Degree of an invertible sheaf on a proper one-dimensional scheme

Assume the Axiom of Choice, inherited from the Euler-characteristic supplier below (The Axiom of Choice). Let $k$ be a field (Field) and let $C$ be a proper $k$-scheme (Proper morphisms) whose underlying topological space is Noetherian of dimension at most one (Chain dimension and the empty-space convention, Locally Noetherian and Noetherian schemes). Write $\chi (-,-)$ for the Euler characteristic of coherent sheaves on a proper $k$-scheme (Euler characteristic of a coherent sheaf).

Degree of an invertible sheaf. For an invertible ${\mathcal{O}}_C$-module $\mathcal{L}$ (Invertible sheaves) set ${\mathrm{deg}}_C(\mathcal{L}):=\chi (C,\mathcal{L})-\chi (C,{\mathcal{O}}_C)\in \mathbb{Z}\mathrm{.}$

Degree of a finite locally free sheaf. For a locally free ${\mathcal{O}}_C$-module $\mathcal{E}$ of finite constant rank $n\ge 0$ (Locally free sheaves of finite rank) set ${\mathrm{deg}}_C(\mathcal{E}):=\chi (C,\mathcal{E})-n\chi (C,{\mathcal{O}}_C)\in \mathbb{Z}\mathrm{.}$

Well-definedness and immediate values.

• The structure sheaf, its dual, and every invertible or locally free finite-rank module on $C$ are coherent, so the Euler characteristic applies. Indeed $C$ is of finite type over the field $k$, hence every affine chart of $C$ is a spectrum of a Noetherian ring (Every algebra of finite type over a Noetherian ring is a Noetherian ring, Locally Noetherian and Noetherian schemes), so $C$ is locally Noetherian; on a locally Noetherian scheme the coherent modules are exactly the finite-type quasi-coherent modules (Coherent sheaves on a locally Noetherian scheme), and ${\mathcal{O}}_C$ and every invertible or finite locally free module are quasi-coherent of finite type (Coherent module sheaves, Invertible sheaves, Locally free sheaves of finite rank). Each alternating sum in the definition is therefore finite and defines an integer (Euler characteristic of a coherent sheaf).
• Both expressions depend only on the isomorphism class of the sheaf: an isomorphism of coherent sheaves induces isomorphisms on all cohomology groups (Euler characteristic of a coherent sheaf), so the Euler characteristic, and with it each degree, is an isomorphism invariant. In particular the degree is a function on isomorphism classes, hence on $\mathrm{Pic}(C)$, and on the isomorphism classes of finite locally free modules; the rank alone does not determine their degree.
• ${\mathrm{deg}}_C({\mathcal{O}}_C)=\chi (C,{\mathcal{O}}_C)-\chi (C,{\mathcal{O}}_C)=0$, and ${\mathrm{deg}}_C(\mathcal{E})=0$ for every finite locally free $\mathcal{E}$ of rank $0$, since such a module is the zero sheaf and $\chi (C,0)=0$.
• If $C=\mathrm{\varnothing }$, then $H^q(C,\mathcal{F})=0$ for every coherent $\mathcal{F}$ and every $q\ge 0$ and $\chi (C,\mathcal{F})=0$ (Euler characteristic of a coherent sheaf); every degree above is therefore $0$.

The dimension bound at most one records that the base is a curve: the definition is applied below to proper curves and to effective Cartier divisors on surfaces, and the degree of a curve in the sense of Degree divisor proper curve agrees with it on smooth proper geometrically integral curves by Riemann-Roch in Euler-characteristic form: the degree shift.