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

  • #1
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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:



Bland - Proposition 2.1.4 ... inc External Direct Sums  ... .png



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
 

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  • #2
andrewkirk
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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.
 
  • #3
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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
 
  • #4
andrewkirk
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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?
 
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  • #5
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Thanks Andrew ... Issues are resolved ...

Peter
 

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