Quasi-isomorphism

In the mathematical sub-area of homological algebra , a quasi-isomorphism is a chain mapping between two chain complexes that induces isomorphisms between the homology groups .

Definitions

Let and two chain complexes of a fixed abelian category , for example, the category of left modules over a fixed ring . Let it be a chain mapping, that is, the diagram commutes for all ${\displaystyle C=(C_{n},d_{n})_{n\in \mathbb {Z} }}$ ${\displaystyle {\tilde {C}}=({\tilde {C}}_{n},{\tilde {d}}_{n})_{n\in \mathbb {Z} }}$ ${\displaystyle f=(f_{n})_{n\in \mathbb {Z} }}$ ${\displaystyle n}$

${\displaystyle {\begin{array}{ccc}C_{n}&{\xrightarrow {d_{n}}}&C_{n-1}\\\downarrow _{f_{n}}&&\downarrow _{f_{n-1}}\\{\tilde {C}}_{n}&{\xrightarrow {{\tilde {d}}_{n}}}&{\tilde {C}}_{n-1}\end{array}}}$.

Chain mapping naturally induces homomorphisms between homology groups. One calls a quasi-isomorphism, if all are even isomorphisms. ${\displaystyle f}$ ${\displaystyle f_{n,*}\colon H_{n}(C)\rightarrow H_{n}({\tilde {C}})}$ ${\displaystyle f}$ ${\displaystyle f_{n,*}}$

For coquette mappings between coquette complexes and one obtains homomorphisms , and one calls a quasi-isomorphism if all isomorphisms are of the cohomology groups. ${\displaystyle f=(f_{n})_{n\in \mathbb {Z} }}$ ${\displaystyle C=(C^{n},d_{n})_{n\in \mathbb {Z} }}$ ${\displaystyle {\tilde {C}}=({\tilde {C}}^{n},{\tilde {d}}_{n})_{n\in \mathbb {Z} }}$ ${\displaystyle f_{n}^{*}\colon H^{n}(C)\rightarrow H^{n}({\tilde {C}})}$ ${\displaystyle f}$ ${\displaystyle f_{n}^{*}}$

properties

• If the chain mappings themselves are already isomorphisms, i.e. have an inverse, then the induced homomorphisms between the (co) homology groups are trivially isomorphisms. Therefore, isomorphisms between (co) chain complexes are quasi-isomorphisms, the converse does not apply, as the next example shows.
• If any acyclic complex is any acyclic complex and 0 denotes the zero complex, which consists only of zero objects , then the zero mapping is trivially a quasi-isomorphism, because all homology groups are 0. For every non-trivial acyclic complex one obtains an example for a quasi-isomorphism that is not an isomorphism of the chain complexes. ${\displaystyle C=(C^{n},d_{n})_{n\in \mathbb {Z} }}$ ${\displaystyle f\colon C\rightarrow 0}$
• Chains of quasi-isomorphisms are again quasi-isomorphisms, as one can easily prove by means of the functoriality of the homology groups. Quasi-isomorphisms usually have no inversions, as the above example of the zero mapping between a non-trivial acyclic complex and the zero complex shows.
• Quasi-isomorphisms play a role in the definition of the derived category , see there.

Individual evidence

1. ^ SI Gelfand, Yu. I. Manin. Methods of Homological Algebra , Springer-Verlag 2000, ISBN 978-3-642-07813-2 , chap. III, definition 5