Derived functor information

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

using the functor G(A)=HomR(A,B). For a fixed module B, this is a contravariant left exact functor, and thus we also have right derived functors RnG, and...

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...

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...

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...

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...

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...

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...

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

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

{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...

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

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

such as exact sequences, and especially short exact sequences, and derived functors. Important theorems that apply in all abelian categories include the...

additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as...

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...

{\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...

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 ,...

{\displaystyle G^{(2)},G^{(3)},\ldots } are called the second derived subgroup, third derived subgroup, and so forth, and the descending normal series ⋯...

total derived functor theorem of Quillen says that the total left derived functor LF: Ho(C) → Ho(D) is a left adjoint to the total right derived functor RG:...

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...

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...

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...

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