In R-Mod Category, example for $Bcong A oplus C nRightarrow 0 to A to B to C to0$ splits.
$begingroup$
https://en.wikipedia.org/wiki/Splitting_lemma
In $R$-Mod Category, short exact sequence
$0 to A stackrel{f}{rightarrow} B stackrel{g}{rightarrow} C to0$ splits
if it satisfies one of the following equivalent conditions:
$(1) exists f_1intext{Hom}(B,A)text{ s.t. } f_1circ f=text{Id}_A$.
$(2) exists g_1intext{Hom}(C,B)text{ s.t. } gcirc g_1=text{Id}_C$.
$(3) text{Im }f=text{Ker }g$ is direct summand of $B$.
$(4) exists text{ isomorphism } h:Bto A oplus C text{ s.t. } $
$h circ f text{ is natural injection and }g circ h^{-1} text{ is natural projection.}$
And
$0 to A stackrel{f}{rightarrow} B stackrel{g}{rightarrow} C to0$ splits $implies Bcong A oplus C$.
So what if we remove the condition:
$ text{isomorphism } h:Bto A oplus C text{ satisfying } $
$h circ f text{ is natural injection and }g circ h^{-1} text{ is natural projection?}$
Is there any counter example? Thanks in advance.
Related questions:
$(1)$ Example of a non-splitting exact sequence $0rightarrow Mrightarrow Moplus Nrightarrow Nrightarrow 0$, question
$(2)$ A nonsplit short exact sequence of abelian groups with $B cong A oplus C$
abstract-algebra modules
edited Dec 26 '18 at 21:28
user26857
39.3k124183
asked Dec 13 '18 at 14:30
AndrewsAndrews
4081317
$endgroup$
add a comment |
$begingroup$
https://en.wikipedia.org/wiki/Splitting_lemma
In $R$-Mod Category, short exact sequence
$0 to A stackrel{f}{rightarrow} B stackrel{g}{rightarrow} C to0$ splits
if it satisfies one of the following equivalent conditions:
$(1) exists f_1intext{Hom}(B,A)text{ s.t. } f_1circ f=text{Id}_A$.
$(2) exists g_1intext{Hom}(C,B)text{ s.t. } gcirc g_1=text{Id}_C$.
$(3) text{Im }f=text{Ker }g$ is direct summand of $B$.
$(4) exists text{ isomorphism } h:Bto A oplus C text{ s.t. } $
$h circ f text{ is natural injection and }g circ h^{-1} text{ is natural projection.}$
And
$0 to A stackrel{f}{rightarrow} B stackrel{g}{rightarrow} C to0$ splits $implies Bcong A oplus C$.
So what if we remove the condition:
$ text{isomorphism } h:Bto A oplus C text{ satisfying } $
$h circ f text{ is natural injection and }g circ h^{-1} text{ is natural projection?}$
Is there any counter example? Thanks in advance.
Related questions:
$(1)$ Example of a non-splitting exact sequence $0rightarrow Mrightarrow Moplus Nrightarrow Nrightarrow 0$, question
$(2)$ A nonsplit short exact sequence of abelian groups with $B cong A oplus C$
abstract-algebra modules
edited Dec 26 '18 at 21:28
user26857
39.3k124183
asked Dec 13 '18 at 14:30
AndrewsAndrews
4081317
$endgroup$
$begingroup$
You mean, there exists an isomorphism, but it just doesn't satisfy the injection/projection conditions?
$endgroup$
– rschwieb
Dec 13 '18 at 14:40
$begingroup$
Possible duplicate of A nonsplit short exact sequence of abelian groups with $B cong A oplus C$
$endgroup$
– yamete kudasai
Dec 13 '18 at 14:42
$begingroup$
What do you mean by "drop the condition"?
$endgroup$
– positrón0802
Dec 13 '18 at 14:57
add a comment |
$begingroup$
https://en.wikipedia.org/wiki/Splitting_lemma
In $R$-Mod Category, short exact sequence
$0 to A stackrel{f}{rightarrow} B stackrel{g}{rightarrow} C to0$ splits
if it satisfies one of the following equivalent conditions:
$(1) exists f_1intext{Hom}(B,A)text{ s.t. } f_1circ f=text{Id}_A$.
$(2) exists g_1intext{Hom}(C,B)text{ s.t. } gcirc g_1=text{Id}_C$.
$(3) text{Im }f=text{Ker }g$ is direct summand of $B$.
$(4) exists text{ isomorphism } h:Bto A oplus C text{ s.t. } $
$h circ f text{ is natural injection and }g circ h^{-1} text{ is natural projection.}$
And
$0 to A stackrel{f}{rightarrow} B stackrel{g}{rightarrow} C to0$ splits $implies Bcong A oplus C$.
So what if we remove the condition:
$ text{isomorphism } h:Bto A oplus C text{ satisfying } $
$h circ f text{ is natural injection and }g circ h^{-1} text{ is natural projection?}$
Is there any counter example? Thanks in advance.
Related questions:
$(1)$ Example of a non-splitting exact sequence $0rightarrow Mrightarrow Moplus Nrightarrow Nrightarrow 0$, question
$(2)$ A nonsplit short exact sequence of abelian groups with $B cong A oplus C$
abstract-algebra modules
edited Dec 26 '18 at 21:28
user26857
39.3k124183
asked Dec 13 '18 at 14:30
AndrewsAndrews
4081317
$endgroup$
https://en.wikipedia.org/wiki/Splitting_lemma
In $R$-Mod Category, short exact sequence
$0 to A stackrel{f}{rightarrow} B stackrel{g}{rightarrow} C to0$ splits
if it satisfies one of the following equivalent conditions:
$(1) exists f_1intext{Hom}(B,A)text{ s.t. } f_1circ f=text{Id}_A$.
