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.

See also

• The dual term is that of injective dissolution .
• Projective resolutions are used in the computation of derived functors .
• Fundamental lemma of homological algebra
• Schanuel's Lemma

Individual evidence

1. ^ Ernst Kunz: Introduction to Commutative Algebra and Algebraic Geometry , Vieweg (1980), ISBN 3-528-07246-6 , Chapter VII, Projective Resolutions
2. ^ PJ Hilton: Lectures in Homological Algebra , American Mathematical Society (1971), ISBN 0821816578 , definition 2.5
3. ^ PJ Hilton: Lectures in Homological Algebra , American Mathematical Society (1971), ISBN 0821816578 , sentence 2.7
4. ^ PJ Hilton: Lectures in Homological Algebra , American Mathematical Society (1971), ISBN 0821816578 , Lemma 2.8 + following comment