How to decide what kind of resolutions should we use to calculate Ext and Tor? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)when to use projective vs. injective resolutionDoes finite projective resolution imply finite free resolution?when to use projective vs. injective resolutionUnderstanding the maps in the long exact sequence of $operatornameExt$Ext and Tor over noncommutative ringsWhy do we impose conditions on the modules in resolutions?Does the adjoint relationship between Tensor and Hom functors give an adjoint relationship between Ext and Tor?Calculating Ext and Tor for Z/5Z and Z/7ZDefinitions of Tor and Ext functors for Abelian groups via resolutions“Good” resolution for calculating Ext and TorBridge between classical and “modern” derived functors
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How to decide what kind of resolutions should we use to calculate Ext and Tor?
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)when to use projective vs. injective resolutionDoes finite projective resolution imply finite free resolution?when to use projective vs. injective resolutionUnderstanding the maps in the long exact sequence of $operatornameExt$Ext and Tor over noncommutative ringsWhy do we impose conditions on the modules in resolutions?Does the adjoint relationship between Tensor and Hom functors give an adjoint relationship between Ext and Tor?Calculating Ext and Tor for Z/5Z and Z/7ZDefinitions of Tor and Ext functors for Abelian groups via resolutions“Good” resolution for calculating Ext and TorBridge between classical and “modern” derived functors
$begingroup$
In general, when we calculate groups like Ext and Tor, how should we choose between free, projective and injective resolutions?
For example, why does Hatcher 3.1 only consider free resolution but not the other two? Is this simply because we can find free resolutions for abelian groups?
abstract-algebra algebraic-topology homological-algebra
asked Mar 31 at 18:50
2010 Kur2010 Kur
204
$endgroup$
add a comment |
$begingroup$
In general, when we calculate groups like Ext and Tor, how should we choose between free, projective and injective resolutions?
For example, why does Hatcher 3.1 only consider free resolution but not the other two? Is this simply because we can find free resolutions for abelian groups?
abstract-algebra algebraic-topology homological-algebra
asked Mar 31 at 18:50
2010 Kur2010 Kur
204
$endgroup$
2
$begingroup$
When we consider abelian groups, free is equivalent to projective since $mathbbZ$ is a PID
$endgroup$
– user289143
Mar 31 at 19:00
$begingroup$
Projective resolutions are "dual" to injective resolutions, so we are often "free" to use both. So the answer to the title question probably is: "We decide by convenience".
$endgroup$
– Dietrich Burde
Mar 31 at 19:11
$begingroup$
You'll probably find this related question helpful.
$endgroup$
– jgon
Apr 1 at 0:30
add a comment |
$begingroup$
In general, when we calculate groups like Ext and Tor, how should we choose between free, projective and injective resolutions?
For example, why does Hatcher 3.1 only consider free resolution but not the other two? Is this simply because we can find free resolutions for abelian groups?
abstract-algebra algebraic-topology homological-algebra
asked Mar 31 at 18:50
2010 Kur2010 Kur
204
$endgroup$
In general, when we calculate groups like Ext and Tor, how should we choose between free, projective and injective resolutions?
For example, why does Hatcher 3.1 only consider free resolution but not the other two? Is this simply because we can find free resolutions for abelian groups?
abstract-algebra algebraic-topology homological-algebra
abstract-algebra algebraic-topology homological-algebra
asked Mar 31 at 18:50
2010 Kur2010 Kur
204
asked Mar 31 at 18:50
2010 Kur2010 Kur
204
edited Mar 31 at 18:56
2010 Kur
asked Mar 31 at 18:50
2010 Kur2010 Kur
204
asked Mar 31 at 18:50
2010 Kur2010 Kur
204
asked Mar 31 at 18:50
2010 Kur2010 Kur
204
204
2
$begingroup$
When we consider abelian groups, free is equivalent to projective since $mathbbZ$ is a PID
$endgroup$
– user289143
Mar 31 at 19:00
$begingroup$
Projective resolutions are "dual" to injective resolutions, so we are often "free" to use both. So the answer to the title question probably is: "We decide by convenience".
$endgroup$
– Dietrich Burde
Mar 31 at 19:11
$begingroup$
You'll probably find this related question helpful.
$endgroup$
– jgon
Apr 1 at 0:30
add a comment |
2
$begingroup$
When we consider abelian groups, free is equivalent to projective since $mathbbZ$ is a PID
$endgroup$
– user289143
Mar 31 at 19:00
$begingroup$
Projective resolutions are "dual" to injective resolutions, so we are often "free" to use both. So the answer to the title question probably is: "We decide by convenience".
$endgroup$
– Dietrich Burde
Mar 31 at 19:11
$begingroup$
You'll probably find this related question helpful.
$endgroup$
– jgon
Apr 1 at 0:30
2
2
$begingroup$
When we consider abelian groups, free is equivalent to projective since $mathbbZ$ is a PID
$endgroup$
– user289143
Mar 31 at 19:00
$begingroup$
When we consider abelian groups, free is equivalent to projective since $mathbbZ$ is a PID
$endgroup$
– user289143
Mar 31 at 19:00
$begingroup$
Projective resolutions are "dual" to injective resolutions, so we are often "free" to use both. So the answer to the title question probably is: "We decide by convenience".
$endgroup$
– Dietrich Burde
Mar 31 at 19:11
$begingroup$
Projective resolutions are "dual" to injective resolutions, so we are often "free" to use both. So the answer to the title question probably is: "We decide by convenience".
$endgroup$
– Dietrich Burde
Mar 31 at 19:11
$begingroup$
You'll probably find this related question helpful.
$endgroup$
– jgon
Apr 1 at 0:30
$begingroup$
You'll probably find this related question helpful.
$endgroup$
– jgon
Apr 1 at 0:30
add a comment |
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2
$begingroup$
When we consider abelian groups, free is equivalent to projective since $mathbbZ$ is a PID
$endgroup$
– user289143
Mar 31 at 19:00
$begingroup$
Projective resolutions are "dual" to injective resolutions, so we are often "free" to use both. So the answer to the title question probably is: "We decide by convenience".
$endgroup$
– Dietrich Burde
Mar 31 at 19:11
$begingroup$
You'll probably find this related question helpful.
$endgroup$
– jgon
Apr 1 at 0:30