LOCALLY FREE SHEAF Meaning and
Definition
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A locally free sheaf is a fundamental concept in algebraic geometry and algebraic topology. It is a sheaf of modules over a topological space that behaves like a vector bundle locally, meaning it is locally isomorphic to a direct sum of copies of the sheaf of local functions.
More precisely, let X be a topological space and E be a sheaf of modules over X. E is called a locally free sheaf if for each point x in X, there exists an open neighborhood U of x such that E|U (the restriction of E to U) is isomorphic to the sheaf of local functions on U, denoted O_U.
This isomorphism captures the notion that E behaves like a vector bundle on U, where the sheaf of local functions serves as the ring of functions. In particular, the sheaf E|U is locally generated by a set of sections, just as a vector bundle is generated by a family of local trivializations.
The concept of a locally free sheaf is highly useful in both algebraic geometry and algebraic topology. It provides a powerful tool for studying geometric objects on X, and plays a significant role in various areas of mathematics, such as algebraic K-theory, algebraic cycles, and the theory of characteristic classes.