$(2) exists g_1intext{Hom}(C,B)text{ s.t. } gcirc g_1=text{Id}_C$.
$(3) text{Im }f=text{Ker }g$ is direct summand of $B$.
$(4) exists text{ isomorphism } h:Bto A oplus C text{ s.t. } $
$h circ f text{ is natural injection and }g circ h^{-1} text{ is natural projection.}$
And
$0 to A stackrel{f}{rightarrow} B stackrel{g}{rightarrow} C to0$ splits $implies Bcong A oplus C$.
So what if we remove the condition:
$ text{isomorphism } h:Bto A oplus C text{ satisfying } $
$h circ f text{ is natural injection and }g circ h^{-1} text{ is natural projection?}$
Is there any counter example? Thanks in advance.
Related questions:
$(1)$ Example of a non-splitting exact sequence $0rightarrow Mrightarrow Moplus Nrightarrow Nrightarrow 0$, question
$(2)$ A nonsplit short exact sequence of abelian groups with $B cong A oplus C$
abstract-algebra modules
abstract-algebra modules
edited Dec 26 '18 at 21:28
user26857
39.3k124183
asked Dec 13 '18 at 14:30
AndrewsAndrews
4081317
edited Dec 26 '18 at 21:28
user26857
39.3k124183
asked Dec 13 '18 at 14:30
AndrewsAndrews
4081317
edited Dec 26 '18 at 21:28
user26857
39.3k124183
edited Dec 26 '18 at 21:28
user26857
39.3k124183
edited Dec 26 '18 at 21:28
user26857
39.3k124183
39.3k124183
asked Dec 13 '18 at 14:30
AndrewsAndrews
4081317
asked Dec 13 '18 at 14:30
AndrewsAndrews
4081317
asked Dec 13 '18 at 14:30
AndrewsAndrews
4081317
4081317
$begingroup$
You mean, there exists an isomorphism, but it just doesn't satisfy the injection/projection conditions?
$endgroup$
– rschwieb
Dec 13 '18 at 14:40
$begingroup$
Possible duplicate of A nonsplit short exact sequence of abelian groups with $B cong A oplus C$
$endgroup$
– yamete kudasai
Dec 13 '18 at 14:42
$begingroup$
What do you mean by "drop the condition"?
$endgroup$
– positrón0802
Dec 13 '18 at 14:57
add a comment |
$begingroup$
You mean, there exists an isomorphism, but it just doesn't satisfy the injection/projection conditions?
$endgroup$
– rschwieb
Dec 13 '18 at 14:40
$begingroup$
Possible duplicate of A nonsplit short exact sequence of abelian groups with $B cong A oplus C$
$endgroup$
– yamete kudasai
Dec 13 '18 at 14:42
$begingroup$
What do you mean by "drop the condition"?
$endgroup$
– positrón0802
Dec 13 '18 at 14:57
$begingroup$
You mean, there exists an isomorphism, but it just doesn't satisfy the injection/projection conditions?
$endgroup$
– rschwieb
Dec 13 '18 at 14:40
$begingroup$
You mean, there exists an isomorphism, but it just doesn't satisfy the injection/projection conditions?
$endgroup$
– rschwieb
Dec 13 '18 at 14:40
$begingroup$
Possible duplicate of A nonsplit short exact sequence of abelian groups with $B cong A oplus C$
$endgroup$
– yamete kudasai
Dec 13 '18 at 14:42
$begingroup$
Possible duplicate of A nonsplit short exact sequence of abelian groups with $B cong A oplus C$
$endgroup$
– yamete kudasai
Dec 13 '18 at 14:42
$begingroup$
What do you mean by "drop the condition"?
$endgroup$
– positrón0802
Dec 13 '18 at 14:57
$begingroup$
What do you mean by "drop the condition"?
$endgroup$
– positrón0802
Dec 13 '18 at 14:57
add a comment |
1 Answer
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$begingroup$
Here ring $R$ is communtative to make Hom$_R(C, - )$ a $R$-module.
answered Dec 13 '18 at 23:50
AndrewsAndrews
4081317
$endgroup$
add a comment |
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$begingroup$
Here ring $R$ is communtative to make Hom$_R(C, - )$ a $R$-module.
answered Dec 13 '18 at 23:50
AndrewsAndrews
4081317
$endgroup$
add a comment |
$begingroup$
Here ring $R$ is communtative to make Hom$_R(C, - )$ a $R$-module.
answered Dec 13 '18 at 23:50
AndrewsAndrews
4081317
$endgroup$
add a comment |
$begingroup$
Here ring $R$ is communtative to make Hom$_R(C, - )$ a $R$-module.
answered Dec 13 '18 at 23:50
AndrewsAndrews
4081317
$endgroup$
Here ring $R$ is communtative to make Hom$_R(C, - )$ a $R$-module.
answered Dec 13 '18 at 23:50
AndrewsAndrews
4081317
answered Dec 13 '18 at 23:50
AndrewsAndrews
4081317
answered Dec 13 '18 at 23:50
AndrewsAndrews
4081317
answered Dec 13 '18 at 23:50
AndrewsAndrews
4081317
4081317
add a comment |
add a comment |
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$begingroup$
You mean, there exists an isomorphism, but it just doesn't satisfy the injection/projection conditions?
$endgroup$
– rschwieb
Dec 13 '18 at 14:40
$begingroup$
Possible duplicate of A nonsplit short exact sequence of abelian groups with $B cong A oplus C$
$endgroup$
– yamete kudasai
Dec 13 '18 at 14:42
$begingroup$
What do you mean by "drop the condition"?
$endgroup$
– positrón0802
Dec 13 '18 at 14:57