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Difference between revisions of "Hyperhomology functor"

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A series of functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h0484801.png" /> on the category of complexes connected with some functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h0484802.png" />. In fact, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h0484803.png" /> be a covariant additive functor from an Abelian category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h0484804.png" /> with a sufficient number of projective objects into an Abelian category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h0484805.png" />. Further, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h0484806.png" /> be a chain complex with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h0484807.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h0484808.png" /> be a Cartan–Eilenberg resolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h0484809.png" />, consisting of projective objects. Then the bicomplex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h04848010.png" /> determines the homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h04848011.png" /> and two spectral sequences (cf. [[Spectral sequence|Spectral sequence]]) converging to them with initial terms
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A series of functors $\mathbf{L}_n F$ on the category of complexes connected with some functor $F$. In fact, let $F\colon A \to B$ be a covariant additive functor from an Abelian category $A$ with a sufficient number of projective objects into an Abelian category $B$. Further, let $K_{\bullet}$ be a chain complex with values in $A$ and let $L_{\bullet \bullet}$ be a Cartan–Eilenberg resolution of $K_{\bullet}$, consisting of projective objects. Then the bicomplex $F(L_{\bullet\bullet})$ determines the homology groups $H_n(F(L_{\bullet\bullet})) = \mathbf{L}_n F(K_{\bullet})$ and two spectral sequences (cf. [[Spectral sequence|Spectral sequence]]) converging to them with initial terms
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h04848012.png" /></td> </tr></table>
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$$
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  \prime E^2_{p,q} = H_p(L_q F(K_{\bullet}))
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  \quad \text{and} \quad
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  \prime\prime E^2_{p,q} = L_p F(H_q(K_{\bullet})) .
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$$
  
These homology groups and spectral sequences depend functorially on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h04848013.png" /> and are known, respectively, as the hyperhomology functors for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h04848014.png" /> and the spectral hyperhomology functors for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h04848015.png" />. The hyperhomology functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h04848016.png" /> is a homology functor on the category of complexes in the following important cases: when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h04848017.png" /> commutes with inductive limits; when the objects in the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h04848018.png" /> have projective resolutions of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048480/h04848019.png" />; or when it is considered on the category of complexes with positive degrees.
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These homology groups and spectral sequences depend functorially on $K_{\bullet}$ and are known, respectively, as the hyperhomology functors for $F$ and the spectral hyperhomology functors for $F$. The hyperhomology functor $\mathbf{L}_{\bullet} F$ is a homology functor on the category of complexes in the following important cases: when $F$ commutes with inductive limits; when the objects in the category $A$ have projective resolutions of length $\le n$; or when it is considered on the category of complexes with positive degrees.
  
 
Hypercohomology functors are defined dually.
 
Hypercohomology functors are defined dually.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck,  "Sur quelques points d'algèbre homologique"  ''Tohoku Math. J.'' , '''9'''  (1957)  pp. 119–221</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Cartan,  S. Eilenberg,  "Homological algebra" , Princeton Univ. Press  (1956)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Grothendieck,  "Sur quelques points d'algèbre homologique"  ''Tohoku Math. J.'' , '''9'''  (1957)  pp. 119–221</TD></TR></table>
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Revision as of 10:44, 12 February 2017

A series of functors $\mathbf{L}_n F$ on the category of complexes connected with some functor $F$. In fact, let $F\colon A \to B$ be a covariant additive functor from an Abelian category $A$ with a sufficient number of projective objects into an Abelian category $B$. Further, let $K_{\bullet}$ be a chain complex with values in $A$ and let $L_{\bullet \bullet}$ be a Cartan–Eilenberg resolution of $K_{\bullet}$, consisting of projective objects. Then the bicomplex $F(L_{\bullet\bullet})$ determines the homology groups $H_n(F(L_{\bullet\bullet})) = \mathbf{L}_n F(K_{\bullet})$ and two spectral sequences (cf. Spectral sequence) converging to them with initial terms

$$ \prime E^2_{p,q} = H_p(L_q F(K_{\bullet})) \quad \text{and} \quad \prime\prime E^2_{p,q} = L_p F(H_q(K_{\bullet})) . $$

These homology groups and spectral sequences depend functorially on $K_{\bullet}$ and are known, respectively, as the hyperhomology functors for $F$ and the spectral hyperhomology functors for $F$. The hyperhomology functor $\mathbf{L}_{\bullet} F$ is a homology functor on the category of complexes in the following important cases: when $F$ commutes with inductive limits; when the objects in the category $A$ have projective resolutions of length $\le n$; or when it is considered on the category of complexes with positive degrees.

Hypercohomology functors are defined dually.

References

[1] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[2] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tohoku Math. J. , 9 (1957) pp. 119–221
How to Cite This Entry:
Hyperhomology functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hyperhomology_functor&oldid=40205
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article