Namespaces
Variants
Actions

Difference between revisions of "Derived functor"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
A functor  "measuring"  the deviation of a given functor from being exact. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d0312901.png" /> be an additive functor from the product of the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d0312902.png" />-modules with the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d0312903.png" />-modules into the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d0312904.png" />-modules that is covariant in the first argument and contravariant in the second argument. From an injective resolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d0312905.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d0312906.png" /> and a projective resolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d0312907.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d0312908.png" /> one obtains a doubly-graded complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d0312909.png" />. The homology of the associated single complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129010.png" /> does not depend on the choice of resolutions, has functorial properties and is called the right derived functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129012.png" />. The basic property of a derived functor is the existence of long exact sequences
+
<!--
 +
d0312901.png
 +
$#A+1 = 33 n = 0
 +
$#C+1 = 33 : ~/encyclopedia/old_files/data/D031/D.0301290 Derived functor
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/d/d031/d031290/d03129013.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
<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/d/d031/d031290/d03129014.png" /></td> </tr></table>
+
A functor  "measuring" the deviation of a given functor from being exact. Let  $  T ( A , C ) $
 +
be an additive functor from the product of the category of  $  R _ {1} $-
 +
modules with the category of  $  R _ {2} $-
 +
modules into the category of  $  R $-
 +
modules that is covariant in the first argument and contravariant in the second argument. From an injective resolution  $  X $
 +
of  $  A $
 +
and a projective resolution  $  Y $
 +
of  $  C $
 +
one obtains a doubly-graded complex  $  T( X , Y ) $.  
 +
The homology of the associated single complex  $  T ( A , C ) $
 +
does not depend on the choice of resolutions, has functorial properties and is called the right derived functor  $  R  ^ {n} T ( A , C ) $
 +
of  $  T ( A , C ) $.  
 +
The basic property of a derived functor is the existence of long exact sequences
  
<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/d/d031/d031290/d03129015.png" /></td> </tr></table>
+
$$
 +
\rightarrow  R  ^ {n} T ( A  ^  \prime  , C )  \rightarrow  R  ^ {n} T ( A , C )  \rightarrow  R  ^ {n}
 +
T ( A  ^ {\prime\prime} , C ) \rightarrow
 +
$$
  
<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/d/d031/d031290/d03129016.png" /></td> </tr></table>
+
$$
 +
\rightarrow \
 +
R  ^ {n+} 1 T ( A  ^  \prime  , C )  \rightarrow \dots
 +
$$
 +
 
 +
$$
 +
\rightarrow  R  ^ {n} T ( A , C  ^ {\prime\prime} )  \rightarrow  R  ^ {n} T (
 +
A , C )  \rightarrow  R  ^ {n} T ( A , C  ^  \prime  ) \rightarrow
 +
$$
 +
 
 +
$$
 +
\rightarrow \
 +
R  ^ {n+} 1 T ( A , C  ^ {\prime\prime} )  \rightarrow \dots ,
 +
$$
  
 
induced by short exact sequences
 
induced by short exact sequences
  
<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/d/d031/d031290/d03129017.png" /></td> </tr></table>
+
$$
 +
0 \rightarrow  A  ^  \prime  \rightarrow  A  \rightarrow  A  ^ {\prime\prime}  \rightarrow  0,
 +
$$
  
<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/d/d031/d031290/d03129018.png" /></td> </tr></table>
+
$$
 +
0  \rightarrow  C  ^  \prime  \rightarrow  C  \rightarrow  C  ^ {\prime\prime}  \rightarrow  0 .
 +
$$
  
The left derived functor is defined analogously. The derived functor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129019.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129020.png" />. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129021.png" /> classifies extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129022.png" /> with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129023.png" /> up to equivalence (cf. [[Baer multiplication|Baer multiplication]]; [[Cohomology of algebras|Cohomology of algebras]]).
+
The left derived functor is defined analogously. The derived functor of $  \mathop{\rm Hom} _ {R} $
 +
is denoted by $  \mathop{\rm Ext} _ {R}  ^ {n} $.  
 +
The group $  \mathop{\rm Ext} _ {R}  ^ {1} ( A , C ) $
 +
classifies extensions of $  A $
 +
with kernel $  C $
 +
up to equivalence (cf. [[Baer multiplication|Baer multiplication]]; [[Cohomology of algebras|Cohomology of algebras]]).
  
