The Hilbert-Samuel polynomial and the multiplicity of a local ring

Since \(A\) is noetherian, the maximal ideal is finitely generated, say \[ \mfm=(x_1,\ldots,x_s). \] The initial forms of the \(x_i\) generate \[ \mfm^q/\mfm^{q+1} \] by degree-\(q\) monomials for every \(q\). Thus \(G\) is a quotient of the standard graded polynomial ring \[ k[X_1,\ldots,X_s] \] with \(X_i\mapsto x_i\bmod\mfm^2\).

The quotient \(A/\mfm^l\) has the finite filtration \[ A/\mfm^l \supset \mfm/\mfm^l \supset\cdots\supset \mfm^{l-1}/\mfm^l \supset0. \] Its successive quotients are \[ \mfm^q/\mfm^{q+1}=G_q, \qquad 0\le q<l. \] Each is a finite-dimensional vector space over the residue field \(k\), hence its \(A\)-module length equals its \(k\)-dimension. Additivity of length gives the formula.

By step 1, \(G\) is a finitely generated standard graded \(k\)-algebra. Hilbert–Serre, equivalently Hartshorne I.7.5 as suggested in the exercise, says that its Hilbert function \[ h_G(q)=\dim_k G_q \] agrees for \(q\gg0\) with a rational polynomial \(Q(q)\).

By step 2, \(\psi(l)\) is the partial sum of this Hilbert function. Finite sums of the polynomial values \(Q(q)\) are polynomial in the upper limit: for each monomial \(q^j\), the sum \[ \sum_{q=0}^{l-1}q^j \] is a polynomial in \(l\) of degree \(j+1\). The finitely many initial values where \(h_G(q)\ne Q(q)\) change only the constant term. Hence a polynomial \(P_A(z)\in\QQ[z]\) satisfies \[ P_A(l)=\psi(l) \] for all \(l\gg0\). This proves part (a).

This is the standard dimension theorem for the associated graded ring of a noetherian local ring. One way to see it is through the extended Rees ring \[ \mathcal R' = A[t^{-1},\mfm t] \subseteq A[t,t^{-1}]. \] The element \(t^{-1}\) is a nonzerodivisor and \[ \mathcal R'/(t^{-1}) \cong \operatorname{gr}_{\mfm}A=G. \] The Rees dimension theorem gives \[ \dim\mathcal R'=\dim A+1; \] quotienting by the homogeneous nonzerodivisor \(t^{-1}\) lowers the dimension by one. Therefore \[ \dim G=\dim A. \]

Put \[ n=\dim A=\dim G \] by step 4.

If \(n=0\), then \(A\) is artinian, so \(\mfm^N=0\) for some \(N\) and \[ \psi(l)=\operatorname{length}(A) \] for \(l\ge N\). Thus \(P_A\) is constant and has degree \(0=n\).

Assume \(n>0\). For a finitely generated standard graded \(k\)-algebra of dimension \(n\), the Hilbert polynomial of its graded pieces has degree \(n-1\). Hence the polynomial \(Q\) in step 3 has degree \(n-1\). Taking partial sums raises the degree by one, so the polynomial \(P_A\) has degree \(n\). This proves part (b).

This is the definition in part (c). For a point \(P\) of a noetherian scheme \(X\), the local multiplicity is consequently \[ \mu_P(X)=\mu(\OO_{X,P}). \] There is no further assertion to prove in part (c).

Because \(X\) is nonsingular of dimension two at \(P\), \(R\) is a regular local ring of dimension two and \[ \operatorname{gr}_{\mfm}R\cong k[u,v]. \] The curve local ring is \[ B=\OO_{C,P}=R/(f), \] with maximal ideal \(\mfm_C=\mfm/(f)\).

The \(\mfm\)-adic order is additive in the regular local ring because its associated graded ring \(k[u,v]\) is a domain. Therefore the initial ideal of the principal ideal \((f)\) is generated by the initial form \(f_r\). Passing to associated graded rings gives \[ \operatorname{gr}_{\mfm_C}B \cong k[u,v]/(f_r). \]

Since \(f_r\ne0\) is homogeneous of degree \(r\), multiplication by \(f_r\) gives an exact sequence of graded \(k[u,v]\)-modules \[ 0 \longrightarrow k[u,v](-r) \xrightarrow{\cdot f_r} k[u,v] \longrightarrow k[u,v]/(f_r) \longrightarrow0. \] Thus the degree-\(q\) piece of the quotient has dimension \[ h(q) = \begin{cases} q+1,&q<r,\\ r,&q\ge r. \end{cases} \] By step 2 applied to the one-dimensional local ring \(B\), \[ \operatorname{length}(B/\mfm_C^l) = \sum_{q=0}^{l-1}h(q). \] For \(l\ge r\) this is \[ \frac{r(r+1)}2+r(l-r) = rl-\frac{r(r-1)}2. \]

The local ring \(\OO_{C,P}\) has dimension one. Step 8 shows that its Hilbert–Samuel polynomial has leading term \[ rz. \] By part (c), its multiplicity is therefore \[ 1!\cdot r=r. \] But \(r\) is exactly the order of a local equation of \(C\) in the maximal ideal of the nonsingular surface, \[ f\in\mfm^r\setminus\mfm^{r+1}, \] which is the multiplicity definition used in the text before (3.5.2). This proves part (d).

Write the projective cone in \(\PP^{N+1}\) with coordinates \[ [x_0:\cdots:x_N:z] \] and the same homogeneous equations \(I_Y\) in the \(x_i\). Its vertex is \[ V=[0:\cdots:0:1]. \] On the affine chart \(z\ne0\), the cone is the affine cone \[ \Spec S, \] and \(V\) corresponds to the homogeneous maximal ideal \[ \mfm=(x_0,\ldots,x_N)S. \] Hence \[ \OO_{V,\operatorname{Cone}(Y)}=S_{\mfm}. \]

Because \(S\) is standard graded, \[ \mfm^q/\mfm^{q+1}\cong S_q. \] Localizing at \(\mfm\) does not change these finite-dimensional residue-field vector spaces, so the associated graded ring of the local ring is \(S\).

For \(q\gg0\), the Hilbert function of the homogeneous coordinate ring is the Hilbert polynomial of \(Y\): \[ \dim_k S_q = \frac{d}{s!}q^s+\text{lower-degree terms} \] D-L6ERW. By steps 2 and 10, the Hilbert–Samuel function at the vertex is the partial sum \[ \sum_{q=0}^{l-1}\dim_kS_q. \] Its leading term is therefore \[ \frac{d}{(s+1)!}l^{s+1}. \] The local ring at the vertex has dimension \(s+1\), so part (c) gives \[ \mu_V(\operatorname{Cone}(Y)) =(s+1)!\frac{d}{(s+1)!} =d. \] This proves part (e).

Steps <1>1–<1>3 prove part (a), steps <1>4–<1>5 prove part (b), step 6 records the definition in part (c), steps <1>7–<1>9 prove part (d), and steps <1>10–<1>11 prove part (e).