Projective resolution

In the mathematical field of category theory and homological algebra , a projective resolution is a long exact sequence of projective objects that ends in a given object.

definition

Let there be an Abelian category (or the category Grp of the groups) and an object . Then a long exact sequence is called the form ${\ displaystyle C}$${\ displaystyle A}$${\ displaystyle C}$

${\ displaystyle \ cdots \ rightarrow P_ {2} \ rightarrow P_ {1} \ rightarrow P_ {0} \ rightarrow A \ rightarrow 0}$

projective resolution of when all are projective. ${\ displaystyle A}$${\ displaystyle P_ {i}}$

If all are even free , one speaks of a free dissolution . ${\ displaystyle P_ {j}}$

existence

In the Abelian category, every object is quotient of a projective object, i. H. there are for each object an epimorphism , which is projective, it is said also possess sufficiently many projective objects . ${\ displaystyle C}$${\ displaystyle X \ in \ operatorname {Ob} (C)}$ ${\ displaystyle P \ rightarrow X}$${\ displaystyle P}$${\ displaystyle C}$

Under these conditions there is also a projective resolution for every object . First of all there is an epimorphism , then an epimorphism on the core of this morphism and then further by induction . ${\ displaystyle A}$${\ displaystyle p_ {0} \ colon P_ {0} \ rightarrow A}$${\ displaystyle p_ {1} \ colon P_ {1} \ rightarrow \ operatorname {ker} (p_ {0})}$${\ displaystyle p_ {n + 1} \ colon P_ {n + 1} \ rightarrow \ operatorname {ker} (p_ {n})}$

The most important category with a sufficient number of projective objects is the category of (left) modules over a ring . If such a module is and is a generating system , then one has a surjective homomorphism by mapping the -th basic element of the free module to . Since free modules are projective, is the quotient of a projective module and thus has enough projective objects. ${\ displaystyle \ mathrm {Mod} _ {R}}$ ${\ displaystyle R}$${\ displaystyle A}$${\ displaystyle (a_ {i}) _ {i \ in I}}$${\ displaystyle R ^ {I} \ rightarrow A}$${\ displaystyle i}$ ${\ displaystyle R ^ {I}}$${\ displaystyle a_ {i}}$${\ displaystyle A}$${\ displaystyle \ mathrm {Mod} _ {R}}$

properties

Is

${\ displaystyle \ cdots \ rightarrow P_ {2} \ rightarrow P_ {1} \ rightarrow P_ {0} \ rightarrow A \ rightarrow 0}$

a projective resolution and

${\ displaystyle \ cdots \ rightarrow A '_ {2} \ rightarrow A' _ {1} \ rightarrow A '_ {0} \ rightarrow A' \ rightarrow 0}$

exactly, every homomorphism (not necessarily unique) can be converted into a commutative diagram ${\ displaystyle C}$${\ displaystyle f \ colon A \ rightarrow A '}$

${\ displaystyle {\ begin {matrix} \ cdots \ rightarrow & P_ {2} & \ rightarrow & P_ {1} & \ rightarrow & P_ {0} & \ rightarrow & A & \ rightarrow 0 \\\ cdots & \ downarrow && \ downarrow && \ downarrow && \ downarrow \\\ cdots \ rightarrow & A '_ {2} & \ rightarrow & A' _ {1} & \ rightarrow & A '_ {0} & \ rightarrow & A' & \ rightarrow 0 \ end {matrix}}}$

complete.