Question

(a) Let \(\mathscr{O}=\mathscr{O}_{P}\left(\mathbb{A}^{2}\right)\) for some \(P \in \mathbb{A}^{2}, \mathfrak{m}=\mathfrak{m}_{P}\left(\mathbb{A}^{2}\right) .\) Calculate \(\chi(n)=\operatorname{dim}_{k}\left(\mathscr{O} / \mathrm{m}^{n}\right)\). (b) Let \(\mathscr{O}=\mathscr{O}_{P}\left(\mathbb{A}^{r}(k)\right)\). Show that \(\chi(n)\) is a polynomial of degree \(r\) in \(n\), with leading coefficient \(1 / r !\) (see Problem 2.36).

Short answer

(b) In part (b) of the solution, we used induction to prove that the dimension of the quotient \(\mathscr{O}/\mathfrak{m}^n\) is a polynomial of degree \(r\) in \(n\) with leading coefficient \(1/r!\) for \(P \in \mathbb{A}^r(k)\). State the formula for the Hilbert function \(\chi(n)\) in terms of \(r\).

Step-by-step solution

Set up the problem

We are given that \(\mathscr{O} = \mathscr{O}_P(\mathbb{A}^2) \) and \(\mathfrak{m} = \mathfrak{m}_P(\mathbb{A}^2)\). We need to compute the dimension of the quotient \(\mathscr{O}/\mathfrak{m}^n\).

Understand the structure of \(\mathscr{O}_P(\mathbb{A}^{2})\) and \(\mathfrak{m}_P(\mathbb{A}^{2})\)

\(\mathscr{O}_P(\mathbb{A}^2)\) is the localization of the coordinate ring \(k[x, y]\) at the prime ideal \(\mathscr{P}_P\), so \(\mathscr{O}_P(\mathbb{A}^2)= k[x,y]_{\mathscr{P}_P}\). Then the maximal ideal \(\mathfrak{m}_P(\mathbb{A}^2)\) consists of all fractions \(\dfrac{f}{g} \in\mathscr{O}_P(\mathbb{A}^2)=k[x,y]_{\mathscr{P}_P}\) with \(f, g \in k[x, y]\) and \(g \notin \mathscr{P}_P\) such that \(f(0,0) = 0\).

Determine the basis for the quotient

Consider the polynomial ring \(k[x, y]\). Let \(f \in k[x, y]\) be a polynomial of degree \(n\). Observe that its leading monomials are of the form \(x^{n-j}y^j\) for \(j=0,1,2,\ldots,n\). Therefore, the possible monomials for a polynomial of degree less than or equal to \(n\) are of the form \(x^iy^j\), where \(i+j \leq n\). These monomials form a basis for \(k[x, y]/(x, y)^{n+1}\). Next, the elements of \(\mathfrak{m}^n\) are of the form \(\dfrac{f}{g}\), where \(f\in(x,y)^n\) and \(g\notin (x,y)\). There are \(n\) basis elements of \(k[x,y]/(x,y)^{n+1}\) going from the diagonal to the left in the following pattern: $$ x^ny^0, x^{n-1}y^1, x^{n-2}y^2, \ldots, x^0y^n $$ Hence, the dimension of the quotient \(\mathscr{O}/\mathfrak{m}^n\) is \(n+1\). Therefore, \(\chi(n)=\operatorname{dim}_{k}\left(\mathscr{O}/\mathfrak{m}^n\right)=n+1\). (b)

Set up the problem

We are given that \(\mathscr{O} = \mathscr{O}_P(\mathbb{A}^r)\) and now we need to show that the dimension of the quotient \(\mathscr{O}/\mathfrak{m}^n\) is a polynomial of degree \(r\) in \(n\) with leading coefficient \(1/r!\).

Use induction to derive the general formula

We will use induction on \(r\). We have already proved the base case for \(r=2\) in part (a). Suppose that the formula holds true for \(r=k\), i.e., \(\chi_k(n) = \frac{1}{k!}\left(n^k + \text{lower order terms}\right)\). Now we need to show that it holds for \(r=k+1\).

