Euler characteristic of a closed point, and invariance under an invertible twist

Set-up. The morphism $i:\mathrm{Spec}\kappa (p)\to X$ is a closed immersion with image the closed point $p$, so $i$ is a homeomorphism onto $\{p\}$; the scheme $Z=\mathrm{Spec}\kappa (p)$ has one point $q$, its structure sheaf is the one-point sheaf $\kappa (p)$, and the stalk of any sheaf on $Z$ at $q$ is its group of global sections. Since $X$ is proper over $k$, it is of finite type over the field $k$, hence locally Noetherian.

Coherence. The structure sheaf ${\mathcal{O}}_Z=\kappa (p)$ is a finite-type quasi-coherent ${\mathcal{O}}_Z$-module, so it is coherent on the locally Noetherian $Z$; the pushforward $i_{\ast }\kappa (p)$ is then a coherent ${\mathcal{O}}_X$-module.

Finite residue degree. The closed point $p$ corresponds to a maximal ideal of a finite-type $k$-algebra, so $\kappa (p)$ is a finite extension of $k$ and $[\kappa (p):k]={\dim }_k\kappa (p)$ is a finite integer.

Cohomology of $i_{\ast }\kappa (p)$. For every $q\ge 0$ there is an isomorphism $H^q(X,i_{\ast }\kappa (p))\cong H^q(Z,\kappa (p))$; the cohomology of $\kappa (p)$ on the one-point space $Z$ vanishes in positive degrees, and $H^0(Z,\kappa (p))\cong \kappa (p)$. Hence $H^q(X,i_{\ast }\kappa (p))=0$ for $q\ge 1$ and $H^0(X,i_{\ast }\kappa (p))\cong \kappa (p)$.

The pullback of $\mathcal{L}$ is trivial. The stalk of $i^{\ast }\mathcal{L}$ at the unique point $q$ is ${\mathcal{O}}_{Z,q}{\otimes }_{{\mathcal{O}}_{X,p}}{\mathcal{L}}_p\cong \kappa (p)$, and this is also the group of global sections: $(i^{\ast }\mathcal{L})(Z)=(i^{\ast }\mathcal{L}{)}_q\cong \kappa (p)={\mathcal{O}}_Z(Z)$. A morphism of sheaves $i^{\ast }\mathcal{L}\to {\mathcal{O}}_Z$ on the one-point space $Z$ is determined by its component on global sections, so a $\kappa (p)$-linear isomorphism $\kappa (p)\to \kappa (p)$ extends to a morphism of ${\mathcal{O}}_Z$-modules $i^{\ast }\mathcal{L}\to {\mathcal{O}}_Z$ whose stalk at $q$ is an isomorphism; by the stalkwise criterion this morphism is an isomorphism, so $i^{\ast }\mathcal{L}\cong {\mathcal{O}}_{\mathrm{Spec}\kappa (p)}$.

Part (1). The Euler characteristic of the coherent module $i_{\ast }\kappa (p)$ on the proper $k$-scheme $X$ is the finite alternating sum ${\sum }_{q\ge 0}(-1{)}^q{\dim }_k{H}^q(X,i_{\ast }\kappa (p))$; by step 1.4 only $q=0$ contributes, with $H^0\cong \kappa (p)$, so $\chi (X,i_{\ast }\kappa (p))={\dim }_k\kappa (p)=[\kappa (p):k]$ by step 1.3.

Part (2). By the projection formula the canonical map $\mathcal{L}\otimes i_{\ast }\kappa (p)\to i_{\ast }(i^{\ast }\mathcal{L}{\otimes }_{{\mathcal{O}}_Z}\kappa (p))$ is an isomorphism; substituting the isomorphism $i^{\ast }\mathcal{L}\cong {\mathcal{O}}_Z$ of step 1.5 and the unit isomorphism ${\mathcal{O}}_Z{\otimes }_{{\mathcal{O}}_Z}\kappa (p)\cong \kappa (p)$ identifies the target with $i_{\ast }\kappa (p)$. Hence $\mathcal{L}\otimes i_{\ast }\kappa (p)\cong i_{\ast }\kappa (p)$.

Consequence for the Euler characteristic. The isomorphism of step 2.2 induces isomorphisms $H^q(X,\mathcal{L}\otimes i_{\ast }\kappa (p))\cong H^q(X,i_{\ast }\kappa (p))$ for every $q\ge 0$, so the two alternating sums agree: $\chi (X,\mathcal{L}\otimes i_{\ast }\kappa (p))=\chi (X,i_{\ast }\kappa (p))=[\kappa (p):k]$ by step 2.1.

Conclusion and choice accounting. Steps 1.4 and 2.1 give statement (1), steps 1.5 and 2.2 give the two isomorphism clauses of statement (2), and step 3.1 gives its Euler-characteristic consequence. The Axiom of Choice is used only through the suppliers recorded in [F7]: the Euler-characteristic and pushforward technology of [F2] and [F4], the closed-immersion and projection-formula machinery of [F6], and the residue-field finiteness of [F3]; the point $p$, the sheaf $\mathcal{L}$ and the morphism $i$ are part of the given data, and no selection is made in steps 1.1–3.1.