 
====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">  S. MacLane,  "Homology" , Springer  (1963)</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">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The above article does not explain the sense in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129024.png" /> measures the deviation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129025.png" /> from being exact. The point is that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129026.png" /> is left exact (i.e. preserves the exactness of sequences of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129027.png" /> in the fist variable and of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129028.png" /> in the second), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129029.png" /> is naturally isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129030.png" />; if further <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129031.png" /> is exact, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129032.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031290/d03129033.png" />. Derived functors may also be defined for additive functors of a single variable between module categories, or, more generally, between arbitrary Abelian categories, provided the necessary injective or projective resolutions exist in the domain category.
+
The above article does not explain the sense in which $  R  ^ {n} T $
 +
measures the deviation of $  T $
 +
from being exact. The point is that if $  T $
 +
is left exact (i.e. preserves the exactness of sequences of the form 0 \rightarrow A  ^  \prime  \rightarrow A \rightarrow A  ^ {\prime\prime} $
 +
in the fist variable and of the form $  C  ^  \prime  \rightarrow C \rightarrow C  ^ {\prime\prime} \rightarrow 0 $
 +
in the second), then $  R  ^ {0} T $
 +
is naturally isomorphic to $  T $;  
 +
if further $  T $
 +
is exact, then $  R  ^ {n} T = 0 $
 +
for all $  n > 0 $.  
 +
Derived functors may also be defined for additive functors of a single variable between module categories, or, more generally, between arbitrary Abelian categories, provided the necessary injective or projective resolutions exist in the domain category.

Latest revision as of 17:33, 5 June 2020


A functor "measuring" the deviation of a given functor from being exact. Let $ T ( A , C ) $ be an additive functor from the product of the category of $ R _ {1} $- modules with the category of $ R _ {2} $- modules into the category of $ R $- modules that is covariant in the first argument and contravariant in the second argument. From an injective resolution $ X $ of $ A $ and a projective resolution $ Y $ of $ C $ one obtains a doubly-graded complex $ T( X , Y ) $. The homology of the associated single complex $ T ( A , C ) $ does not depend on the choice of resolutions, has functorial properties and is called the right derived functor $ R ^ {n} T ( A , C ) $ of $ T ( A , C ) $. The basic property of a derived functor is the existence of long exact sequences

$$ \rightarrow R ^ {n} T ( A ^ \prime , C ) \rightarrow R ^ {n} T ( A , C ) \rightarrow R ^ {n} T ( A ^ {\prime\prime} , C ) \rightarrow $$

$$ \rightarrow \ R ^ {n+} 1 T ( A ^ \prime , C ) \rightarrow \dots $$

$$ \rightarrow R ^ {n} T ( A , C ^ {\prime\prime} ) \rightarrow R ^ {n} T ( A , C ) \rightarrow R ^ {n} T ( A , C ^ \prime ) \rightarrow $$

$$ \rightarrow \ R ^ {n+} 1 T ( A , C ^ {\prime\prime} ) \rightarrow \dots , $$

induced by short exact sequences

$$ 0 \rightarrow A ^ \prime \rightarrow A \rightarrow A ^ {\prime\prime} \rightarrow 0, $$

$$ 0 \rightarrow C ^ \prime \rightarrow C \rightarrow C ^ {\prime\prime} \rightarrow 0 . $$

The left derived functor is defined analogously. The derived functor of $ \mathop{\rm Hom} _ {R} $ is denoted by $ \mathop{\rm Ext} _ {R} ^ {n} $. The group $ \mathop{\rm Ext} _ {R} ^ {1} ( A , C ) $ classifies extensions of $ A $ with kernel $ C $ up to equivalence (cf. Baer multiplication; Cohomology of algebras).

References

[1] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
[2] S. MacLane, "Homology" , Springer (1963)

Comments

The above article does not explain the sense in which $ R ^ {n} T $ measures the deviation of $ T $ from being exact. The point is that if $ T $ is left exact (i.e. preserves the exactness of sequences of the form $ 0 \rightarrow A ^ \prime \rightarrow A \rightarrow A ^ {\prime\prime} $ in the fist variable and of the form $ C ^ \prime \rightarrow C \rightarrow C ^ {\prime\prime} \rightarrow 0 $ in the second), then $ R ^ {0} T $ is naturally isomorphic to $ T $; if further $ T $ is exact, then $ R ^ {n} T = 0 $ for all $ n > 0 $. Derived functors may also be defined for additive functors of a single variable between module categories, or, more generally, between arbitrary Abelian categories, provided the necessary injective or projective resolutions exist in the domain category.

How to Cite This Entry:
Derived functor. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Derived_functor&oldid=15566
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article