## Pullbacks and pushouts of modules

Let be a ring with 1 and let , , and be left unital -modules.

- Let and be module homomorphisms. Prove that there exists a module and two module homomorphisms and such that and which have the following property: if is a module and and are module homomorphisms such that , then there exists a unique module homomorphism such that and . That is, given the following diagram of modules,
The pullback of two modules

there exists a unique which makes the diagram commute. Deduce that is unique up to isomorphism. We will call this the *pullback* or *fiber product* of and , and sometimes denote it by .

- Let and be module homomorphisms. Prove that there exists a module and two module homomorphisms and such that and which have the following property: If is a module and and are module homomorphisms such that , then there exists a unique module homomorphism such that and . That is, given the following diagram of modules,
The pushout of two module homomorphisms

there exists a unique which makes the diagram commute. Deduce that is unique up to isomorphism. We will call this the *pushout* or *fiber sum* of and and , and sometimes denote it by .

- Let , and let and . Note that if and , then , so that . Since , by the submodule criterion, is a submodule of . Certainly if , then , so that . Thus . Now suppose we have , , and . Note that for all , we have , so that . Define by . Clearly is a module homomorphism. Moreover, and . The uniqueness of follows easily.
- Define . Note that , and if and , then . By the submodule criterion, is a submodule. Let , and define and by and . Note that for all , we have . Thus . Now suppose we have , , and . Define by . To see that is well-defined, suppose . Then we have and for some . Applying and , we see that . Thus . Clearly is a module homomorphism. Moreover, we see that and . To see uniqueness, suppose there exists such that and . Now , so that .

### Like this:

Like Loading...

*Related*

or leave a trackback:

Trackback URL.