Bridge between classical and “modern” derived functors
$begingroup$
This is a question for a reference.
What I would call the classical approach to derived functors, is the following:
Let $F:mathcal{A}to mathcal{B}$ be a right exact functor between abelian categories. When $mathcal{A}$ has enough projectives then we can define a projective resolution $P^bullet$ of any object $A$. We call $H^i(F(P^bullet))$ the derived functors of $F$.
This works mainly because projectives have a bunch of nice properties. For example if $Q^bullet$ is another projective resolution it is not difficult to construct a map between the two and check that we obtain the same results.
The more modern approach uses triangulated categories and localizations to construct $LF:D(mathcal{A})to D(mathcal{B})$. Here it is enough that a family of objects (not necessarily the family of projectives) is sufficiently large. Via $H^i(LF)$ we get much the same results.
However in practice one often uses non-projective resolutions but merely $F$-acyclic resolutions, e.g. flat resolutions in the case of $F=Motimes -$.
My question is if there is a nice source defining derived functors in the 'classical' way, without projectives?
For example it seems natural to me to define $Tor_1(M,N)$ by saying "pick a flat resolution of $N$, apply $Motimes-$ to the resolution and take homology of the resulting complex". But this clearly does not work so nicely, because given two different flat resolutions there is no reason, why there should be a map between them.
reference-request homological-algebra
asked Jan 15 at 14:32
Rene RecktenwaldRene Recktenwald
1808
$endgroup$
add a comment |
$begingroup$
This is a question for a reference.
What I would call the classical approach to derived functors, is the following:
Let $F:mathcal{A}to mathcal{B}$ be a right exact functor between abelian categories. When $mathcal{A}$ has enough projectives then we can define a projective resolution $P^bullet$ of any object $A$. We call $H^i(F(P^bullet))$ the derived functors of $F$.
This works mainly because projectives have a bunch of nice properties. For example if $Q^bullet$ is another projective resolution it is not difficult to construct a map between the two and check that we obtain the same results.
The more modern approach uses triangulated categories and localizations to construct $LF:D(mathcal{A})to D(mathcal{B})$. Here it is enough that a family of objects (not necessarily the family of projectives) is sufficiently large. Via $H^i(LF)$ we get much the same results.
However in practice one often uses non-projective resolutions but merely $F$-acyclic resolutions, e.g. flat resolutions in the case of $F=Motimes -$.
My question is if there is a nice source defining derived functors in the 'classical' way, without projectives?
For example it seems natural to me to define $Tor_1(M,N)$ by saying "pick a flat resolution of $N$, apply $Motimes-$ to the resolution and take homology of the resulting complex". But this clearly does not work so nicely, because given two different flat resolutions there is no reason, why there should be a map between them.
reference-request homological-algebra
asked Jan 15 at 14:32
Rene RecktenwaldRene Recktenwald
1808
$endgroup$
add a comment |
$begingroup$
This is a question for a reference.
What I would call the classical approach to derived functors, is the following:
Let $F:mathcal{A}to mathcal{B}$ be a right exact functor between abelian categories. When $mathcal{A}$ has enough projectives then we can define a projective resolution $P^bullet$ of any object $A$. We call $H^i(F(P^bullet))$ the derived functors of $F$.
This works mainly because projectives have a bunch of nice properties. For example if $Q^bullet$ is another projective resolution it is not difficult to construct a map between the two and check that we obtain the same results.
The more modern approach uses triangulated categories and localizations to construct $LF:D(mathcal{A})to D(mathcal{B})$. Here it is enough that a family of objects (not necessarily the family of projectives) is sufficiently large. Via $H^i(LF)$ we get much the same results.
However in practice one often uses non-projective resolutions but merely $F$-acyclic resolutions, e.g. flat resolutions in the case of $F=Motimes -$.
My question is if there is a nice source defining derived functors in the 'classical' way, without projectives?
For example it seems natural to me to define $Tor_1(M,N)$ by saying "pick a flat resolution of $N$, apply $Motimes-$ to the resolution and take homology of the resulting complex". But this clearly does not work so nicely, because given two different flat resolutions there is no reason, why there should be a map between them.
reference-request homological-algebra
asked Jan 15 at 14:32
Rene RecktenwaldRene Recktenwald
1808
$endgroup$
This is a question for a reference.
What I would call the classical approach to derived functors, is the following:
Let $F:mathcal{A}to mathcal{B}$ be a right exact functor between abelian categories. When $mathcal{A}$ has enough projectives then we can define a projective resolution $P^bullet$ of any object $A$. We call $H^i(F(P^bullet))$ the derived functors of $F$.
This works mainly because projectives have a bunch of nice properties. For example if $Q^bullet$ is another projective resolution it is not difficult to construct a map between the two and check that we obtain the same results.
The more modern approach uses triangulated categories and localizations to construct $LF:D(mathcal{A})to D(mathcal{B})$. Here it is enough that a family of objects (not necessarily the family of projectives) is sufficiently large. Via $H^i(LF)$ we get much the same results.
However in practice one often uses non-projective resolutions but merely $F$-acyclic resolutions, e.g. flat resolutions in the case of $F=Motimes -$.
My question is if there is a nice source defining derived functors in the 'classical' way, without projectives?
For example it seems natural to me to define $Tor_1(M,N)$ by saying "pick a flat resolution of $N$, apply $Motimes-$ to the resolution and take homology of the resulting complex". But this clearly does not work so nicely, because given two different flat resolutions there is no reason, why there should be a map between them.
reference-request homological-algebra
reference-request homological-algebra
asked Jan 15 at 14:32
Rene RecktenwaldRene Recktenwald
1808
asked Jan 15 at 14:32
Rene RecktenwaldRene Recktenwald
1808
asked Jan 15 at 14:32
Rene RecktenwaldRene Recktenwald
1808
asked Jan 15 at 14:32
Rene RecktenwaldRene Recktenwald
1808
asked Jan 15 at 14:32
Rene RecktenwaldRene Recktenwald
1808
1808
add a comment |
add a comment |
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