91Ó°ÊÓ

Glueing Lemma. Generalize the glueing procedure described in the text \((2.3 .5)\) as follows. Let \(; X_{i}\); be a family of schemes (possible infinite). For each \(i \neq j\). suppose given an open subset \(U_{i j} \subseteq X_{i}\), and let it have the induced scheme structure (Ex. 2.2 ). Suppose also given for each \(i \neq j\) an isomorphism of schemes \(\varphi_{i j}: U_{i j} \rightarrow U_{l^{\prime}}\) such that (1) for each \(i, j, \varphi_{i j}=\varphi_{11}^{-1},\) and (2) for each \(i, j, k\).\(\varphi_{i j}\left(U_{i j} \cap U_{i k}\right)=U_{j i} \cap C_{j k},\) and \(\varphi_{i k}=\varphi_{j k} \quad \varphi_{i j}\) on \(U_{i j} \cap L_{i k} .\) Then show that there is a scheme \(X\), together with morphisms \(\psi_{1}: X_{1} \rightarrow X\) for each \(i,\) such that (1) \(\psi_{i}\) is an isomorphism of \(X\), onto an open subscheme of \(X,(2)\) the \(\psi,(X,)\) cover \(X,(3) \psi_{i}\left(U_{i j}\right)=\psi_{i}\left(X_{i}\right) \cap \psi_{j}\left(X_{j}\right)\) and \((4) \psi_{i}=\psi_{j} \quad \varphi_{i j}\) on \(U_{i j}, \quad\) We say that \(X\) is obtained by glueing the schemes \(X_{i}\) along the isomorphisms \(\varphi_{y} .\) An interesting special case is when the family \(X_{i}\) is arbitrary, but the \(U_{i j}\) and \(\varphi_{i j}\) are all empty. Then the scheme \(X\) is called the disjoint union of the \(X_{1},\) and is denoted ||\(X\).

Step by step solution

01

Introduction to variables

First, identify each variable: we have a family of schemes \(X_i\), open subsets \(U_{ij}\) within \(X_i\), and isomorphisms \(\varphi_{ij}: U_{ij}\rightarrow U_{li^\prime}\) that meet certain conditions.

02

Proving the existence of X

We need to prove that there is a scheme \(X\). Start by considering the disjoint union \(\sqcup X_i\) of all \(X_i\). Suppose we have an equivalence relation \(\sim\) on \(\sqcup X_i\) such that for any \(x\in U_{ij} \subseteq X_i, y\in\varphi_{ij}(U_{ij}) \subseteq X_j, x \sim y\) if and only if \(\varphi_{ij}(x)=y\). It can be shown that \(X =\sqcup X_i/\sim\), and \(\psi_i: X_i \rightarrow X\) can be defined as the quotient map, thus proving the existence of \(X\).

03

Proving the properties of X

Now we have to prove four properties of \(X\). For the first property, it follows directly from the construction that \(\psi_i\) is an isomorphism of \(X_i\) onto an open subscheme of \(X\). For the second property, the cover of \(X\) by \(\psi_i(X_i)\) also follows directly from the construction. For the third property, note that \(\psi_i(U_{ij})\neq\emptyset\) if and only if \(U_{ij}\cap U_{ji}\neq\emptyset\), which implies \(\psi_i(X_i)\cap\psi_j(X_j)\neq\emptyset\). Finally, for the fourth property, \(\psi_i = \psi_j \varphi_{ij}\) on \(U_{ij}\) by construction.

04

Considering the special case

When the \(U_{ij}\) and \(\varphi_{ij}\) are empty, the scheme \(X\) is the disjoint union of the \(X_i\), denoted by \(\sqcup X_i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Schemes in Algebraic Geometry

Understanding schemes is crucial in algebraic geometry as they extend the concept of algebraic varieties to accommodate singularities and complex structures. A scheme is a mathematical construct that generalizes varieties by patching together local data from the spectrum of a ring, known as affine schemes. Think of schemes as the "glue" that holds components together in a geometric space, providing a coherent structure even in areas where singularities might occur.

Schemes are not limited to describing simple curves but can model more elaborate structures, including higher-dimensional varieties. By working with the concepts of open subsets, gluing data, and morphisms, schemes enable mathematicians to tackle more complex and subtle properties of geometric objects. The idea of covering a scheme by smaller, more manageable open affines simplifies understanding of their global properties through local inspections.

In this way, schemes serve as a bridge between algebraic and geometric viewpoints, enriching both fields by allowing algebraic techniques to explore geometric properties and vice versa, thus revealing the deep interplay between these mathematical realms.

Isomorphisms of Schemes

An isomorphism in schemes is a critical concept in algebraic geometry, where it denotes a bi-directional morphism that preserves the structure of schemes. Isomorphisms ensure that two schemes are algebraically the same, meaning they have identical properties and structures when viewed as geometric objects.

To establish an isomorphism, one must find a morphism from one scheme to another and a reverse morphism such that both compositions (going out and back) yield the identity on each scheme. This not only guarantees that there's no loss of information but also that all intrinsic properties like topology and algebraic structure are preserved.

The role of isomorphisms is instrumental when considering the gluing lemma, where open subsets of schemes are stitched together. The isomorphisms act as the formal correspondences between these pieces, ensuring compatibility and coherence in the larger structure being assembled. Familiarity with this concept allows for deeper explorations into the transformative properties of complex geometric spaces.

Disjoint Union of Schemes

The disjoint union in the context of schemes is a method that constructs a new scheme from a collection of individual schemes without overlap. It's a straightforward concept but incredibly powerful when considering complex compositions of geometric entities.

Imagine having multiple geometric figures that you wish to consider as parts of a larger entity without altering their original properties or interactions. The disjoint union becomes the tool that keeps these individual schemes distinct yet part of a single comprehensive scheme.

In formal terms, the disjoint union of schemes refers to taking an arbitrary set of schemes and creating a "sum" scheme where each component scheme remains unchanged in its structure and identity, retaining their independence.

• This concept is particularly useful in algebraic geometry as it allows flexible manipulations of schemes without losing their unique characteristics.
• It's also pivotal in understanding more advanced constructs, like glued schemes, where the disjoint union provides a baseline from which we can build even more interconnected systems.

By mastering the idea of disjoint unions, students can better appreciate how algebraic geometry systematically integrates diverse scheme components into harmonious wholes.