The normalization defect is an Euler characteristic and a weighted sum of local lengths

The normalization sequence $0\to {\mathcal{O}}_C\to {\nu }_{\ast }{\mathcal{O}}_{\tilde{C}}\to {\mathcal{Q}}_C\to 0$ is short exact: ${\mathcal{Q}}_C$ is the cokernel by definition [F1], and ${\mathcal{O}}_C\to {\nu }_{\ast }{\mathcal{O}}_{\tilde{C}}$ is injective because on an affine chart it is the inclusion of $A$ into its integral closure in the total ring of fractions [F2]. All three terms are coherent (${\mathcal{O}}_C$, ${\nu }_{\ast }{\mathcal{O}}_{\tilde{C}}$ by [F2], and ${\mathcal{Q}}_C$ by [F1]), so the sequence satisfies the hypotheses of [F3].

One has $\chi (C,{\nu }_{\ast }{\mathcal{O}}_{\tilde{C}})=\chi (\tilde{C},{\mathcal{O}}_{\tilde{C}})$: the finite morphism $\nu$ is affine by [F2] and its inverse image of an affine open is affine, so [F5] identifies $H^q(C,{\nu }_{\ast }{\mathcal{O}}_{\tilde{C}})$ with $H^q(\tilde{C},{\mathcal{O}}_{\tilde{C}})$ for every $q\ge 0$, and the two alternating sums of [F4] agree.

Let $F$ be the finite closed support of ${\mathcal{Q}}_C$. The germ maps define a sheaf map ${\mathcal{Q}}_C\to {⨁}_{p\in F}{i}_{p,\ast }{\mathcal{Q}}_{C,p}$, where each summand is the skyscraper of the underlying abelian group. It is an isomorphism on stalks: at $p\in F$ its $p$-component is the identity and the other summands have zero stalk since their points are closed; outside $F$ both stalks are zero. Hence it is an isomorphism of abelian sheaves. A finite sum of skyscrapers is flasque, because every restriction is a direct sum of identities and maps onto zero. Flasque acyclicity proves $H^q(C,{\mathcal{Q}}_C)=0$ for $q\ge 1$ and also proves directly the degree-zero stalk sum used below.

Consequently ${\delta }_k(C)={\dim }_k{H}^0(C,{\mathcal{Q}}_C)=\chi (C,{\mathcal{Q}}_C)$ by [F4], and applying [F3] to the sequence of step 1.1 gives $\chi (C,{\nu }_{\ast }{\mathcal{O}}_{\tilde{C}})=\chi (C,{\mathcal{O}}_C)+\chi (C,{\mathcal{Q}}_C)$, so the two identities combined with step 1.2 yield ${\delta }_k(C)=\chi (\tilde{C},{\mathcal{O}}_{\tilde{C}})-\chi (C,{\mathcal{O}}_C)$; this proves (1).

For the length formula of (2): by [F1], $H^0(C,{\mathcal{Q}}_C)$ is the direct sum of the stalks ${\mathcal{Q}}_{C,p}$ over the finite non-normal locus, and each ${\mathcal{Q}}_{C,p}$ has finite length over the Noetherian local ring ${\mathcal{O}}_{C,p}$; fixing a composition series [F6] with successive quotients $\kappa (p)$ and using additivity of $k$-dimension in short exact sequences together with [F7], one gets ${\dim }_k{\mathcal{Q}}_{C,p}={\mathrm{\ell }}_{{\mathcal{O}}_{C,p}}({\mathcal{Q}}_{C,p})\cdot [\kappa (p):k]$. Summing gives ${\delta }_k(C)={\sum }_p[\kappa (p):k]{\mathrm{\ell }}_{{\mathcal{O}}_{C,p}}({\mathcal{Q}}_{C,p})$ over the closed points of the finite non-normal locus.

For (3): the sum in step 3.1 has nonnegative terms, so ${\delta }_k(C)\ge 0$; if ${\delta }_k(C)=0$ then every local length vanishes, hence ${\mathcal{Q}}_C=0$ and the injective map ${\mathcal{O}}_C\to {\nu }_{\ast }{\mathcal{O}}_{\tilde{C}}$ of step 1.1 is an isomorphism, so the affine morphism $\nu$ is an isomorphism and $C\cong \tilde{C}$ is regular of dimension one by [F2]. Conversely, if $C$ is regular, then each one-dimensional local ring ${\mathcal{O}}_{C,p}$ is a discrete valuation ring by [F9], hence an integrally closed domain by [F10], and the zero-dimensional stalks are fields, so $C$ is normal in the sense of [F11]; the identity $C\to C$ is then a finite morphism from a normal curve that is an isomorphism over the regular locus, so the uniqueness clause of [F2] makes the normalization isomorphic to the identity over $C$, whence ${\mathcal{Q}}_C=0$ and ${\delta }_k(C)=0$. Finally, in the regular case every local ring is a discrete valuation ring or a field, hence a domain, so each point of $C$ lies in a unique irreducible component and distinct components are disjoint; a component, having everywhere the local ring of $C$, is itself regular and hence normal.