Contact order of two regular components at a point

Assume the Axiom of Choice (The Axiom of Choice), inherited from the regular-local prerequisites. Let $S$ be a regular Noetherian scheme of pure dimension two over a field $k$ (embedding dimension and regular local ring, Left and right Noetherian rings, Chain dimension and the empty-space convention). Let $Y,Z\subseteq S$ be reduced closed subschemes of pure dimension one, with no common irreducible component, and suppose every branch of either curve through the chosen closed point $p$ is regular at $p$. Write $A={\mathcal{O}}_{S,p}$ and $I_Y,I_Z\subseteq A$ for their germ ideals. If $p$ misses either curve, set $n_p(Y,Z)=0$. Otherwise the contact order is $n_p(Y,Z)={\mathrm{length}}_{A\mathrm{/}{I}_Y}(A\mathrm{/}(I_Y+I_Z))\mathrm{.}$ Length means composition-series length (Composition series and length of a module).

Here the required local dimension follows from the geometry, rather than from the global dimension alone. A closed point of a pure one-dimensional Noetherian curve has local dimension one: a zero-dimensional local ring would make it the generic point of a zero-dimensional component, since the point is closed. At a closed point on $Y$, the ambient regular local ring cannot have dimension zero. If it had dimension one, it would be a DVR (one dimensional regular local rings are dvrs), and a branch prime with one-dimensional quotient would be zero. The closed curve would then contain the generic point, hence the whole two-dimensional ambient component, contradicting its pure dimension one. Thus $\dim A=2$ at every actual contact point. Generic local rings of $S$ need not have dimension two.

The displayed length is finite. The minimal primes of $A\mathrm{/}{I}_Y$ are the branch primes of $Y$ through $p$. No common component means that $I_Z$ is contained in none of these primes: an inclusion would make a one-dimensional branch of $Y$ a component of $Z$. Thus the quotient has no generic point of a curve branch in its support, and its support is only the closed point. A finite module over a Noetherian local ring with this support has finite length. This uses noncontainment in every branch prime, not the weaker assertion that the image ideal is merely nonzero.

Local equations. A regular branch prime $P\subset A$ is principal. Indeed, its regular quotient $A\mathrm{/}P$ has cotangent dimension one, so choose $f\in P$ with nonzero class in $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2$. Then $A\mathrm{/}(f)$ is regular of dimension one (regular local quotient by parameter is regular) and hence a DVR. The prime $P\mathrm{/}(f)$ must be zero since its quotient still has dimension one. Therefore $P=(f)$, with $f$ a prime element of the regular local domain $A$ (regular local rings are domains and cohen macaulay). The reduced curve ideal is the intersection of its finitely many distinct branch primes, so it is their product: if an element divisible by a product of some distinct prime elements is also divisible by a new prime element, primality forces divisibility of its remaining factor by that new element. Induction gives the intersection/product equality. Consequently $I_Y=(y)$ and $I_Z=(z)$, with $y,z$ products of the respective branch equations, and $n_p(Y,Z)={\mathrm{length}}_{A\mathrm{/}(y)}(A\mathrm{/}(y,z))\mathrm{.}$ These nonzero equations are regular sections; they define effective Cartier data. Changing an equation by a unit does not change the quotient or its length (Effective cartier divisor, Cartier divisor, Cartier divisor local equation equivalence).

Symmetry and transversality. The length equals the length of $A\mathrm{/}(I_Y+I_Z)$ as an $A$-module, and similarly as an $A\mathrm{/}{I}_Z$-module, since all composition factors are the same residue field. Thus contact is symmetric. It equals one precisely when $(y,z)=\mathfrak{m}$: a nonzero local quotient has length one precisely when it is the residue field. In that case the classes of $y,z$ form a basis of the two-dimensional cotangent space. Their regular parameter quotients are one-dimensional regular local rings, and their tangent lines are distinct. Conversely, if both curve germs are regular and their tangent lines are distinct, their equations have independent cotangent classes and generate $\mathfrak{m}$ by Nakayama (Assuming the Axiom of Choice, generators modulo an ideal in the Jacobson radical lift to generators). Hence their contact is one. This is exactly transversal meeting at $p$. If either curve has at least two branches through $p$, its product equation lies in ${\mathfrak{m}}^2$; its cotangent class cannot be part of a parameter basis, so the positive contact length is at least two, even if individual pairs of branches have distinct tangents.

The total contact order is $n(Y,Z)={\sum }_{p\in Y\cap Z}{n}_p(Y,Z)\mathrm{.}$ The intersection is a zero-dimensional closed subscheme of a Noetherian scheme because there is no common component, so it has finitely many closed points. The sum is therefore a finite nonnegative integer. This definition includes all regular finite-type surface cases and uses no perfectness or rationality assumption on the residue fields.