- #1
NihilTico
- 32
- 2
A couple of notes first:
1.
\hom_{A}(-,N) is the left-exact functor I'm referring to; Lang gives an exercise in the section preceeding to show this.
2.
This might be my own idiosyncrasy but I write TFDC to mean 'The following diagram commutes'
3.
Titles are short, so I know that the hom-functor here isn't actually taking a coproduct to a product in the same category. The coproduct \bigoplus M_i lies in the category \text{Mod}(A) while the product \prod\hom_{A}\left(M_i,N\right) lies in \bf{Ab}.
I don't think I saw this before, but on page 131 of Lang's Algebra (3rd edition) he writes (NOTE: I'm dropping the indexing set I unless I feel it is necessary for clarity) what generalizes naturally to \hom_{A}\left(\bigoplus M_{i},N\right)\approx\prod\hom_{A}\left(M_i,N\right). Where the M_i and N are $A$-modules. But he goes about demonstrating it for the case of two A-modules, curiously.
To be a thorough as possible, is it the case, since the coproduct is the initial object in the category of tuples \left(C,\left\{f_{i}\colon{M_i}\to{C}\right\}_{i\in{I}}\right) (where the f_i are A-homomorphisms) meaning that a morphism f\colon\left(\bigoplus M_i,\left\{\jmath_{i}\colon{M_i}\to{\bigoplus {M_i}}\right\}_{i\in{I}}\right)\to\left(C,\left\{f_{i}\colon{M_i}\to{C}\right\}_{i\in{I}}\right) is a unique A homomorphism h\colon \bigoplus M_i\to{C} such that \forall i\in{I} TFDC:
<br /> \begin{array}{cccc} & \jmath_{i} & \bigoplus M_{i}\\ M_{i} & \nearrow & \downarrow\\ & \searrow & \ \downarrow h\\ & f_{i} & C \end{array}
that this is totally obvious, since h is uniquely determined by the family \left\{f_i\right\}_{i\in{I}}? In particular, the problem becomes—more or less—one of the existence of a group homomorphsim q\colon\hom_{A}\left(\bigoplus M_{i},N\right)\to\prod\hom_{A}\left(M_i,N\right).
[/B]
Not applicable.
[/B]
Not so much as an attempt, as an observation.
Clearly, any h\colon\bigoplus M_i\to{N} induces a unique family of A-homomorphisms given by \left\{h_i\colon M_i\to{N}\right\}_{i\in{I}} ; after all, if another family had this h as well, then the families would be equal as well by definition of the coprouct. Similarly, for any family in the product on the right, there is a unique h from the coproduct to N. Isomorphisms in \bf{Ab} are bijective so this is enough by defining q(h)=\left(h_i\right)_{i\in{I}}.
Right?
If the above is correct, does it speak to something deeper about hom-functors that I'd be better served finding in Mac Lane's CWM?Thanks
1.
\hom_{A}(-,N) is the left-exact functor I'm referring to; Lang gives an exercise in the section preceeding to show this.
2.
This might be my own idiosyncrasy but I write TFDC to mean 'The following diagram commutes'
3.
Titles are short, so I know that the hom-functor here isn't actually taking a coproduct to a product in the same category. The coproduct \bigoplus M_i lies in the category \text{Mod}(A) while the product \prod\hom_{A}\left(M_i,N\right) lies in \bf{Ab}.
Homework Statement
I don't think I saw this before, but on page 131 of Lang's Algebra (3rd edition) he writes (NOTE: I'm dropping the indexing set I unless I feel it is necessary for clarity) what generalizes naturally to \hom_{A}\left(\bigoplus M_{i},N\right)\approx\prod\hom_{A}\left(M_i,N\right). Where the M_i and N are $A$-modules. But he goes about demonstrating it for the case of two A-modules, curiously.
To be a thorough as possible, is it the case, since the coproduct is the initial object in the category of tuples \left(C,\left\{f_{i}\colon{M_i}\to{C}\right\}_{i\in{I}}\right) (where the f_i are A-homomorphisms) meaning that a morphism f\colon\left(\bigoplus M_i,\left\{\jmath_{i}\colon{M_i}\to{\bigoplus {M_i}}\right\}_{i\in{I}}\right)\to\left(C,\left\{f_{i}\colon{M_i}\to{C}\right\}_{i\in{I}}\right) is a unique A homomorphism h\colon \bigoplus M_i\to{C} such that \forall i\in{I} TFDC:
<br /> \begin{array}{cccc} & \jmath_{i} & \bigoplus M_{i}\\ M_{i} & \nearrow & \downarrow\\ & \searrow & \ \downarrow h\\ & f_{i} & C \end{array}
that this is totally obvious, since h is uniquely determined by the family \left\{f_i\right\}_{i\in{I}}? In particular, the problem becomes—more or less—one of the existence of a group homomorphsim q\colon\hom_{A}\left(\bigoplus M_{i},N\right)\to\prod\hom_{A}\left(M_i,N\right).
Homework Equations
[/B]
Not applicable.
The Attempt at a Solution
[/B]
Not so much as an attempt, as an observation.
Clearly, any h\colon\bigoplus M_i\to{N} induces a unique family of A-homomorphisms given by \left\{h_i\colon M_i\to{N}\right\}_{i\in{I}} ; after all, if another family had this h as well, then the families would be equal as well by definition of the coprouct. Similarly, for any family in the product on the right, there is a unique h from the coproduct to N. Isomorphisms in \bf{Ab} are bijective so this is enough by defining q(h)=\left(h_i\right)_{i\in{I}}.
Right?
If the above is correct, does it speak to something deeper about hom-functors that I'd be better served finding in Mac Lane's CWM?Thanks