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In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties: such that [1] (i) The map π {\displaystyle \pi } is surjective, and its fibers are exactly the G-orbits in X.
The direct approach can be made, by means of the function field of a variety (i.e. rational functions): take the G-invariant rational functions on it, as the function field of the quotient variety. Unfortunately this — the point of view of birational geometry — can only give a first approximation to the answer. As Mumford put it in the ...
Denote the subspace of all functions f ∈ C[0,1] with f(0) = 0 by M. Then the equivalence class of some function g is determined by its value at 0, and the quotient space C[0,1]/M is isomorphic to R. If X is a Hilbert space, then the quotient space X/M is isomorphic to the orthogonal complement of M.
An effective quotient orbifold, e.g., [/] where the action has only finite stabilizers on the smooth space , is an example of a quotient stack. [2]If = with trivial action of (often is a point), then [/] is called the classifying stack of (in analogy with the classifying space of ) and is usually denoted by .
In linear algebra, a quotient space is a vector space formed by taking a quotient group, where the quotient homomorphism is a linear map. By extension, in abstract algebra, the term quotient space may be used for quotient modules, quotient rings, quotient groups, or any quotient algebra. However, the use of the term for the more general cases ...
This definition corresponds to the ordinary understanding of a subobject outside category theory. When the category's objects are sets (possibly with additional structure, such as a group structure) and the morphisms are set functions (preserving the additional structure), one thinks of a monomorphism in terms of its image.
In general a modular function field is a function field of a modular curve (or, occasionally, of some other moduli space that turns out to be an irreducible variety). Genus zero means such a function field has a single transcendental function as generator: for example the j-function generates the function field of X (1) = PSL(2, Z )\ H *.
[5] [6] The difference quotient is a measure of the average rate of change of the function over an interval (in this case, an interval of length h). [ 7 ] [ 8 ] : 237 [ 9 ] The limit of the difference quotient (i.e., the derivative) is thus the instantaneous rate of change.