Intersection multiplicity of closed subschemes at a point

Let $X$ be a locally Noetherian scheme, let $Y,Z\subseteq X$ be closed subschemes, and let $p\in Y\cap Z$ be a closed point. Assume that $Y$ is integral of dimension one (Integral schemes, Chain dimension and the empty-space convention) and that the generic point of $Y$ is not contained in $Z$; since $Z$ is closed, this means $Y\subseteq ̸Z$, and it excludes every one-dimensional component of $Y\cap Z$. The local intersection multiplicity of $Y$ and $Z$ at $p$ is

$m_p(Y\cap Z):={\mathrm{length}}_{{\mathcal{O}}_{X,p}}\left({\mathcal{O}}_{Y\cap Z,p}\right),$

where ${\mathcal{O}}_{X,p}$ is the local ring of $X$ at $p$ (A local ring is a nonzero commutative ring with a unique maximal ideal), where $I_{Y\cap Z}=I_Y+I_Z$ is the sum of the ideal sheaves of $Y$ and $Z$, and where

${\mathcal{O}}_{Y\cap Z,p}={\mathcal{O}}_{X,p}\mathrm{/}{I}_{Y\cap Z,p}$

is the local ring at $p$ of the scheme-theoretic intersection (Scheme-theoretic fibre).

The definition is meaningful. Write $\eta$ for the generic point of the integral one-dimensional scheme $Y$. By hypothesis $\eta \mathrm{\notin }Z$, so $\eta \mathrm{\notin }Y\cap Z$. An irreducible component of the closed subset $Y\cap Z$ is the closure of one of its points, hence a closed subset of $Y$ not containing $\eta$. Such a component is a single closed point of $Y$: if a point $y\mathrm{\ne }\eta$ of $Y$ had a closure containing a further point $z\mathrm{\ne }y$, then the closures of $z$, of $y$ and of $\eta$ would form a strict chain of irreducible closed subsets of length two, contradicting $\dim Y=1$ (Chain dimension and the empty-space convention); therefore $Y\cap Z$ is supported in the closed points of $Y$ that lie on $Z$. In particular ${\mathcal{O}}_{Y\cap Z,p}$, being a quotient of the Noetherian local ring ${\mathcal{O}}_{X,p}$ (Locally Noetherian and Noetherian schemes, Noetherian commutative rings and modules), is a zero-dimensional Noetherian local ring, and such a ring has a finite composition series over itself (Composition series and length of a module); so the length is a well-defined natural number (Remarks). The value depends only on the local data at $p$: pulling back along the canonical localization morphism $\mathrm{Spec}{\mathcal{O}}_{X,p}\to X$ does not change ${\mathcal{O}}_{X,p}$, $I_{Y,p}$ or $I_{Z,p}$, so replacing $X$ by an open neighbourhood of $p$ leaves $m_p(Y\cap Z)$ unchanged. Finally $m_p(Y\cap Z)\ge 1$, because $p\in Y\cap Z$ makes ${\mathcal{O}}_{Y\cap Z,p}$ a nonzero ring.

The invariant is the one used for strict transforms: after blowing up a closed point, one computes $m_q(Y^{\mathrm{\prime }}\cap Z^{\mathrm{\prime }})$ in the blown-up ambient scheme at a closed point $q$ lying over $p$.

[R1] A zero-dimensional Noetherian local ring has finite length. Let $(A,\mathfrak{m})$ be Noetherian local with $\dim A=0$. Every prime ideal of $A$ is then equal to $\mathfrak{m}$, so the nilradical $\mathrm{Nil}(A)$, which is the intersection of all prime ideals, equals $\mathfrak{m}$; hence $\mathfrak{m}$ is nilpotent, say ${\mathfrak{m}}^N=0$ (The nilradical of a Noetherian ring is nilpotent). This gives a finite filtration $A={\mathfrak{m}}^0\supseteq \mathfrak{m}\supseteq \cdots \supseteq {\mathfrak{m}}^N=0$ by ideals of $A$. Each successive quotient ${\mathfrak{m}}^i\mathrm{/}{\mathfrak{m}}^{i+1}$ is a module over the field $A\mathrm{/}\mathfrak{m}$ that is generated by the images of a finite generating set of the ideal ${\mathfrak{m}}^i$, so it is a finite-dimensional $A\mathrm{/}\mathfrak{m}$-vector space and has finite length; the length of $A$ is the sum of these finitely many lengths (Module length is additive in short exact sequences). No choice principle beyond the ambient definition is used.

[R2] On this page, $Y$ may be the strict transform of a curve and $Z$ another curve component or an exceptional divisor; $Y\cap Z$ is their zero-dimensional contact locus. When both components are regular curves on a regular surface, $m_p(Y\cap Z)\ge 2$ is exactly the tangency that the resolution process must remove: the point-blowup drop lemma compares $m_p(Y\cap Z)$ with the multiplicities $m_q(Y^{\mathrm{\prime }}\cap Z^{\mathrm{\prime }})$ at points of the blown-up surface lying over $p$.