Coprime tangent cones force a power of the maximal ideal into the local ideal

Given: AC, an algebraically closed field $k$, the local ring $O$ of ${\mathbf{A}}^2$ at the origin, its maximal ideal $\mathfrak{m}=(x,y)$, elements $f,g\in \mathfrak{m}$ with orders $m,n\ge 1$, and their lowest-degree forms $f^{\ast },g^{\ast }$ (the initial forms in the $\mathfrak{m}$-adic filtration).

$O$ is a two-dimensional regular local ring; its associated graded ring is ${\mathrm{gr}}_{\mathfrak{m}}O\cong k[X,Y]$ with standard grading, in particular a domain, so initial forms multiply: $(h_1{h}_2{)}^{\ast }=h_1^{\ast }{h}_2^{\ast }$ associated graded ring of a regular local ring, embedding dimension and regular local ring, A local ring is a nonzero commutative ring with a unique maximal ideal, The Axiom of Choice.

$k[x,y]$ is a unique factorisation domain and every nonzero homogeneous form of degree $d$ in two variables has a $k$-basis of the $d+1$ monomials of total degree $d$, so ${\dim }_{k}k[x,y{]}_d=d+1$ Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes, Monomials, coefficients, degree in each variable and total degree in $F[x_1,\dots ,x_n]$, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Vector space over a field.

For coprime forms $f^{\ast },g^{\ast }$ of degrees $m,n$ the graded multiplication map $k[x,y{]}_{t-m}\oplus k[x,y{]}_{t-n}\to k[x,y{]}_t$, $(A,B)\mapsto A{f}^{\ast }+B{g}^{\ast }$, is surjective whenever $t\ge m+n-1$. Indeed, if $A{f}^{\ast }+B{g}^{\ast }=0$, then $f^{\ast }\mid B$ and $g^{\ast }\mid A$ by coprimality in the UFD, so $(A,B)=(g^{\ast }h,-f^{\ast }h)$ for $h\in k[x,y{]}_{t-m-n}$ when $t\ge m+n$, and there is no nonzero syzygy when $t=m+n-1$; by [F2] and rank-nullity the kernel has dimension $\max (0,t-m-n+1)$, while the domain has dimension $2t-m-n+2$ and the target has dimension $t+1$, so surjectivity follows for $t\ge m+n-1$ Rank-nullity: ${\dim }_{F}V=\mathrm{nullity}T+\mathrm{rank}T$, Module homomorphism and isomorphism, kernel, image and cokernel, Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis.

Coprime lowest-degree forms force $f,g$ to be coprime in $O$: if a nonunit irreducible $h\in O$ divided both, then $h$ has order $\ge 1$ and $h^{\ast }$ is a nonconstant form, and by [F1] $h^{\ast }\mid f^{\ast }$ and $h^{\ast }\mid g^{\ast }$, contradicting coprimality Irreducible and prime elements of an integral domain, Prime ideals and maximal ideals in a commutative ring. Clear the unit denominators of $f,g$ to apply Finite local length exactly when no common local branch to polynomial numerators. Their ideal is $\mathfrak{m}$-primary; Proof 1.3 of that lemma gives ${\mathfrak{m}}^N\subseteq (f,g)O$ for some $N$.

AC is assumed; it enters through the associated-graded, Nullstellensatz and primarity suppliers The Axiom of Choice, Classical affine algebraic sets correspond to radical ideals, and irreducible sets to prime ideals. The orders and initial forms are those of Multiplicity of a plane curve at a point, Tangent cone and tangent lines at a point, projective hypersurface affine pieces.