Resolution (algebra)

In mathematics, and more specifically in homological algebra, a resolution (or left resolution; dually a coresolution or right resolution) is an exact sequence of modules (or, more generally, of objects of an abelian category) that is used to define invariants characterizing the structure of a specific module or object of this category. When, as usually, arrows are oriented to the right, the sequence is supposed to be infinite to the left for (left) resolutions, and to the right for right resolutions. However, a finite resolution is one where only finitely many of the objects in the sequence are non-zero; it is usually represented by a finite exact sequence in which the leftmost object (for resolutions) or the rightmost object (for coresolutions) is the zero-object.

Generally, the objects in the sequence are restricted to have some property P (for example to be free). Thus one speaks of a P resolution. In particular, every module has free resolutions, projective resolutions and flat resolutions, which are left resolutions consisting, respectively of free modules, projective modules or flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules.

Resolutions of modules

Definitions

Given a module

M

{\displaystyle M}

over a ring

R

{\displaystyle R}

, a left resolution (or simply resolution) of

M

{\displaystyle M}

is an exact sequence (possibly infinite) of

R

{\displaystyle R}

-modules

⋯

⟶

d

n

+

1

E

n

⟶

d

n

⋯

⟶

d

3

E

2

⟶

d

2

E

1

⟶

d

1

E

0

⟶

ε

M

⟶

0.

{\displaystyle \cdots {\overset {d_{n+1}}{\longrightarrow }}E_{n}{\overset {d_{n}}{\longrightarrow }}\cdots {\overset {d_{3}}{\longrightarrow }}E_{2}{\overset {d_{2}}{\longrightarrow }}E_{1}{\overset {d_{1}}{\longrightarrow }}E_{0}{\overset {\varepsilon }{\longrightarrow }}M\longrightarrow 0.}

The homomorphisms

d

i

{\displaystyle d_{i}}

are called boundary maps. The map

ε

{\displaystyle \varepsilon }

is called an augmentation map.