Lineaarskeemen

Lineaarskeemen, also known as linear schemes, are a type of mathematical structure used in algebraic geometry and commutative algebra. They were introduced by Grothendieck in the 1960s as a generalization of schemes, which are themselves generalizations of algebraic varieties. A linear scheme is defined as a pair (X, O_X), where X is a topological space and O_X is a sheaf of commutative rings on X, satisfying certain conditions that ensure it behaves like a scheme but with additional linear structure.

The defining feature of a linear scheme is the presence of a linear structure on the sheaf of rings O_X. This means that for any open subset U of X, the ring O_X(U) is a module over a fixed commutative ring R, and the restriction maps O_X(U) -> O_X(V) for U ⊆ V are R-linear. This additional structure allows for the development of linear algebra techniques in the study of linear schemes.

Linear schemes have applications in various areas of mathematics, including algebraic geometry, commutative algebra, and representation theory. They provide a unified framework for studying linear algebraic structures and have been used to prove important theorems and develop new techniques in these fields. Some notable examples of linear schemes include linear algebraic groups, linear algebraic varieties, and linear schemes over a field.

In summary, lineaarskeemen are a type of mathematical structure that generalizes schemes by incorporating a linear structure on the sheaf of rings. They have applications in algebraic geometry, commutative algebra, and representation theory, and provide a unified framework for studying linear algebraic structures.