\). and that it has center \(x_{0}\) on \(X .\) When is \(R\) a discrete valuation ring? Note. We will see later (V.Ex. 5.6) that in fact the \(R_{0}\) of (3) is already a valuation ring itself, so \(R_{0}=R\). Furthermore, every valuation ring of \(K, k\) (except for \(K\) itself) is one of the three kinds just described.

Extension of Coherent Sheaves. We will prove the following theorem in several steps: Let \(X\) be a noetherian scheme, let \(U\) be an open subset, and let \(\mathscr{F}\) be a coherent sheaf on \(U\). Then there is a coherent sheaf \(\mathscr{F}^{\prime}\) on \(X\) such that \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F}\) (a) On a noetherian affine scheme, every quasi-coherent sheaf is the union of its coherent subsheaves. We say a sheaf \(\mathscr{F}\) is the union of its subsheaves \(\mathscr{F}\) if for every open set \(U\), the group \(\mathscr{F}(U)\) is the union of the subgroups ?\((U)\) (b) Let \(X\) be an affine noetherian scheme, \(U\) an open subset, and \(\mathscr{F}\) coherent on \(U .\) Then there exists a coherent sheaf \(\mathscr{F}^{\prime}\) on \(X\) with \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F} .\) [Hint: Let \(\left.i: U \rightarrow X \text { be the inclusion map. Show that } i_{*} \mathscr{F} \text { is quasi-coherent, then use }(a) .\right]\) (c) With \(X, U, \mathscr{F}\) as in (b), suppose furthermore we are given a quasi-coherent sheaf \(\mathscr{G}\) on \(X\) such that \(\left.\mathscr{F} \subseteq \mathscr{G}\right|_{v} .\) Show that we can find \(\mathscr{F}^{\prime}\) a coherent subsheaf of \(\mathscr{G},\) with \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F}\). [Hint: Use the same method, but replace \(i_{*} \mathscr{F}\) by \(\left.\rho^{-1}\left(i_{*} \mathscr{F}\right) \text { , where } \rho \text { is the natural } \operatorname{map} \mathscr{G} \rightarrow i_{*}\left(\left.\mathscr{G}\right|_{U}\right) .\right]\) (d) Now let \(X\) be any noetherian scheme, \(U\) an open subset, \(\mathscr{F}\) a coherent sheaf on \(U,\) and \(\mathscr{G}\) a quasi-coherent sheaf on \(X\) such that \(\left.\mathscr{F} \subseteq \mathscr{G}\right|_{V} .\) Show that there is a coherent subsheaf \(\mathscr{F}^{\prime} \subseteq \mathscr{G}\) on \(X\) with \(\left.\mathscr{F}^{\prime}\right|_{v} \cong \mathscr{F}\). Taking \(\mathscr{I}=i_{*} \mathscr{F}\) proves the result announced at the beginning. [Hint: Cover \(X\) with open affines, and extend over one of them at a time. (e) As an extra corollary, show that on a noetherian scheme, any quasi- coherent sheaf \(\mathscr{F}\) is the union of its coherent subsheaves. [Hint: If \(s\) is a section of \(\mathscr{F}\) over an open set \(U,\) apply (d) to the subsheaf of \(\left.\mathscr{F}\right|_{v}\) generated by s.]

The real importance of the notion of constructible subsets derives from the following theorem of Chevalley-see Cartan and Chevalley [1, expos茅 7] and see also Matsumura \([2, \mathrm{Ch} .2, \$ 6]:\) let \(f: X \rightarrow Y\) be a morphism of finite type of noetherian schemes. Then the image of any constructible subset of \(X\) is a constructible subset of \(Y\). In particular, \(f(X),\) which need not be either open or closed, is a constructible subset of \(Y\). Prove this theorem in the following steps. (a) Reduce to showing that \(f(X)\) itself is constructible, in the case where \(X\) and \(Y\) are affine, integral noetherian schemes, and \(f\) is a dominant morphism. (b) In that case, show that \(f(X)\) contains a nonempty open subset of \(Y\) by using the following result from commutative algebra: let \(A \subseteq B\) be an inclusion of noetherian integral domains, such that \(B\) is a finitely generated \(A\) -algebra. Then given a nonzero element \(b \in B,\) there is a nonzero element \(a \in A\) with the following property: if \(\varphi: A \rightarrow K\) is any homomorphism of \(A\) to an algebraically closed field \(K,\) such that \(\varphi(a) \neq 0,\) then \(\varphi\) extends to a homomorphism \(\varphi^{\prime}\) of \(B\) into \(K,\) such that \(\varphi^{\prime}(b) \neq 0 .[\) Hint: Prove this algebraic result by induction on the number of generators of \(B\) over \(A\). For the case of one generator, prove the result directly. In the application, take \(b=1 .]\) (c) Now use noetherian induction on \(Y\) to complete the proof. (d) Give some examples of morphisms \(f: X \rightarrow Y\) of varieties over an algebraically closed field \(k,\) to show that \(f(X)\) need not be either open or closed.
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