SUBSCHEME Meaning and
Definition
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A subscheme refers to a concept used in algebraic geometry. It is a fundamental object that arises when studying algebraic varieties or schemes. In this context, a subscheme represents a geometric structure that is contained within a larger geometric structure.
More precisely, given a scheme or an algebraic variety, a subscheme is a subset of it that possesses its own structure of a scheme or variety. This means that a subscheme not only consists of its underlying points, but it also includes additional data such as its own coordinate ring, specifying how functions on the subscheme are defined.
To qualify as a subscheme, the underlying set must not only be closed under the topological structure induced by the larger geometry, but it must also be equipped with its own algebraic structure. This algebraic structure is described by an ideal sheaf, which encodes how equations and functions on the subscheme interact.
Subschemes are essential in algebraic geometry as they allow for a deeper understanding and analysis of the overall geometry. By studying the substructures within a larger scheme or variety, mathematicians can uncover properties and relationships that would not be immediately apparent from the global perspective. Subschemes play a vital role in classifying and characterizing geometric objects, allowing for a more nuanced comprehension of complex geometrical spaces.
Etymology of SUBSCHEME
The word "subscheme" is derived from the combination of two words: "sub" and "scheme".
1. "Sub" is a prefix derived from Latin and means "under" or "below". It is commonly used in English to indicate something smaller, subordinate, or lower in rank or position. For example, in words like "subway", "subordinate", or "submarine".
2. "Scheme" comes from the Old French word "esqueme" which means "figure" or "shape". The term evolved from the Latin word "schema", which signifies a "plan" or "scheme".
When combined, "sub" and "scheme" form the word "subscheme", which generally refers to a smaller or subordinate part of a larger system or plan. In the context of mathematics, a subscheme is a subset of a scheme with its own structural properties, often used in algebraic geometry.
