Homological algebra

area of mathematics

collection of maps in which all map compositions starting from the same set and ending with the same set give the same result

sequence of homomorphisms such that each kernel equals the preceding image

in homological algebra, a structure consisting of a sequence of modules and a sequence of homomorphisms between consecutive modules such that the image of each homomorphism is included in the kernel of the next

preadditive category with a zero object and all binary biproducts, kernels and cokernels in which all monomorphisms and epimorphisms are normal

tool used in mathematics

theorem

module such that taking the tensor product with it induces an exact functor

lemma in category theory about commutative diagrams

Abelian group constructed universally from a commutative monoid

in algebra, a module that is the direct summand of a free module

elements of a module space sent to 0 by regular elements of a ring

module that makes certain exact sequences split

homological construction in category theory

in homological algebra, the left derived functor of the tensor product of modules over a ring

algebraic structure

theorem

two kinds of possible inverses of a morphism in a category

functor that preserves short exact sequences

object 𝑃 in an abelian category such that hom(𝑃,–) is an exact functor to the category of abelian groups

Group comohology of Galois modules

derived functors of the Hom functor

conjectural objects in algebraic geometry that provide a universal cohomology theory of varieties

in algebra, an exact sequence which is used to describe the structure of an object

category theory lemma about commutative diagrams

right derived functors of the global sections functor Γ: AbSh → Ab

algebraic construction of the cohomology of (the underlying manifold of) a simply connected Lie group in terms of its Lie algebra

Establish relationships between homology and cohomology theories

homotopy category of chain complexes in an abelian category with inverses to quasi-isomorphisms adjoined

cohomology theory associated to a group 𝐺 and a 𝐺‐module

homology theory for associative algebras over rings

homomorphism G → PGL(V) from a group G to a projective linear group PGL(V) over a vector space V

sheaf cohomology on the étale site

lemma that, in an Abelian category, a short exact sequence, one of whose two morphisms admits a section or retraction into the middle term, is a direct sum

In homological algebra, a branch of mathematics, a quasi-isomorphism or quism is a morphism A → B of chain complexes (respectively, cochain complexes) such that the induced morphisms

operation that pairs a left and a right 𝑅‐module into an abelian group

Weil cohomology theory for schemes over a base field, whose values are modules over the ring of Witt vectors over the base field, that replaces Zariski open sets by infinitesimal thickenings of Zariski open sets with divided power structures

category with the additional structure of a translation functor and a class of distinguished triangles

homological property of mathematical rings

Additive category in homological algebra

construction of homological algebra used in commutative agebra

the theorem that extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, which sends all ideal classes to the class of a principal ideal

algebraic statement

differential associative algebra with integer grading in which the differential has grading +1 (cohomological convention) or −1 (homological convention)

theorem