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Chapter 3: Problem 1

Let \(X\) be a projective scheme over a field \(k,\) and let \(\overline{\mathscr{I}}\) be a coherent sheaf on \(X\) We define the Euler characteristic of \(\mathscr{F}\) by $$\chi(\mathscr{F})=\sum(-1)^{i} \operatorname{dim}_{k} H^{i}(X, \overline{\mathscr{F}})$$ If$$0 \rightarrow \mathscr{F}^{\prime} \rightarrow \mathscr{F} \rightarrow \mathscr{F}^{\prime \prime} \rightarrow 0$$ is a short exact sequence of coherent sheaves on \(X,\) show that \(\chi(\mathscr{F})=Y\left(\mathscr{F}^{\prime}\right)+\) \(x\left(\mathscr{F}^{\prime \prime}\right)\)

Short Answer

Expert verified
The equality \(\chi(\mathscr{F})=\chi \left(\mathscr{F'}\right)+\chi \left(\mathscr{F''}\right)\) holds because the Euler characteristic of a sheaf is additive over short exact sequences of sheaves.

Step by step solution

01

Recall the long exact sequence in cohomology

Given a short exact sequence of sheaves \(0 \rightarrow \mathscr{F}^{\prime} \rightarrow \mathscr{F} \rightarrow \mathscr{F}^{\prime \prime} \rightarrow 0\), it induces a long exact sequence in cohomology: \[...\rightarrow H^{i}(X, \mathscr{F}^{\prime}) \rightarrow H^{i}(X, \mathscr{F}) \rightarrow H^{i}(X, \mathscr{F}^{\prime \prime}) \rightarrow H^{i+1}(X, \mathscr{F}^{\prime}) \rightarrow ...\]
02

Define the Euler characteristic for each sheaf

The Euler characteristic for the sheaves \(\mathscr{F}\), \(\mathscr{F'}\) and \(\mathscr{F''}\) are defined as \(\chi(\mathscr{F}) = \sum(-1)^{i} \dim_{k} H^{i}(X, \mathscr{F})\), \(\chi(\mathscr{F'}) = \sum(-1)^{i} \dim_{k} H^{i}(X, \mathscr{F'})\) and \(\chi(\mathscr{F''}) = \sum(-1)^{i} \dim_{k} H^{i}(X, \mathscr{F''})\) respectively.
03

Apply alternating sum to the long exact sequence

Applying an alternating sum to the long exact sequence in cohomology gives: \[\sum_i (-1)^i \dim_k [H^i(X, \mathscr{F'}) \rightarrow H^i(X, \mathscr{F}) \rightarrow H^i(X, \mathscr{F''})] = 0\] The connecting homomorphisms imply that each term in the sum cancels with the next, all except for three terms that correspond to the Euler characteristics of the three sheaves.
04

Conclude the equality

Because of the cancellation in the alternating sum, we arrive at the equality \[ \chi(\mathscr{F})=\chi \left(\mathscr{F'}\right)+\chi \left(\mathscr{F''}\right) \] which is the additivity property of the Euler characteristic in context of a short exact sequence of sheaves.

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Key Concepts

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

Coherent Sheaves
In the realm of algebraic geometry, a coherent sheaf is a fundamental concept allowing mathematicians to study the properties and functions on algebraic varieties and schemes. Coherent sheaves are particularly important because they generalize the idea of vector bundles and module theory to arbitrary algebraic varieties, including those that might be singular or not smooth.

A coherent sheaf on a projective scheme, such as \( \mathscr{F} \), is a sheaf of O-modules that is locally of finite presentation and its kernel is also finitely generated. More intuitively, you can think of a coherent sheaf as a way to organize the local algebraic data of geometric spaces into a global framework. Such sheaves come equipped with morphisms that reflect the coherent properties of the geometric structures they are associated with. They are crucial for defining complex structures, property checks, and global sections that are instrumental in understanding the geometric object in question.

In the exercise, the relevance of coherent sheaves is underscored when considering the Euler characteristic. The way in which these sheaves behave under short exact sequences demonstrates their algebraic structure and helps unveil the underlying geometry of the scheme.
Long Exact Sequence in Cohomology
The long exact sequence in cohomology is an essential tool when working with sheaves in algebraic geometry. It is the backbone that allows for the manipulation and understanding of the cohomological properties of sheaves. When a short exact sequence of coherent sheaves is given, cohomology creates a bridge to a long exact sequence that spreads all the way to infinity in both directions.

The long exact sequence can be perceived as an 'unfolded' version of the original short sequence, revealing additional information hidden in between the terms. Each cohomology group \(H^i(X, \mathscr{F})\) captures different layers of global algebraic information of the sheaf \(\mathscr{F}\) over the projective scheme \(X\). The alternating signs in the sequence are due to the change of direction of the morphisms between these groups.

