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In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the six-functor formalism. [1] It originally sprang from the relations in étale cohomology that arise from a morphism of schemes f : X → Y.
Good Luck!! topped the Japanese drama ratings with 35% of the Japanese viewership in 2003. [3] The first episode aired on January 19, 2003, with ratings of 31.6%, the third highest ratings of an opening episode, behind Beautiful Life and HERO, also starring Takuya Kimura. The single episode ratings are as follows: [4]
A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness. Torsionless A module is called torsionless if it embeds into its algebraic dual. Simple A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible. [5 ...
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 .
In mathematics, the annihilator of a subset S of a module over a ring is the ideal formed by the elements of the ring that give always zero when multiplied by each element of S. Over an integral domain, a module that has a nonzero annihilator is a torsion module, and a finitely generated torsion module has a nonzero annihilator.
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contains all the cycles of length 4, 6, ..., 2 n and is thus a bipancyclic graph. can be drawn as a unit distance graph in the Euclidean plane by using the construction of the hypercube graph from subsets of a set of n elements, choosing a distinct unit vector for each set element, and placing the vertex corresponding to the set S at the sum of ...
The integers, considered as a module over the ring of integers, is a Noetherian module. If R = M n (F) is the full matrix ring over a field, and M = M n 1 (F) is the set of column vectors over F, then M can be made into a module using matrix multiplication by elements of R on the left of elements of M. This is a Noetherian module.