COHERENT SHEAF Meaning and
Definition
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A coherent sheaf, in the field of mathematics called algebraic geometry, refers to a mathematical object that captures the behavior of an algebraic variety using tools from sheaf theory. A sheaf, in general, is a mathematical structure that associates, for each open set of a topological space, a collection of mathematical objects called sections that satisfy certain compatibility conditions.
A coherent sheaf, specifically, is a sheaf that smoothly varies over the open sets of a given topological space. In the context of algebraic geometry, the underlying space is typically a variety defined over a field, which is a geometric object described by polynomial equations. Coherent sheaves reflect the coherent nature of the algebraic equations defining the variety.
Formally, a coherent sheaf is a sheaf with additional properties. Firstly, it is finitely generated, which means that it is constructed from a finite number of sections. Secondly, it is of finite type, indicating that certain local descriptions of the sheaf can be preserved. Finally, a coherent sheaf satisfies a condition known as the coherence condition, ensuring that its sections behave well under restriction to smaller open sets.
Coherent sheaves serve as fundamental building blocks in algebraic geometry, allowing the study of geometric properties of varieties through the lens of sheaf theory. By applying tools from commutative algebra, these sheaves enable precise investigations of the geometry and topology of algebraic varieties, contributing to the rich and fascinating field of algebraic geometry.
Etymology of COHERENT SHEAF
The word "coherent" in the context of sheaves originates from algebraic geometry. It comes from the French word "coherent", meaning "coherent" or "logical". In this mathematical context, a sheaf is said to be coherent if it satisfies certain finiteness conditions, which ensure that local information can be patched together consistently. This notion was introduced by Jean-Pierre Serre in the 1950s.
The term "sheaf" itself comes from the German word "Garbe", meaning "bundle" or "sheaf". The concept of sheaves was developed by mathematicians such as Jean Leray and Henri Cartan in the 1940s as a way to generalize the concept of a function defined on a topological space. They are fundamental objects in algebraic geometry and are extensively used in many areas of mathematics.
