Spitzenkörper

Spitzenkörper is a term used in mathematics, particularly in algebraic geometry, to describe a specific type of algebraic object. It can be understood as a generalization of a vector bundle. Informally, a Spitzenkörper can be thought of as a "vector bundle on a space with singularities." More precisely, it is a coherent sheaf on a scheme, often with additional properties that make it behave like a vector bundle in some situations.

The concept of a Spitzenkörper arises when studying algebraic varieties that are not smooth. In such cases, traditional vector bundles might not be well-behaved. Spitzenkörper provide a framework to extend the notion of vector bundles to these more general settings. They play a role in understanding the geometry of singular varieties, including their tangent spaces and related structures.

The formal definition of a Spitzenkörper can vary slightly depending on the context, but it generally involves a coherent sheaf that satisfies certain conditions. These conditions often relate to the sheaf's behavior near the singular points of the underlying scheme. The study of Spitzenkörper is an advanced topic within algebraic geometry, contributing to a deeper understanding of the structure and properties of algebraic varieties.