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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.
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 the category of sets, the coequalizer of two functions f, g : X → Y is the quotient of Y by the smallest equivalence relation ~ such that for every x ∈ X, we have f(x) ~ g(x). [1] In particular, if R is an equivalence relation on a set Y, and r 1, r 2 are the natural projections (R ⊂ Y × Y) → Y then the coequalizer of r 1 and r 2 is ...
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 ...
In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V.In classical algebraic geometry they are ratios of polynomials; in complex geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let () = (), where both f and g are differentiable and () The quotient rule states that the derivative of h(x) is
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.
The cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y / im(f) of the codomain of f by the image of f. The dimension of the cokernel is called the corank of f . Cokernels are dual to the kernels of category theory , hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokernel ...