External Direct Sums and the Sum of a Family of Mappings ....

[Math Amateur]

I have an issue/problem that relates to Bland initial treatment of external direct sums including Proposition 2.1.5 ... especially Bland's definition of the sum of a family of mappings ...

Bland's text on this is as follows:

In the above text by Bland we read the following:

" ... ... We now need the concept of a family of mappings. If $f_\alpha \ : \ M_\alpha \rightarrow N$ is an R-linear mapping for each $\alpha \in \Delta$, where $N$ is a fixed R-module, then $f \ : \ \bigoplus_\Delta M_\alpha \rightarrow N$ defined by $f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x)$ ... ... "

But in the last sentence of the proof of Proposition 2.1.5 ( ... again, see above text by Bland ... ) we read:

" ... ... If $( x_\alpha ) \in \bigoplus_\Delta M_\alpha$, then $f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x_\alpha)$ ... ... "So ... in the text above the Proposition we have ... $f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x)$ ... ... and in the proof of the proposition we have $f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x_\alpha)$ ...

... which of these is correct ... or in some strange way, are they both correct ...

... I note that x is mentioned in the definition of the canonical injections above ..

Can someone please clarify ... ?
But ... if $f$ is defined by $f( ( x_\alpha ) ) = \sum_\Delta f_\alpha (x_\alpha)$ ... then I have a further problem ...

... we know that $f \ : \ M_\alpha \rightarrow N$ ... that is the domain of $f_\alpha$ is $M_\alpha$ ... BUT ...PROBLEM ... $( x_\alpha ) \in \bigoplus_\Delta M_\alpha$ and $( x_\alpha ) \notin M_\alpha$ ...

... can someone please clarify ?
Hope someone can help ...

Peter

 

[andrewkirk]

The second one is correct. I think the $\alpha$ subscript was left off the $x$ in the first one by mistake.

Re the further problem. You say we know that $f:M_\alpha\to N$. I can't see that the author has said that. Right now, because of an internet problem, I can only see your latex, not the png from the text. But in the latex you have quoted the author as writing that the domain of $f$ is $\bigoplus_\Delta M_\alpha$, not $M_\alpha$ which is the domain of $f_\alpha$ (note the subscript on the $f$). As long as we stick to that, I think we'll be OK.

 

[Math Amateur]

The second one is correct. I think the $\alpha$ subscript was left off the $x$ in the first one by mistake.

Re the further problem. You say we know that $f:M_\alpha\to N$. I can't see that the author has said that. Right now, because of an internet problem, I can only see your latex, not the png from the text. But in the latex you have quoted the author as writing that the domain of $f$ is $\bigoplus_\Delta M_\alpha$, not $M_\alpha$ which is the domain of $f_\alpha$ (note the subscript on the $f$). As long as we stick to that, I think we'll be OK.

Sorry Andrew ... it was a typo ...

I wrote:

" ... ... ... we know that $f \ : \ M_\alpha \rightarrow N$ ... that is the domain of $f_\alpha$ is $M_\alpha$ ... "

but I meant

" ... ... $f_\alpha \ : \ M_\alpha \rightarrow N$ ... that is the domain of $f_\alpha$ is $M_\alpha$ ... ... "

Peter

 

[andrewkirk]

Internet is mended now so I can see the png. Is your problem solved? If not can you elaborate on what the remaining difficulty is?

 

[Math Amateur]

Thanks Andrew ... Issues are resolved ...

Peter