Injective dissolution

In the mathematical field of category theory and homological algebra , an injective resolution is a long exact sequence of injective objects that begins with a given object.

definition

Formally, an Abelian category and an object are selected . Then a long exact sequence is called the form ${\displaystyle C}$ ${\displaystyle A}$ ${\displaystyle C}$

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

injective resolution of when all are injective. ${\displaystyle A}$ ${\displaystyle I_{i}}$

existence

In the Abelian category, each object is a sub-object of an injective object, i. H. if there is a monomorphism for every object , where injective is, it is also said, have enough injective objects . An important example of such categories is the category of links modules over a ring . ${\displaystyle C}$ ${\displaystyle X\in \operatorname {Ob} (C)}$ ${\displaystyle X\rightarrow I}$ ${\displaystyle I}$ ${\displaystyle C}$

Under these conditions there is also an injective resolution for every object . First there is a monomorphism , then a monomorphism and then further by induction . ${\displaystyle A}$ ${\displaystyle i_{0}:A\rightarrow I_{0}}$ ${\displaystyle i_{1}:\operatorname {coker} (i_{0})\rightarrow I_{1}}$ ${\displaystyle i_{n+1}:\operatorname {coker} (i_{n})\rightarrow I_{n+1}}$

properties

Is

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

an injective resolution and

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

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

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

complete. An important consequence of this property is that every two injective resolutions of an object are of the same homotopy type .

See also

• The dual term is that of projective resolution .
• Injective resolutions are used in the computation of derived functors .

Individual evidence

1. ^ PJ Hilton: Lectures in Homological Algebra , American Mathematical Society (1971), ISBN 0821816578 , definition 2.6
2. ↑ Peter Hilton, Urs Stammbach: A course in homological algebra , 1st edition 1970, ISBN 3-540-90032-2 , Chapter IV, Theorem 4.4 and Theorem 4.5