Lecture

Tangent Spaces Insights

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

Description

This lecture delves into the relationships between tangent spaces, dimensions, and sub-varieties in algebraic geometry. It covers the definition of dimension, the connection between the dimension of a variety and its tangent space, and the tangent spaces of sub-varieties. The lecture also explores how derivations induce subspaces within tangent spaces, providing valuable insights into the structure of algebraic varieties.

Official source

Ontological neighbourhood

Mathematics

Geometry: Algebraic geometry

Related concepts (22)

Algebraic variety

Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly.

Morphism of algebraic varieties

In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials. It is also called a regular map. A morphism from an algebraic variety to the affine line is also called a regular function. A regular map whose inverse is also regular is called biregular, and the biregular maps are the isomorphisms of algebraic varieties.

Algebraic geometry

Algebraic geometry is a branch of mathematics which classically studies zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations.

Derived algebraic geometry

Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over ), simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements.

Complex algebraic variety

In algebraic geometry, a complex algebraic variety is an algebraic variety (in the scheme sense or otherwise) over the field of complex numbers. Chow's theorem Chow's theorem states that a projective analytic variety; i.e., a closed analytic subvariety of the complex projective space is an algebraic variety; it is usually simply referred to as a projective variety. Not every complex analytic variety is algebraic, though.

Related lectures (49)

Weil Conjectures: Rationality and Function Equation

Covers the Weil conjectures on rationality, functional equation, and the Riemann hypothesis, exploring properties of varieties in algebraic geometry.

Tangent Spaces in Algebraic Geometry

Explores derivations and the Zariski tangent space in algebraic geometry, emphasizing their importance and practical applications.

Affine Algebraic Sets

Covers affine algebraic sets, hypersurfaces, elliptic curves, ideals, and Noetherian rings in algebraic geometry.

Riemann Zeta Function

Explores the Riemann zeta function, its properties, applications, and analogies in number theory and algebraic geometry.

Lie Algebras: Introduction and Structure