Exact functor

In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that fail to be exact, but in ways that can still be controlled. Let P and Q be abelian categories, and let F: P→Q be a covariant additive functor (so that, in particular, F(0) = 0). We say that F is an exact functor if whenever is a short exact sequence in P then is a short exact sequence in Q. (The maps are often omitted and implied, and one says: "if 0→A→B→C→0 is exact, then 0→F(A)→F(B)→F(C)→0 is also exact".) Further, we say that F is left-exact if whenever 0→A→B→C→0 is exact then 0→F(A)→F(B)→F(C) is exact; right-exact if whenever 0→A→B→C→0 is exact then F(A)→F(B)→F(C)→0 is exact; half-exact if whenever 0→A→B→C→0 is exact then F(A)→F(B)→F(C) is exact. This is distinct from the notion of a topological half-exact functor. If G is a contravariant additive functor from P to Q, we similarly define G to be exact if whenever 0→A→B→C→0 is exact then 0→G(C)→G(B)→G(A)→0 is exact; left-exact if whenever 0→A→B→C→0 is exact then 0→G(C)→G(B)→G(A) is exact; right-exact if whenever 0→A→B→C→0 is exact then G(C)→G(B)→G(A)→0 is exact; half-exact if whenever 0→A→B→C→0 is exact then G(C)→G(B)→G(A) is exact. It is not always necessary to start with an entire short exact sequence 0→A→B→C→0 to have some exactness preserved. The following definitions are equivalent to the ones given above: F is exact if and only if A→B→C exact implies F(A)→F(B)→F(C) exact; F is left-exact if and only if 0→A→B→C exact implies 0→F(A)→F(B)→F(C) exact (i.e. if "F turns kernels into kernels"); F is right-exact if and only if A→B→C→0 exact implies F(A)→F(B)→F(C)→0 exact (i.e. if "F turns cokernels into cokernels"); G is left-exact if and only if A→B→C→0 exact implies 0→G(C)→G(B)→G(A) exact (i.e.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (37)

Related courses (13)

Related lectures (238)

Category of modules

In algebra, given a ring R, the category of left modules over R is the whose are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the . The category of right modules is defined in a similar way. One can also define the category of bimodules over a ring R but that category is equivalent to the category of left (or right) modules over the enveloping algebra of R (or over the opposite of that).

Category of abelian groups

In mathematics, the Ab has the abelian groups as and group homomorphisms as morphisms. This is the prototype of an : indeed, every can be embedded in Ab. The zero object of Ab is the trivial group {0} which consists only of its neutral element. The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms. Ab is a of Grp, the .

Exact functor

MATH-436: Homotopical algebra

This course will provide an introduction to model category theory, which is an abstract framework for generalizing homotopy theory beyond topological spaces and continuous maps. We will study numerous

MATH-506: Topology IV.b - cohomology rings

Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative a

ME-523: Nonlinear Control Systems

Les systèmes non linéaires sont analysés en vue d'établir des lois de commande. On présente la stabilité au sens de Lyapunov, ainsi que des méthodes de commande géométrique (linéarisation exacte). Div

Exact Linearization: Earth Dynamics and Stabilization ME-523: Nonlinear Control Systems

Explores exact linearization techniques for transforming non-linear systems into linear ones, emphasizing system stability.

Exact Linearization: Determining Conditions and Transformations ME-523: Nonlinear Control Systems

Explores exact linearization, determining conditions and transformations using Lie bracket and hook of Lie.

Group Cohomology MATH-506: Topology IV.b - cohomology rings

Covers the concept of group cohomology, focusing on chain complexes, cochain complexes, cup products, and group rings.