Question

Suppose \(X, Y\) varieties, \(P \in X, Q \in Y\), with \(\mathscr{O}_{P}(X)\) isomorphic (over \(k\) ) to \(\mathscr{O}_{Q}(Y) .\) Then there are neighborhoods \(U\) of \(P\) on \(X, V\) of \(Q\) on \(Y\), such that \(U\) is isomorphic to \(V\). This is another justification for the assertion that properties of \(X\) near \(P\) should be determined by the local ring \(\mathscr{O}_{P}(X)\).

Short answer

Question: Given that the local rings \(\mathscr{O}_{P}(X)\) and \(\mathscr{O}_{Q}(Y)\) are isomorphic over some field \(k\), show that there exist neighborhoods \(U\) of \(P\) on \(X\) and \(V\) of \(Q\) on \(Y\) such that \(U\) is isomorphic to \(V\). Answer: To show that \(U\) is isomorphic to \(V\), we first find affine charts \(\phi: U \to k^n\) and \(\psi: V \to k^m\), and then define the isomorphism \(\Phi: \mathscr{O}_P(X) \to \mathscr{O}_Q(Y)\). Restrict \(\Phi\) to open affine subsets \(U\) and \(V\). Then we construct the morphism \(\psi^{-1} \circ \Phi \circ \phi: k^n \to k^m\) which induces isomorphism between neighborhoods \(U\) and \(V\). Thus, \(U\) is isomorphic to \(V\).

Step-by-step solution

Consider isomorphism and find an affine chart

Firstly, since \(\mathscr{O}_P(X) \cong \mathscr{O}_Q(Y)\), there must exist an isomorphism between a neighborhood of \(P\) in \(X\) and a neighborhood of \(Q\) in \(Y\). Let's find an affine chart \(\phi: U \to k^n\) containing \(P\) on \(X\), and \(\psi: V \to k^m\) containing \(Q\) on \(Y\).

Define the isomorphism and restrict to neighborhoods

Now define the isomorphism \(\Phi: \mathscr{O}_P(X) \to \mathscr{O}_Q(Y)\). Since this is an isomorphism, we can restrict \(\Phi\) to the open affine subsets \(U\) and \(V\) such that \(\Phi: \mathscr{O}(U) \to \mathscr{O}(V)\).

Constructing the isomorphism between U and V

Now, we use the affine charts \(\phi\) and \(\psi\). If we set \(\psi^{-1} \circ \Phi \circ \phi: k^n \to k^m\), then this is a morphism which induces an isomorphism between neighborhoods \(U\) and \(V\). That is, \(\psi^{-1} \circ \Phi \circ \phi(U) = V\).

Conclude that U and V are isomorphic

Since we have constructed an isomorphism between the neighborhoods \(U\) and \(V\), we can conclude that \(U\) is isomorphic to \(V\). This result highlights that the properties of \(X\) near the point \(P\) are indeed determined by the local ring \(\mathscr{O}_{P}(X)\), as stated in the original question.

Key concepts

Affine Varieties

Affine varieties are fundamental objects in algebraic geometry. They are sets of solutions to systems of polynomial equations, which can be visualized as geometric shapes in a defined space. For instance, the familiar parabolas and circles we study in algebra are simple examples. But in algebraic geometry, these shapes exist in multidimensional spaces, opening up a fascinating world of shapes and curves.
An affine variety can be mathematically defined as a subset of an affine space over a field. It is represented by the vanishing of a set of polynomials. In the context of our exercise, affine varieties play a crucial role because they provide a framework to understand the neighborhood structure around a given point on a variety. It helps in conceptualizing how local aspects are tied to global geometry, making them indispensable for assessing isomorphism between varieties.

Isomorphism

In mathematical terms, an isomorphism is a mapping between two structures that shows one can be transformed into the other without any loss of information. It's like a perfectly reversible transformation. If two algebraic structures have an isomorphism between them, they are essentially the same in structure, even if they appear different.
The exercise underlines the importance of isomorphism between local rings to ensure that the properties around a point on one variety can determine those on another variety. When local rings are isomorphic, there exists a corresponding relationship between small neighborhoods of two varieties, ensuring that their algebraic and geometric properties align perfectly, point by point.

Local Rings

Local rings are a fundamental concept in understanding the properties of varieties at specific points. A local ring is a commutative ring with a unique maximal ideal. It can be thought of as capturing the 'local' behavior of a variety at a particular point.
In the problem, the local ring \( \mathscr{O}_P(X) \) at point \( P \) in \( X \) reflects the functions that are defined and analytical around \( P \). The similar structure of \( \mathscr{O}_Q(Y) \) at \( Q \) in \( Y \) implies that the local behaviors around these points can be equated. This equivalence suggests the presence of an isomorphic local structure, which is a critical insight into the relationship between affine varieties at specified locations. The structural analysis of local rings aids in drawing parallels between seemingly disparate varieties.

Affine Charts

Affine charts are like a map that helps us explore and understand specific parts of a variety. Think of an affine chart as a way of "zooming in" on a neighborhood of a variety, allowing it to be studied as an open subset of simple affine space. These charts are instrumental for dealing with complex algebraic structures by breaking them down into simpler, more manageable parts.
In our exercise, affine charts are utilized to isolate neighborhoods \( U \) and \( V \) of points \( P \) and \( Q \), respectively. By establishing isomorphism through these affine charts, we can show a one-to-one correspondence that preserves the algebraic structure. Affine charts thus serve as essential tools for navigating the intricate web of relationships in algebraic geometry, particularly when demonstrating isomorphisms between neighborhoods on complex varieties.