Calculate the dimension of the quotient for \(r=k+1\)

Consider a polynomial of degree less than or equal to \(n\) in the polynomial ring \(k[x_1, \ldots, x_{k+1}]\). This polynomial can be written as a linear combination of monomials \(x_1^{i_1}\cdots x_{k+1}^{i_{k+1}}\) with \(\sum_{j=1}^{k+1}i_j \leq n\). It is known that the number of basis elements for this space is the binomial coefficient \(\binom{n+k}{k}\). By the induction hypothesis, \(\chi_{k+1}(n) - \chi_k(n) = \binom{n+k}{k} - \frac{1}{k!}\left(n^k + \text{lower order terms}\right)\). By induction, we need to show that: $$ \chi_{k+1}(n) = \frac{1}{(k+1)!}\left(n^{k+1}+\text{lower order terms}\right) $$ We will proceed by using an algebraic trick.

Use an algebraic trick to derive the formula

Observe that \(\binom{n+k}{k} = \frac{n(n-1)\cdots(n-k+1)}{k!}\). We can write \(\binom{n+k}{k} - \frac{1}{k!}\left(n^k + \text{lower order terms}\right)\) as: $$ \frac{n(n-1)\cdots(n-k+1)}{k!}-\frac{1}{k!}\left(n^k+\text{lower order terms}\right) $$ $$ \frac{n^{k+1}+\text{lower order terms}}{k!} $$ Multiplying by a factor of \((k+1)\), we obtain: $$ \chi_{k+1}(n) = \frac{1}{(k+1)!}\left(n^{k+1}+\text{lower order terms}\right) $$ which is what we wanted to show. Therefore, the result holds for \(r=k+1\) by induction.

Key concepts

Dimension of Quotient

Understanding the dimension of a quotient space in algebraic geometry is essential for grasping more complex topics. When we talk about the dimension of the quotient \( \mathscr{O} / \mathfrak{m}^n \), we're counting the number of linearly independent elements that can exist in this space.
In our specific problem, we determine the dimension by finding a basis for the quotient \(k[x, y]/(x, y)^{n+1}\) and counting the number of monomials in it, which ends up being \(n+1\), reflecting the polynomial's degree and the binomial coefficient. This conceptual understanding also lays the groundwork for tackling more complex dimensions in higher-dimensional spaces.

Localization of Coordinate Ring

Localization is a process that allows us to focus on the properties of algebraic objects at a particular point or set of points. For a point \(P\) in the affine space \(\mathbb{A}^2\), the localization of the coordinate ring, often denoted as \( \mathscr{O}_P(\mathbb{A}^2) \), is essentially the coordinate ring \(k[x, y]\) but with an altered structure that takes into account the localization at a prime ideal. This provides a magnified look at the algebraic structure near that point, offering a way to understand the behaviour of functions within a local neighbourhood.

Maximal Ideal

In algebraic structures, an ideal is a subset that retains certain characteristics that align with the ring it's part of.
A maximal ideal is particularly interesting because it's an ideal that is not contained in any larger ideal (except for the ring itself), which indicates that it plays a significant role in the underlying algebraic structure.
In our case, the maximal ideal \(\mathfrak{m}_P(\mathbb{A}^2)\) relates to the point \(P\), embodying functions that vanish at \(P\). This concept is pivotal in defining local properties and facilitating a deeper understanding of algebraic curves at specific points.

Polynomial Degree

The degree of a polynomial is a fundamental concept in algebra that refers to the highest power of the variable in the expression.
It's crucial in determining the polynomial's properties, such as the number of roots it might have or its end behaviour. Investigating the polynomial degree connects closely to understanding the structure of the function it represents.
In our exercise, the degree dictated the number of basis elements in the space and related directly to the dimension of the quotient, showcasing the intersection of algebraic and geometric reasoning in the problem-solving process.

Binomial Coefficient

The binomial coefficient, symbolically represented as \( \binom{n}{k} \), quantifies the number of ways to choose \(k\) elements from a set of \(n\) items without regard to the order of selection.
An essential element in combinatorics, it also arises in algebraic contexts, such as in our solution where it's used to count the number of monomials of a certain degree. This number then feeds into the determination of the dimension of our quotient space — illustrating that these coefficients are more than just combinatorial tools; they have important algebraic implications.