An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category)...

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...

mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations...

and cokernels. The most common type of exact sequence is the short exact sequence. This is an exact sequence of the form A ↪ f B ↠ g C {\displaystyle...

spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and...

cofibration is the mapping cone, then the resulting exact (or dually, coexact) sequence is given by the Puppe sequence. There are many realizations of spheres as...

an obvious generalization of cohomological long exact sequences associated to short exact sequences 0 → M ′ → M → M ″ → 0 {\displaystyle 0\to M'\to M\to...

mathematics, specifically in category theory, an exact category is a category equipped with short exact sequences. The concept is due to Daniel Quillen and is...

short exact sequences exact after tensoring, a pure submodule defines a short exact sequence (known as a pure exact sequence) that remains exact after...

< q < n {\displaystyle 0<q<n} hold, an exact sequence exists (also known under the name Serre exact sequence): H m + n − 1 ( F ) → i ∗ H m + n − 1 (...

compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a natural long exact sequence, whose entries...

An exact sequence (or exact complex) is a chain complex whose homology groups are all zero. This means all closed elements in the complex are exact. A...

Therefore, ASVs represent a finer distinction between sequences. ASVs are also referred to as exact sequence variants (ESVs), zero-radius OTUs (ZOTUs), sub-OTUs...

(f_{i+1})=\operatorname {ker} (f_{i})} . A special case of an exact sequence is a short exact sequence: 0 → A → f B → g C → 0 {\displaystyle 0\to A{\overset {f}{\to...

spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and...

mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the...

surgery exact sequence is the long exact sequence induced by a fibration sequence of spectra. This implies that all the sets involved in the sequence are...

In the mathematical surgery theory the surgery exact sequence is the main technical tool to calculate the surgery structure set of a compact manifold in...

inflation-restriction exact sequence is an exact sequence occurring in group cohomology and is a special case of the five-term exact sequence arising from the...

In mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of...

short exact sequence of modules of the form 0 → A → B → P → 0 {\displaystyle 0\rightarrow A\rightarrow B\rightarrow P\rightarrow 0} is a split exact sequence...

mathematics, five-term exact sequence or exact sequence of low-degree terms is a sequence of terms related to the first step of a spectral sequence. More precisely...

triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology. Much of homological...

In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry. Let M be a complex manifold,...

only if O K {\displaystyle {\mathcal {O}}_{K}} is a UFD. There is an exact sequence 0 → O K ∗ → K ∗ → I K → C K → 0 {\displaystyle 0\to {\mathcal {O}}_{K}^{*}\to...

field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the...

holds, the sequence is called a split exact sequence, and the sequence is said to split. In the above short exact sequence, where the sequence splits, it...

inflation-restriction exact sequence in group cohomology, and in integration in fibers. It also naturally arises in many spectral sequences; see spectral sequence#Edge...

{\displaystyle \Longrightarrow } ' means convergence of spectral sequences. The exact sequence of low degrees reads 0 → R 1 G ( F A ) → R 1 ( G F ) ( A ) →...

various quite different settings that a short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and...