91Ó°ÊÓ

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 \(X \oplus \mathbf{Z}\)

Step by step solution

01

Define the map and show it is well-defined

For any divisor \(D=\sum n_{i} P_{i}\) on \(X\), let \(\psi(D)=\sum n_{i}.k\left(P_{i}\right)\) in \(K(X)\). If \(D\) is an effective divisor, \(\mathrm{C}_{D}\) is the structure sheaf of the associated subscheme of codimension \(1\), so \(\psi(D)=\dot{\gamma}\left(C_{D}\right)\). Using (6.18) we can say that for any \(D, \psi(D)\) depends only on the linear equivalence class of \(D\), so \(\psi\) defines a homomorphism \(\psi: \mathrm{Cl} X \rightarrow K(X)\)

02

Exact sequence and homomorphism

For any coherent sheaf \(\mathscr{F}\) on \(X\), there exist locally free sheaves \(\delta_{0}\) and \(\mathscr{E}_{1}\) and exact sequence $0\rightarrow \mathscr{E}_{1} \rightarrow \mathscr{B}_{0}\rightarrow\mathscr{H} \rightarrow 0$. Let \(r_{0}=\operatorname{rank} \delta_{0}\) and \(r_{1}=\operatorname{rank} \delta_{1}\). Define \(\operatorname{det}\tilde{\mathscr{H}}=\left(\bigwedge^{r_{0}} \mathscr{E}_{0}\right)\otimes\left(\bigwedge^{r_{1}} \delta_{1}\right)^{-1} \in \operatorname{Pic} X\). We can show that det \(\mathscr{H}\) is independent of the resolution chosen, and that it gives a homomorphism det: \(K(X) \rightarrow\) Pic \(X\). If \(D\) is a divisor, then \(\operatorname{det}(\psi(D))=\mathscr{L}(D)\)

03

Exact sequence and torsion sheaf

For a coherent sheaf \(\mathscr{F}\) of rank \(r\), there is 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. We can conclude that if \(\mathscr{F}\) is sheaf of rank \(r,\) then \(\gamma(\mathscr{F})-r \gamma\left(\mathcal{O}_{X}\right) \in \operatorname{Im} \psi\)

04

Show that the Grothendieck group is isomorphic to Picard group plus integers

Using the maps \(\psi,\) det, rank,and \(1 \mapsto \gamma\left(C_{X}\right)\) from \(\mathbf{Z} \rightarrow K(X),\) we can demonstrate that \(K(X) \cong\) Pic \(X \oplus\mathbf{Z}\)

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Key Concepts

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

Algebraic Geometry

Algebraic geometry is a branch of mathematics that studies geometrical properties and structures using algebraic tools. It involves the study of solutions to algebraic equations and their generalizations, encompassing topics such as varieties, schemes, and spaces defined over a field.

For instance, in the context of a nonsingular curve over an algebraically closed field, one wants to understand the curve's geometric properties through algebraic invariants. Fundamental to this exploration is the concept of divisors and sheaves, which we will discuss in more detail in subsequent sections. Algebraic geometry provides a powerful framework for analyzing the relationships between different mathematical objects and their symmetries, making it crucial for solving problems like the one in the exercise where we explore the structure of the Grothendieck Group of a nonsingular curve.

Divisor Theory

Divisor theory is a pivotal concept within algebraic geometry, dealing with the formal sum of points on algebraic varieties, especially curves. A divisor on a nonsingular curve is expressed as a sum \(D=\sum n_{i} P_{i}\) where \(n_i\) are integers and \(P_i\) are points on the curve.

When looking at nonsingular algebraic curves, divisors play a role in the definitions of line bundles and rational functions. Effective divisors correspond to subschemes of codimension 1, and the class of a divisor under linear equivalence provides crucial information about the curve's structure. Divisor theory interlinks with other mathematical objects such as line bundles and sheaves, leading to the study of the Picard Group, which is the group of isomorphism classes of line bundles on a variety.

Coherent Sheaves

Coherent sheaves are essential tools for understanding the structure of algebraic varieties in algebraic geometry. They are generalizations of vector bundles and locally look like finitely generated modules over the structure sheaf.

On a nonsingular curve \(X\), every coherent sheaf can be associated with an exact sequence of locally free sheaves. This property allows us to define invariants like the determinant line bundle, denoted in the exercise as \(\operatorname{det}\tilde{\mathscr{H}}\), which is constructed via exterior powers of the sheaves involved in an exact sequence. The computation of the determinant line bundle is part of resolving the structure of the Grothendieck group of \(X\) and provides a link to understanding the class group of divisors on the curve.

Picard Group

The Picard Group, often denoted as \(\operatorname{Pic} X\), is a group of line bundles or divisors modulo linear equivalence on an algebraic variety or curve. In the context of a nonsingular curve, it classifies line bundles up to isomorphism, which can be seen as a generalization of the divisor class group.

The Picard Group is fundamental for the exercise because it connects divisors and coherent sheaves with the Grothendieck Group \(K(X)\) of the curve \(X\). This group, \(\operatorname{Pic} X\), via the determinant map from the exercise, reveals the deep relationship between algebraic and geometric structures. An isomorphism between \(K(X)\) and \(\operatorname{Pic} X \oplus \mathbf{Z}\) encapsulates the notion that line bundles on \(X\), along with a numeric invariant, can effectively catalog all coherent sheaves, disclosing crucial information about the curve's geometry.