Concept

Split exact sequence

In mathematics, a split exact sequence is a short exact sequence in which the middle term is built out of the two outer terms in the simplest possible way. A short exact sequence of abelian groups or of modules over a fixed ring, or more generally of objects in an is called split exact if it is isomorphic to the exact sequence where the middle term is the direct sum of the outer ones: The requirement that the sequence is isomorphic means that there is an isomorphism such that the composite is the natural inclusion and such that the composite equals b. This can be summarized by a commutative diagram as: The splitting lemma provides further equivalent characterizations of split exact sequences. A trivial example of a split short exact sequence is where are R-modules, is the canonical injection and is the canonical projection. Any short exact sequence of vector spaces is split exact. This is a rephrasing of the fact that any set of linearly independent vectors in a vector space can be extended to a basis. The exact sequence (where the first map is multiplication by 2) is not split exact. Pure exact sequences can be characterized as the filtered colimits of split exact sequences.

Source officielle

https://en.wikipedia.org/wiki/Split_exact_sequence

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Source officielle

https://en.wikipedia.org/wiki/Split_exact_sequence

À propos de ce résultat

Séances de cours associées (6)

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Couvre le lemme Zig Zag et la longue séquence exacte de l'homologie relative.

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Explore les séquences exactes divisées et construisant des homomorphismes entre les modules.

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Publications associées (4)

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arXiv2025

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2012

On Degraded Two Message Set Broadcast

We consider the two message set problem, where a source broadcasts a common message

W_1

to an arbitrary set of receivers

\mathcal{U}

and a private message

W_2

to a subset of the receivers

\mathcal{P}\subseteq \mathcal{U}

. Transmissions occur over linear deterministic channels. For the case where at most two receivers do not require the private message, we give an exact characterization of the capacity region, where achievability is through linear coding.

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Publications associées (4)

W_1

to an arbitrary set of receivers

\mathcal{U}

and a private message

W_2

to a subset of the receivers

\mathcal{P}\subseteq \mathcal{U}

EPFL Technical Report2009

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