Question

Let \(P=(0, \ldots, 0) \in \mathbb{A}^{n}, \mathscr{O}=\mathscr{O}_{P}\left(\mathbb{A}^{n}\right), \mathfrak{m}=\mathfrak{m}_{P}\left(\mathbb{A}^{n}\right)\). Let \(I \subset k\left[X_{1}, \ldots, X_{n}\right]\) be the ideal generated by \(X_{1}, \ldots, X_{n}\). Show that \(I \mathscr{O}=m\), so \(I^{r} \mathscr{O}=m^{r}\) for all \(r\).

Short answer

Based on the given information, we have shown that the product of ideal \(I\) and local ring \(\mathscr{O}\) of an affine space \(\mathbb{A}^n\) at a point \(P\) is equal to the maximal ideal \(\mathfrak{m}\). Moreover, we have proven that \(I^r \mathscr{O} = \mathfrak{m}^r\) for any integer \(r\). This result demonstrates the relationship between ideals, local rings, and maximal ideals in the context of affine spaces.

Step-by-step solution

Recall the definitions of the local ring and maximal ideal

Recall that the local ring \(\mathscr{O}_P(\mathbb{A}^n)\) is the set of all rational functions \(f/g\) where \(f, g \in k[X_1, \ldots, X_n]\) and \(g(P) \neq 0\). The maximal ideal \(\mathfrak{m}_P(\mathbb{A}^n)\) is the set of all elements in \(\mathscr{O}_P(\mathbb{A}^n)\) that vanish at \(P\), i.e., those elements of the form \(\frac{f}{g}\) with \(f(P) = 0\) and \(g(P) \neq 0\).

Recall the definition of the ideal \(I\)

The ideal \(I\) is generated by the variables \(X_1, \ldots, X_n\), which means that any element in \(I\) can be written as a linear combination of these variables with coefficients in the field \(k\). In other words, \(I = \{ \sum_{i=1}^{n} c_i X_i \mid c_i \in k\}\).

Compute the product of \(I\) and \(\mathscr{O}\)

The product of the ideal \(I\) and the local ring \(\mathscr{O}\), denoted by \(I\mathscr{O}\), is the set of all elements of the form \(a \cdot \frac{f}{g}\) where \(a \in I\) and \(\frac{f}{g} \in \mathscr{O}\). Let's take an arbitrary element of \(I\mathscr{O}\): \(a \cdot \frac{f}{g} = \left(\sum_{i=1}^{n} c_i X_i\right) \cdot \frac{f}{g} = \frac{\left(\sum_{i=1}^{n} c_i X_i\right) \cdot f}{g}\). Notice that since \(a \in I\), \(a(P) = 0\), and thus \(a \cdot \frac{f}{g}(P) = 0\). Hence, \(I\mathscr{O} \subseteq \mathfrak{m}\).

Show that \(I\mathscr{O} = \mathfrak{m}\)

To show that \(I\mathscr{O} = \mathfrak{m}\), we need to show that \(\mathfrak{m} \subseteq I\mathscr{O}\). Since \(I \subseteq k[X_1, \ldots, X_n]\), it's clear that \(\mathfrak{m} \subseteq I\mathscr{O}\). Hence, \(I\mathscr{O} = \mathfrak{m}\).

Prove that \(I^{r} \mathscr{O}=m^{r}\) for all \(r\)

To show the general case, we will proceed by induction. The base case \(r=1\) has already been proven in Steps 3 and 4. Now, assume that \(I^k \mathscr{O} = \mathfrak{m}^k\) for some integer \(k\). We need to show that \(I^{k+1} \mathscr{O} = \mathfrak{m}^{k+1}\). Observe the following: \(I^{k+1} \mathscr{O} = I^k I \mathscr{O} = (I^k \mathscr{O})(I \mathscr{O}) = \mathfrak{m}^k \mathfrak{m} = \mathfrak{m}^{k+1}\). The last equality follows because the product of two elements of a maximal ideal is also in the maximal ideal. Therefore, the induction step has been proven, and we can conclude that \(I^{r} \mathscr{O}=\mathfrak{m}^{r}\) for all \(r\).

Key concepts

Local Ring

In algebraic geometry and commutative algebra, a local ring is a ring that has a unique maximal ideal. This concept is crucial when studying algebraic curves and other geometric structures, as local rings reflect the properties of functions near a point on a curve or space. In the exercise, we focus on the local ring \( \mathscr{O}_P(\mathbb{A}^n) \) at the point P on an affine space \( \mathbb{A}^n \).

Imagine this as narrowing down our view to just the neighborhood around the point P, and looking at all possible rational functions we can create that are defined at P. These functions, \( f/g \), where neither the numerator \( f \) nor the denominator \( g \) is zero at P, form the local ring. The 'local' aspect comes into play because we are not interested in the behavior of these functions everywhere, just at the specific point P or in a small region around it. By focusing on the local perspective, mathematicians can gain deep insights into the structure and properties of algebraic curves right at P.

Maximal Ideal

A maximal ideal in a ring is an ideal that is maximal with respect to the inclusion relation among proper ideals, meaning that it is contained in no other proper ideal of the ring. In the context of the exercise, \( \mathfrak{m}_P(\mathbb{A}^n) \) is the maximal ideal of the local ring \( \mathscr{O}_P(\mathbb{A}^n) \).

The elements of \( \mathfrak{m}_P(\mathbb{A}^n) \) are all the rational functions that vanish at the point P, which is akin to looking at behaviors that 'disappear' or have no effect when we are precisely at P. Learning about maximal ideals helps students understand the concept of 'vanishing' at a point—which is a powerful idea in studying the topology and geometry of spaces. In algebraic terms, this can be seen as a condition that helps us isolate where a function has a specific kind of behavior—such as where it might have a root or a singularity—and to work within the confines of that behavior.

Inductive Proof

The inductive proof is a mathematical technique used to prove a statement is true for all natural numbers. It consists of two steps: showing the base case holds, and then proving the inductive step—that if the statement holds for one case, it also holds for the next. This method is applied in our exercise to show that \( I^{r} \mathscr{O}=\mathfrak{m}^{r} \) for all positive integers \( r \).

The idea fits beautifully into the fabric of algebraic curves and their local rings and ideals. By demonstrating that the concept works for \( r=1 \), we create a strong foundation. Then, we leverage the property that in a maximal ideal, the product of any two elements also belongs to the ideal. By assuming the statement for \( k \) and showing it holds for \( k+1 \) (the inductive step), we firmly establish its validity for every natural number. This sort of argument is a powerful way to handle infinite cases with finite work and is a cornerstone of higher mathematics, especially in algebra.