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One method of copying an object is the shallow copy.In that case a new object B is created, and the fields values of A are copied over to B. [3] [4] [5] This is also known as a field-by-field copy, [6] [7] [8] field-for-field copy, or field copy. [9]
Let C be a curve of degree d in P 3, then consider all the lines in P 3 that intersect the curve C. This is a degree d divisor D C in G(2, 4), the Grassmannian of lines in P 3. When C varies, by associating C to D C, we obtain a parameter space of degree d curves as a subset of the space of degree d divisors of the Grassmannian: Chow(d,P 3).
In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke [2] and Marc Rieffel, the latter in a paper that used Hilbert C*-modules to construct a theory of induced representations of C*-algebras. [3] Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory, [4] and provide the ...
A module P is projective if and only if every short exact sequence of modules of the form . is a split exact sequence.That is, for every surjective module homomorphism f : B ↠ P there exists a section map, that is, a module homomorphism h : P → B such that f h = id P .
if K is a number field, the subgroup of the group of fractional ideals generated by ideals coprime to m f; [12] if K is a function field of an algebraic curve over k, the group of divisors, rational over k, with support away from m. [13] In both case, there is a group homomorphism i : K m,1 → I m obtained by sending a to the principal ideal ...
In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) → F(V) are compatible with the restriction maps O(U) → O(V): the restriction of fs is the restriction of f times the restriction of s for any f in O(U ...
An object X of C can be considered a trivial inverse system, where all objects are equal to X and all arrow are the identity of X. This defines a "trivial functor" from C to . The inverse limit, if it exists, is defined as a right adjoint of this trivial functor.
In abstract algebra, Kaplansky's theorem on projective modules, first proven by Irving Kaplansky, states that a projective module over a local ring is free; [1] where a not-necessarily-commutative ring is called local if for each element x, either x or 1 − x is a unit element. [2]