# Difference between revisions of "Hyperhomology functor"

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− | A series of functors | + | 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 |

− | + | $$ | |

+ | \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 | + | 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=12890