Local analytic geometry

Algebraic geometry focuses on the study of zero sets of polynomials over an algebraically closed field. Developing the basics of this field requires a solid understanding of commutative algebra, which can be challenging without the geometric insights that often inform its concepts. Local analytic geometry, on the other hand, examines germs of zero sets of analytic functions near a point. The foundational theories of local analytic and algebraic geometry share many similarities, suggesting that a simultaneous development of both could be beneficial. However, this book does not follow that path, as the commutative algebra necessary for local analytic geometry is more complex due to convergence issues. A key example of this complexity is the Weierstrass Division Theorem, which replaces the division with remainder concept in local analytic contexts. To address these themes, the first four chapters are structured to introduce the necessary algebra. Chapter 1 focuses on the algebraic foundations, assuming the reader has prior knowledge of linear algebra and abstract algebra, including some exposure to Galois theory. This approach sets the stage for a deeper exploration of both algebraic and local analytic geometry.