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One of the cohomological characteristics of a module over a commutative ring. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311801.png" /> be a Noetherian ring, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311802.png" /> be an ideal in it and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311803.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311804.png" />-module of finite type. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311806.png" />-depth of the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311807.png" /> is the least integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d0311808.png" /> for which
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<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/d031180/d0311809.png" /></td> </tr></table>
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The depth of a module is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118010.png" /> or by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118011.png" />. A different definition can be given in terms of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118013.png" />-regular sequence, i.e. a sequence of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118015.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118016.png" /> is not a [[Zero divisor|zero divisor]] in the module
+
One of the cohomological characteristics of a module over a commutative ring. Let  $  A $
 +
be a Noetherian ring, let  $  I $
 +
be an ideal in it and let  $  M $
 +
be an $  A $-
 +
module of finite type. Then the  $  I $-
 +
depth of the module  $  M $
 +
is the least integer  $  n $
 +
for which
  
<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/d031180/d03118017.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ext} _ {A}  ^ {n} ( A/I, M)  \neq  0.
 +
$$
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118018.png" />-depth of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118019.png" /> is equal to the length of the largest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118020.png" />-regular sequence consisting of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118021.png" />. The maximal ideal is usually taken for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118022.png" /> in the case of a local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118023.png" />. The following formula is valid:
+
The depth of a module is denoted by  $  \mathop{\rm depth} _ {I} ( M) $
 +
or by  $  \mathop{\rm prof} _ {I} ( M) $.  
 +
A different definition can be given in terms of an  $  M $-
 +
regular sequence, i.e. a sequence of elements $  a _ {1} \dots a _ {k} $
 +
of $  A $
 +
such that  $  a _ {i} $
 +
is not a [[Zero divisor|zero divisor]] in the module
  
<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/d031180/d03118024.png" /></td> </tr></table>
+
$$
 +
M/( a _ {1} \dots a _ {i - 1 }  ) M.
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118025.png" /> denotes a prime ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118026.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118027.png" /> is considered as a module over the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118028.png" />.
+
The  $  I $-
 +
depth of  $  M $
 +
is equal to the length of the largest  $  M $-
 +
regular sequence consisting of elements of  $  I $.  
 +
The maximal ideal is usually taken for  $  I $
 +
in the case of a local ring $  A $.
 +
The following formula is valid:
  
The concept of the depth of a module was introduced in [[#References|[1]]] under the name of homological codimension. If the projective dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118029.png" /> of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118030.png" /> over a local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118031.png" /> is finite, then
+
$$
 +
\mathop{\rm prof} _ {I} ( M)  = \
 +
\inf _ {\mathfrak p \supset I } \
 +
(  \mathop{\rm prof} ( M _ {\mathfrak p} )),
 +
$$
  
<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/d031180/d03118032.png" /></td> </tr></table>
+
where  $  \mathfrak p $
 +
denotes a prime ideal in  $  A $,
 +
while  $  M _ {\mathfrak p} $
 +
is considered as a module over the local ring  $  A _ {\mathfrak p} $.
  
In general <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118033.png" /> is not larger than the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118034.png" />.
+
The concept of the depth of a module was introduced in [[#References|[1]]] under the name of homological codimension. If the projective dimension $  \mathop{\rm dh} ( M) $
 +
of a module  $  M $
 +
over a local ring  $  A $
 +
is finite, then
  
The depth of a module is one of the basic tools in the study of modules. Thus, Cohen–Macaulay modules and rings (cf. [[Cohen–Macaulay ring|Cohen–Macaulay ring]]) have been defined in terms of the depth of modules. The Serre criterion (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118036.png" />) for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118037.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118038.png" />:
+
$$
 +
\mathop{\rm dh} ( M) +  \mathop{\rm prof} ( M)  =   \mathop{\rm prof} ( A).
 +
$$
  
<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/d031180/d03118039.png" /></td> </tr></table>
+
In general  $  \mathop{\rm prof} ( M) $
 +
is not larger than the dimension of  $  M $.
  
for all prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118041.png" />, often proves to be useful. Finally, the depth of modules is closely connected with local cohomology modules: The statement
+
The depth of a module is one of the basic tools in the study of modules. Thus, Cohen–Macaulay modules and rings (cf. [[Cohen–Macaulay ring|Cohen–Macaulay ring]]) have been defined in terms of the depth of modules. The Serre criterion ( $  S _ {k} $)
 +
for an  $  A $-
 +
module  $  M $:
  
