In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies...

for fixed A in ModR. This is a left exact functor and thus has right derived functors RnT. The Ext functor is defined by Ext R n ⁡ ( A , B ) = ( R n T...

so does CI, and the right derived functors of the inverse limit functor can thus be defined. The nth right derived functor is denoted R n lim ← : C I...

introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects...

morphisms that satisfy properties generalising those of derived functors. A universal δ-functor is a δ-functor satisfying a specific universal property related...

only defined on the level of derived categories, i.e., the functor is not obtained as the derived functor of some functor between abelian categories. If...

In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological...

exact but not necessarily right exact. We may therefore form its right derived functors. Their values are abelian groups and they are denoted by H n ( G ,...

In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental...

operations are six functors. Usually these are functors between derived categories and so are actually left and right derived functors. the direct image...

formulate homology theories as derived functors on appropriate abelian categories, measuring the failure of an appropriate functor to be exact. One can describe...

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

mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central...

functors from the derived category of sheaves on X to abelian groups. In a broad sense of the word, "cohomology" is often used for the right derived functors...

adjoint functors, as does the unicity. The notation Rf! is an abuse of notation insofar as there is in general no functor f! whose derived functor would...

derived functor cohomology of an object and the homology of a chain complex since hypercohomology corresponds to the derived global sections functor R...

Combinatorial species Exact functor Derived functor Dominant functor Enriched functor Kan extension of a functor Hom functor Product (category theory) Equaliser...

the groups Hi(X,E) for integers i are defined as the right derived functors of the functor E ↦ E(X). This makes it automatic that Hi(X,E) is zero for...

derived functors of left exact functors. The étale cohomology groups Hi(F) of the sheaf F of abelian groups are defined as the right derived functors...

computes the derived functors of the composition of two functors G ∘ F {\displaystyle G\circ F} , from knowledge of the derived functors of F {\displaystyle...

theory has yet to be developed.) The term "derived" is used in the same way as derived functor or derived category, in the sense that the category of...

the derived functors RiF(En) vanish for all i > 0 and n ≥ 0. Dually, a left resolution is acyclic with respect to a right exact functor if its derived functors...

{\displaystyle \operatorname {H} ^{i}(X,-)} as the i-th right derived functor of the global section functor Γ ( X , − ) {\displaystyle \Gamma (X,-)} . Given a ringed...

category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These...

{F}})} in the derived category of sheaves on S', similarly as mentioned above. Here L g ∗ {\displaystyle Lg^{*}} is the (total) derived functor of the pullback...

construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There is a further group of related concepts applied...

G])\subseteq [H,H]} . This shows that the commutator subgroup can be viewed as a functor on the category of groups, some implications of which are explored below...

Ext functors, the derived functors of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level...

algebraic situations involving derived functors. While their theoretical importance has decreased since the introduction of derived categories, they are still...

the homotopy category (i.e., derived category). By definition, it is the left derived functor of the tensor product functor − ⊗ A − : M A × A M → R M {\displaystyle...