Total length of a zero-dimensional projective scheme

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field, let $I\subseteq k[x_0,\dots ,x_n]$ be a homogeneous ideal, put $S=k[x_0,\dots ,x_n]\mathrm{/}I$ with its standard grading, and let $X=\mathrm{Proj}S$ with standard charts $D_{+}(x_i)=\mathrm{Spec}(A_i)$, $A_i=(S_{x_i}{)}_0$ (Projective scheme of a homogeneous quotient and its standard affine charts). Assume that $X$ is zero-dimensional, meaning that every chart ring $A_i$ is either zero or of Krull dimension $0$ (Krull dimension of a nonzero ring); this is the chartwise form of zero-dimensionality used throughout this pair.

Under this hypothesis A zero-dimensional projective scheme has finitely many closed points with finite-dimensional local rings provides exactly the data needed for a finite total sum: the point set $\mathrm{\mid }X\mathrm{\mid }$ is finite; for each $x\in \mathrm{\mid }X\mathrm{\mid }$ the local ring ${\mathcal{O}}_{X,x}$ has finite length as a module over itself (Composition series and length of a module); and the residue field $\kappa (x)={\mathcal{O}}_{X,x}\mathrm{/}{\mathfrak{m}}_x$ (The residue field at a point of an affine scheme) is a finite extension of $k$, so the degree $[\kappa (x):k]={\dim }_k\kappa (x)$ is a natural number (The degree $[K:F]={\dim }_{F}K$ of a finite field extension, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis). The total length of $X$ over $k$ is the finite sum

${\mathrm{len}}_k(X):={\sum }_{x\in \mathrm{\mid }X\mathrm{\mid }}{\mathrm{\ell }}_{{\mathcal{O}}_{X,x}}\left({\mathcal{O}}_{X,x}\right)\cdot [\kappa (x):k]\in \mathbb{N}\mathrm{.}$

Its summands are the composition lengths of the local rings, taken as modules over themselves, multiplied by the degrees of the residue field extensions. The empty sum is the natural number $0$, so ${\mathrm{len}}_k(\mathrm{\varnothing })=0$.

1. Length taken in $X$, not in an ambient plane. The factor ${\mathrm{\ell }}_{{\mathcal{O}}_{X,x}}({\mathcal{O}}_{X,x})$ is the length of the local ring of $X$ at $x$ as a module over itself. For $X=\mathrm{Proj}S$ this is the local factor of the chart ring of any standard chart containing $x$ (A zero-dimensional projective scheme has finitely many closed points with finite-dimensional local rings), so it is an invariant of the pair $(X,x)$; it is not the length of any ring attached to an ambient projective space into which $X$ might be embedded.
2. No closedness of $k$ is assumed. Over a general field the residue field $\kappa (x)$ may be a proper finite extension of $k$, and the factor $[\kappa (x):k]$ records that degree; over an algebraically closed field this factor is $1$ for every point, but no such equality is built into the definition.
3. Finiteness is inherited, not assumed. Finiteness of $\mathrm{\mid }X\mathrm{\mid }$, of each length, and of each residue degree all come from A zero-dimensional projective scheme has finitely many closed points with finite-dimensional local rings, whose proof uses the Axiom of Choice through the prime-lifting and Artinian-structure suppliers; the definition itself performs no selection beyond that inherited hypothesis.

Consistency with the affine case. Suppose $X=\mathrm{Spec}A$ is an affine scheme whose coordinate ring $A$ is a finite-dimensional $k$-algebra, which is the situation of a standard chart above. Then the points of $X$ are the finitely many maximal ideals ${\mathfrak{m}}_1,\dots ,{\mathfrak{m}}_r$ of $A$, with ${\mathcal{O}}_{X,{\mathfrak{m}}_j}=A_{{\mathfrak{m}}_j}$ and $\kappa ({\mathfrak{m}}_j)=A\mathrm{/}{\mathfrak{m}}_j$, and

${\mathrm{len}}_k(X)={\dim }_{k}A\mathrm{.}$

Indeed, An Artinian ring is canonically the finite product of its localizations at its maximal ideals writes $A\cong {\prod }_{j=1}^r{A}_{{\mathfrak{m}}_j}$; additivity of the dimension over a direct sum (If $V={⨁}_{i<n}{U}_i$ with every $U_i$ finite-dimensional, then $V$ is finite-dimensional and ${\dim }_{F}V={\sum }_{i<n}{\dim }_F{U}_i$; in particular ${\dim }_F(U\oplus W)={\dim }_{F}U+{\dim }_{F}W$) gives ${\dim }_{k}A={\sum }_{j=1}^r{\dim }_k{A}_{{\mathfrak{m}}_j}$. Each local factor has nilpotent maximal ideal (A zero-dimensional projective scheme has finitely many closed points with finite-dimensional local rings), so its filtration by powers of the maximal ideal has $\kappa ({\mathfrak{m}}_j)$-vector space factors and finite length; additivity of length in short exact sequences (Module length is additive in short exact sequences) and additivity of dimension over such a filtration give ${\dim }_k{A}_{{\mathfrak{m}}_j}={\mathrm{\ell }}_{A_{{\mathfrak{m}}_j}}(A_{{\mathfrak{m}}_j})\cdot [\kappa ({\mathfrak{m}}_j):k]$ for every $j$. Summing the equalities yields the displayed identity. This computation is a consistency check on the definition and is never used in place of the local intersection computations of this pair.