<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/d031180/d03118042.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm prof}  M _ {\mathfrak p}  \geq  \inf \
 +
( k,  \mathop{\rm dim}  M _ {\mathfrak p} )
 +
$$
  
is equivalent to saying that the local cohomology modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118043.png" /> vanish if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d031/d031180/d03118044.png" />.
+
for all prime ideals  $  \mathfrak p $
 +
in  $  A $,
 +
often proves to be useful. Finally, the depth of modules is closely connected with local cohomology modules: The statement
 +
 
 +
$$
 +
\mathop{\rm prof} _ {I} ( M)  \geq  n
 +
$$
 +
 
 +
is equivalent to saying that the local cohomology modules $  H _ {I}  ^ {i} ( M) $
 +
vanish if $  i < n $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Auslander,  D.A. Buchsbaum,  "Homological dimension in Noetherian rings"  ''Proc. Nat. Acad. Sci. USA'' , '''42'''  (1956)  pp. 36–38</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,  "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Grothendieck,  "Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux" , ''SGA 2'' , IHES  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Auslander,  D.A. Buchsbaum,  "Homological dimension in Noetherian rings"  ''Proc. Nat. Acad. Sci. USA'' , '''42'''  (1956)  pp. 36–38</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.-P. Serre,  "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer  (1965)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Grothendieck,  "Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux" , ''SGA 2'' , IHES  (1965)</TD></TR></table>

Latest revision as of 17:32, 5 June 2020


One of the cohomological characteristics of a module over a commutative ring. Let $ A $ be a Noetherian ring, let $ I $ be an ideal in it and let $ M $ be an $ A $- module of finite type. Then the $ I $- depth of the module $ M $ is the least integer $ n $ for which

$$ \mathop{\rm Ext} _ {A} ^ {n} ( A/I, M) \neq 0. $$

The depth of a module is denoted by $ \mathop{\rm depth} _ {I} ( M) $ or by $ \mathop{\rm prof} _ {I} ( M) $. A different definition can be given in terms of an $ M $- regular sequence, i.e. a sequence of elements $ a _ {1} \dots a _ {k} $ of $ A $ such that $ a _ {i} $ is not a zero divisor in the module

$$ M/( a _ {1} \dots a _ {i - 1 } ) M. $$

The $ I $- depth of $ M $ is equal to the length of the largest $ M $- regular sequence consisting of elements of $ I $. The maximal ideal is usually taken for $ I $ in the case of a local ring $ A $. The following formula is valid:

$$ \mathop{\rm prof} _ {I} ( M) = \ \inf _ {\mathfrak p \supset I } \ ( \mathop{\rm prof} ( M _ {\mathfrak p} )), $$

where $ \mathfrak p $ denotes a prime ideal in $ A $, while $ M _ {\mathfrak p} $ is considered as a module over the local ring $ A _ {\mathfrak p} $.

The concept of the depth of a module was introduced in [1] under the name of homological codimension. If the projective dimension $ \mathop{\rm dh} ( M) $ of a module $ M $ over a local ring $ A $ is finite, then

$$ \mathop{\rm dh} ( M) + \mathop{\rm prof} ( M) = \mathop{\rm prof} ( A). $$

In general $ \mathop{\rm prof} ( M) $ is not larger than the dimension of $ M $.

The depth of a module is one of the basic tools in the study of modules. Thus, Cohen–Macaulay modules and rings (cf. Cohen–Macaulay ring) have been defined in terms of the depth of modules. The Serre criterion ( $ S _ {k} $) for an $ A $- module $ M $:

$$ \mathop{\rm prof} M _ {\mathfrak p} \geq \inf \ ( k, \mathop{\rm dim} M _ {\mathfrak p} ) $$

for all prime ideals $ \mathfrak p $ in $ A $, often proves to be useful. Finally, the depth of modules is closely connected with local cohomology modules: The statement

$$ \mathop{\rm prof} _ {I} ( M) \geq n $$

is equivalent to saying that the local cohomology modules $ H _ {I} ^ {i} ( M) $ vanish if $ i < n $.

References

[1] M. Auslander, D.A. Buchsbaum, "Homological dimension in Noetherian rings" Proc. Nat. Acad. Sci. USA , 42 (1956) pp. 36–38
[2] J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965)
[3] A. Grothendieck, "Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux" , SGA 2 , IHES (1965)
How to Cite This Entry:
Depth of a module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Depth_of_a_module&oldid=17353
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article