In the solution provided, using the long exact sequence allowed us to pinpoint the places where dimensions, added with alternating signs, would cancel out, leaving behind a simple expression with just the Euler characteristics of the sheaves involved.
Projective Scheme
A projective scheme serves as a cornerstone of study in algebraic geometry. It generalizes the concept of projective space, extending it to incorporate richer algebraic structures. It is created by 'gluing together' spectra of graded rings along their common parts and has proved to be an apt setting for examining coherent sheaves and their cohomology.

Projective schemes benefit from powerful geometric properties, such as the ampleness of the twisting sheaf, and these properties play a pivotal role when investigating the behavior of coherent sheaves on these spaces. The projective nature implies that the schemes are well-suited for compactifying affine varieties, and thus providing a way to extend properties of affine pieces to a complete projective variety.

The projective scheme \(X\) in the given exercise is the space upon which the coherent sheaves \(\mathscr{F}\), \(\mathscr{F}'\), and \(\mathscr{F}''\) live. By understanding the scheme, one grasps the geometric stage on which all the cohomological action takes place 鈥� a necessity for comprehending the deeper algebraic and geometric phenomena at play.

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Most popular questions from this chapter

Let \(X\) be a Zariski space (II, Ex. 3.17). Let \(P \in X\) be a closed point, and let \(X_{P}\) be the subset of \(X\) consisting of all points \(Q \in X\) such that \(P \in\left\\{Q ;^{-} . \text {We call } X_{P}\right.\) the local space of \(X\) at \(P\), and give it the induced topology. Let \(j: X_{P} \rightarrow X\) be the inclusion, and for any sheaf \(\mathscr{F}\) on \(X\), let \(\mathscr{F}_{P}=j^{*} \mathscr{H}\). Show that for all \(i, \mathscr{F},\) we have \\[ H_{P}^{i}(X, \mathscr{F})=H_{P}^{i}\left(X_{P}, \widetilde{\mathscr{H}}_{P}\right) \\]

Let \(Y\) be an integral scheme of finite type over an algebraically closed field \(k\) Let \(f: X \rightarrow Y\) be a fiat projective morphism whose fibres are all integral schemes. Let \(\mathscr{L},\) // be invertible sheaves on \(X,\) and assume for each \(y \in Y\) that \(\mathscr{Y}_{r} \cong .1 /\) on the fibre \(X_{r} .\) Then show that there is an invertible sheaf. 1 on \(Y\) such that \(\mathscr{L} \cong . / / \otimes f^{*} .1:\left[\text { Hint: Use the results of this section to show that } f_{*}\left(\mathscr{L} \otimes .1 /^{-1}\right)\right.\) is locally free of rank \(1 \text { on } Y .]\)

Let \(X=\mathbf{P}_{k}^{1},\) with \(k\) an infinite field. (a) Show that there does not exist a projective object \(\mathscr{P} \in \mathfrak{M}\) od \((X)\), together with a surjective map \(\mathscr{P} \rightarrow \mathscr{O}_{X} \rightarrow 0 .\) [Hint: Consider surjections of the form \(\mathscr{O}_{V} \rightarrow\) \(k(x) \rightarrow 0,\) where \(x \in X\) is a closed point, \(V\) is an open neighborhood of \(x\) and \(\mathscr{O}_{V}=j_{!}\left(\left.\mathcal{O}_{X}\right|_{V}\right),\) where \(j: V \rightarrow X\) is the inclusion. (b) Show that there does not exist a projective object \(\mathscr{P}\) in either \(\mathbb{Z} \operatorname{co}(X)\) or \(\operatorname{Cob}(X)\) together with a surjection \(\mathscr{P} \rightarrow \mathscr{O}_{X} \rightarrow 0 .\) [Hint: Consider surjections of the form \(\mathscr{L} \rightarrow \mathscr{L} \otimes k(x) \rightarrow 0,\) where \(x \in X\) is a closed point, and \(\mathscr{L}\) is an invertible sheaf on \(X .]\)

Let \(X\) be a reduced noetherian scheme. Show that \(X\) is affine if and only if each irreducible component is affine.

Let \(X=\operatorname{Spec} k\left[x_{1}, x_{2}, x_{3}, x_{4}\right]\) be affine four-space over a field \(k .\) Let \(Y_{1}\) be the plane \(x_{1}=x_{2}=0\) and let \(Y_{2}\) be the plane \(x_{3}=x_{4}=0 .\) Show that \(Y=Y_{1} \cup Y_{2}\) is not a set-theoretic complete intersection in \(X\), Therefore the projective closure \(\bar{Y}\) in \(\mathbf{P}_{k}^{4}\) is also not a set-theoretic complete intersection. [Hints: Use an affine analogue of (Ex. 4.8e). Then show that \(H^{2}\left(X-Y, \mathcal{O}_{X}\right) \neq 0,\) by using (Ex. 2.3) and (Ex. 2.4). If \(\left.P=Y_{1} \cap Y_{2}, \text { imitate (Ex. 4.3) to show } H^{3}\left(X-P, O_{X}\right) \neq 0 .\right]\)
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