Truncation of an injective resolution

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This piece comes from the proof of corollary 10.5.11 in Weibel's book on homological algebra.



There we start with a cochain complex $X^bullet$ and we are trying to construct a quasi-isomorphism $X^bullet to X'^bullet$ with $X'^bullet$ being complex of $F$-acyclic objects, where $F$ has cohomological dimension $n$.



We choose Cartan-Eilenberg resolution $X^bullet to I_CE^bullet, bullet$ and then Weibel suggests we take a (good) truncation $tau_leq n(I_CE^p bullet)$ to get a finite resolution by $F$-acyclic objects.



So I cannon see why the last object we are truncating at is actually $F$ acyclic. I initially wanted to splice the exact sequence $X^p to I_CE^p bullet$ and apply that with $A'$ and $A$ $F$-acyclic in $$0 to A' to A to A'' to 0,$$ $A''$ is also $F$-acyclic.



But than this doesn't work because $I^p,0 to I^p,1$ isn't injective. Can't quite see how cohomological dimension $n$ comes into play. Any thoughts would be appreciated.




homological-algebra




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    This piece comes from the proof of corollary 10.5.11 in Weibel's book on homological algebra.



    There we start with a cochain complex $X^bullet$ and we are trying to construct a quasi-isomorphism $X^bullet to X'^bullet$ with $X'^bullet$ being complex of $F$-acyclic objects, where $F$ has cohomological dimension $n$.



    We choose Cartan-Eilenberg resolution $X^bullet to I_CE^bullet, bullet$ and then Weibel suggests we take a (good) truncation $tau_leq n(I_CE^p bullet)$ to get a finite resolution by $F$-acyclic objects.



    So I cannon see why the last object we are truncating at is actually $F$ acyclic. I initially wanted to splice the exact sequence $X^p to I_CE^p bullet$ and apply that with $A'$ and $A$ $F$-acyclic in $$0 to A' to A to A'' to 0,$$ $A''$ is also $F$-acyclic.



    But than this doesn't work because $I^p,0 to I^p,1$ isn't injective. Can't quite see how cohomological dimension $n$ comes into play. Any thoughts would be appreciated.




    homological-algebra




    share|cite|improve this question























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      This piece comes from the proof of corollary 10.5.11 in Weibel's book on homological algebra.



      There we start with a cochain complex $X^bullet$ and we are trying to construct a quasi-isomorphism $X^bullet to X'^bullet$ with $X'^bullet$ being complex of $F$-acyclic objects, where $F$ has cohomological dimension $n$.



      We choose Cartan-Eilenberg resolution $X^bullet to I_CE^bullet, bullet$ and then Weibel suggests we take a (good) truncation $tau_leq n(I_CE^p bullet)$ to get a finite resolution by $F$-acyclic objects.



      So I cannon see why the last object we are truncating at is actually $F$ acyclic. I initially wanted to splice the exact sequence $X^p to I_CE^p bullet$ and apply that with $A'$ and $A$ $F$-acyclic in $$0 to A' to A to A'' to 0,$$ $A''$ is also $F$-acyclic.



      But than this doesn't work because $I^p,0 to I^p,1$ isn't injective. Can't quite see how cohomological dimension $n$ comes into play. Any thoughts would be appreciated.




      homological-algebra




      share|cite|improve this question













      This piece comes from the proof of corollary 10.5.11 in Weibel's book on homological algebra.



      There we start with a cochain complex $X^bullet$ and we are trying to construct a quasi-isomorphism $X^bullet to X'^bullet$ with $X'^bullet$ being complex of $F$-acyclic objects, where $F$ has cohomological dimension $n$.



      We choose Cartan-Eilenberg resolution $X^bullet to I_CE^bullet, bullet$ and then Weibel suggests we take a (good) truncation $tau_leq n(I_CE^p bullet)$ to get a finite resolution by $F$-acyclic objects.



      So I cannon see why the last object we are truncating at is actually $F$ acyclic. I initially wanted to splice the exact sequence $X^p to I_CE^p bullet$ and apply that with $A'$ and $A$ $F$-acyclic in $$0 to A' to A to A'' to 0,$$ $A''$ is also $F$-acyclic.



      But than this doesn't work because $I^p,0 to I^p,1$ isn't injective. Can't quite see how cohomological dimension $n$ comes into play. Any thoughts would be appreciated.





      homological-algebra





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      Bananeen

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