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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, 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.
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
More formally, a quotient graph is a quotient object in the category of graphs. The category of graphs is concretizable – mapping a graph to its set of vertices makes it a concrete category – so its objects can be regarded as "sets with additional structure", and a quotient graph corresponds to the graph induced on the quotient set V / R of ...
In order for the definition to make sense the category in question must have zero morphisms. The cokernel of a morphism f : X → Y is defined as the coequalizer of f and the zero morphism 0 XY : X → Y. Explicitly, this means the following. The cokernel of f : X → Y is an object Q together with a morphism q : Y → Q such that the diagram
The question is of a great interest since the GIT approach produces an explicit quotient, as opposed to an abstract quotient, which is hard to compute. One known partial answer to this question is the following: [2] let be a locally factorial algebraic variety (for example, a smooth variety) with an action of .
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 quotient by a homogeneous ideal of a multivariate polynomial ring, graded by the total degree. The quotient by an ideal of a multivariate polynomial ring, filtered by the total degree. The filtration of a local ring by the powers of its maximal ideal. In this case the Hilbert polynomial is called the Hilbert–Samuel polynomial.