Let \(X\) be a complete nonsingular curve over an algebraically closed field \(k\). Show that there is a unique way to define the degree of any coherent sheaf on \(X\), \(\deg \mcf \in \ZZ\), such that
if \(D\) is a divisor, then \(\deg \mcl(D) = \deg D\);
if \(\mcf\) is a torsion sheaf, meaning a sheaf whose stalk at the generic point is zero, then \(\deg \mcf = \sum_{P \in X} \length(\mcf_P)\);
if \(0 \to \mcf' \to \mcf \to \mcf'' \to 0\) is exact, then \(\deg \mcf = \deg \mcf' + \deg \mcf''\).
Solution
Write \(\gamma(\mathcal F)\) for the class of a coherent sheaf in the group \(K(X)\) of P-AGH2610KGROUP. For a divisor \(D\), put \(\OO_X(D)=\mathcal L(D)\) and \(\psi(D)=\sum_P n_P\gamma(k(P))\) when \(D=\sum_P n_PP\). The determinant homomorphism is denoted by \(\det:K(X)\to\Pic(X)\).
1. The degree of divisors induces a homomorphism \(d:\Pic(X)\to\ZZ\) with \(d([\OO_X(D)])=\deg D\).
Every principal divisor on the complete nonsingular curve has degree zero [@Har10a, Proposition II.6.10]. Thus the additive map \(D=\sum_P n_PP\mapsto\sum_P n_P\) descends to \(\Cl(X)\). The Cartier and Weil comparison identifies \(\Cl(X)\) with \(\Pic(X)\) by \([D]\mapsto[\OO_X(D)]\) [@Har10a, Corollary II.6.16]. Transporting degree through this isomorphism gives \(d\). In particular, \(d([\OO_X])=0\).
2. Define the degree of a coherent sheaf by\[ \boxed{\deg\mathcal F\coloneqq d\bigl(\det(\gamma(\mathcal F))\bigr).} \]This satisfies (a) and (c).
For an invertible sheaf \(\OO_X(D)\), its determinant class is \([\OO_X(D)]\), so step 1 gives condition (a). For a short exact sequence of coherent sheaves, the defining relation in \(K(X)\) is \[ \gamma(\mathcal F)=\gamma(\mathcal F')+\gamma(\mathcal F''). \] Both determinant and \(d\) are homomorphisms, by P-AGH2611KCURVE, step 5, and step 1 of this proof. Their composite is additive, proving (c).
3. The degree in step 2 satisfies the torsion formula (b).
1.The same card’s step 5 gives \(\det(\gamma(k(P)))=[\OO_X(P)]\).Every closed point has degree one because \(k\) is algebraically closed.Applying \(d\circ\det\) therefore gives \(\deg\mathcal T=\sum_P\ell_P\), as required.
4. Any degree assignment satisfying (a) and (c) equals the assignment in step 2; in particular, all three requirements determine it uniquely.
Let \(\delta\) be such an assignment. Condition (c) applied to the zero sequence gives \(\delta(0)=0\); applied to a sheaf isomorphism regarded as a short exact sequence, it gives invariance under isomorphism. It therefore induces a homomorphism \(\bar\delta:K(X)\to\ZZ\). Condition (a) gives \(\bar\delta(\gamma(\OO_X))=0\). For every divisor \(D\), P-AGH2611KCURVE, step 2, gives \[ \psi(D)=\gamma(\OO_X(D))-\gamma(\OO_X), \qquad \bar\delta(\psi(D))=\deg D. \] The isomorphism in step 7 of that card says that every element of \(K(X)\) is of the form \(\psi(D)+n\gamma(\OO_X)\). Thus \(\bar\delta\) is determined on a generating set and agrees there with \(d\circ\det\). The homomorphisms, and hence the degree assignments, coincide.
5. Q.E.D.
Steps <1>1–<1>3 construct a degree satisfying all three requirements, and step 4 proves uniqueness.