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''with respect to an ideal''
 
''with respect to an ideal''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m0654901.png" /> be a commutative ring with unit. A module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m0654902.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m0654903.png" /> is said to be of finite length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m0654904.png" /> if there is a sequence of submodules (a Jordan–Hölder sequence) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m0654905.png" /> such that each of the quotients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m0654906.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m0654907.png" />, is a simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m0654908.png" />-module. (The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m0654909.png" /> does not depend on the sequence chosen, by the [[Jordan–Hölder theorem|Jordan–Hölder theorem]].) Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549010.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549011.png" />-module of finite type and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549012.png" /> an ideal contained in the radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549013.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549014.png" /> is of finite length, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549015.png" /> be of Krull dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549016.png" />. (The Krull dimension of a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549017.png" /> is equal to the dimension of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549018.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549019.png" /> is the annihilator of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549020.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549021.png" />.) Then there exists a unique integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549022.png" /> such that
+
Let $  A $
 +
be a commutative ring with unit. A module $  M $
 +
over $  A $
 +
is said to be of finite length $  n $
 +
if there is a sequence of submodules (a Jordan–Hölder sequence) $  M _ {0} \subset  \cdots \subset  M _ {n} $
 +
such that each of the quotients $  M _ {i} / M _ {i+ 1} $,  
 +
$  i = 0, \dots, n - 1 $,
 +
is a simple $  A $-module. (The number $  n $
 +
does not depend on the sequence chosen, by the [[Jordan–Hölder theorem|Jordan–Hölder theorem]].) Now let $  M $
 +
be an $  A $-module of finite type and $  \mathfrak a $
 +
an ideal contained in the radical of $  A $
 +
and such that $  M / \mathfrak a M $
 +
is of finite length, and let $  M \neq 0 $
 +
be of Krull dimension $  d $.  
 +
(The Krull dimension of a module $  M $
 +
is equal to the dimension of the ring $  A / \mathfrak q ( M) $
 +
where $  \mathfrak q ( M) $
 +
is the annihilator of $  M $,  
 +
i.e. $  \mathfrak q ( M) = \{ {a \in A } : {a M = 0 } \} $.)  
 +
Then there exists a unique integer $  e _ {A} ( \mathfrak a ;  M ) $
 +
such that
  
<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/m/m065/m065490/m06549023.png" /></td> </tr></table>
+
$$
 +
\textrm{length} _ {A} ( M / \mathfrak a  ^ {n+ 1} M )  = \
 +
e _ {A} ( \mathfrak a ; M )
 +
\frac{n  ^ {d} }{d!}
 +
+ \textrm{(lower degree terms)}
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549024.png" /> large enough. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549025.png" /> is called the multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549026.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549027.png" />. The multiplicity of an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549028.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549029.png" />. Thus, the multiplicity of the maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549030.png" /> of a local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549031.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549032.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549033.png" /> times the leading coefficient of the Hilbert–Samuel polynomial of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549034.png" />, cf. [[Local ring|Local ring]].
+
for $  n $
 +
large enough. The number $  e _ {A} ( \mathfrak a ;  M ) $
 +
is called the multiplicity of $  M $
 +
with respect to $  \mathfrak a $.  
 +
The multiplicity of an ideal $  \mathfrak a $
 +
is $  e ( \mathfrak a ) = e _ {A} ( \mathfrak a ;  A ) $.  
 +
Thus, the multiplicity of the maximal ideal $  \mathfrak m $
 +
of a local ring $  A $
 +
of dimension $  d $
 +
is equal to $  ( d - 1 ) ! $
 +
times the leading coefficient of the Hilbert–Samuel polynomial of $  A $,  
 +
cf. [[Local ring|Local ring]].
  
There are some mild terminological discrepancies in the literature with respect to the Hilbert–Samuel polynomial. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549036.png" />. Then both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549038.png" /> are sometimes called Hilbert–Samuel functions. For both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549040.png" /> there are polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549041.png" /> (of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549043.png" />, respectively) such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549045.png" /> coincide with these polynomials for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549046.png" />. Both these polynomials occur in the literature under the name Hilbert–Samuel polynomial.
+
There are some mild terminological discrepancies in the literature with respect to the Hilbert–Samuel polynomial. Let $  \psi ( n) = \textrm{length} _ {A} ( M / \mathfrak a  ^ {n+ 1} M ) $
 +
and $  \chi ( n) = \textrm{length} _ {A} ( \mathfrak a  ^ {n} M / \mathfrak a  ^ {n+ 1} M ) $.  
 +
Then both $  \psi ( n) $
 +
and $  \chi ( n) $
 +
are sometimes called Hilbert–Samuel functions. For both $  \psi ( n) $
 +
and $  \chi ( n) $
 +
there are polynomials in $  n $ (of degree $  d $
 +
and $  d - 1 $,  
 +
respectively) such that $  \psi ( n) $
 +
and $  \chi ( n) $
 +
coincide with these polynomials for large $  n $.  
 +
Both these polynomials occur in the literature under the name Hilbert–Samuel polynomial.
  
 
For a more general set-up cf. [[#References|[a1]]].
 
For a more general set-up cf. [[#References|[a1]]].
  
The multiplicity of a local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549047.png" /> is the multiplicity of its maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065490/m06549049.png" />.
+
The multiplicity of a local ring $  A $
 +
is the multiplicity of its maximal ideal $  \mathfrak m $,  
 +
$  e _ {A} ( \mathfrak m ;  A ) $.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki,   "Algèbre commutative" , Masson (1983) pp. Chapt. 8, §4: Dimension</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Nagata,   "Local rings" , Interscience (1962) pp. Chapt. III, §23</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Mumford,   "Algebraic geometry" , '''1. Complex projective varieties''' , Springer (1976) pp. Appendix to Chapt. 6</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> O. Zariski,   P. Samuel,   "Commutative algebra" , '''2''' , v. Nostrand (1960) pp. Chapt. VIII, §10</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N. Bourbaki, "Algèbre commutative" , Masson (1983) pp. Chapt. 8, §4: Dimension {{MR|2333539}} {{MR|2284892}} {{MR|0260715}} {{MR|0194450}} {{MR|0217051}} {{MR|0171800}} {{ZBL|0579.13001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Nagata, "Local rings" , Interscience (1962) pp. Chapt. III, §23 {{MR|0155856}} {{ZBL|0123.03402}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Mumford, "Algebraic geometry" , '''1. Complex projective varieties''' , Springer (1976) pp. Appendix to Chapt. 6 {{MR|0453732}} {{ZBL|0356.14002}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , v. Nostrand (1960) pp. Chapt. VIII, §10 {{MR|0120249}} {{ZBL|0121.27801}} </TD></TR></table>

Latest revision as of 06:47, 16 June 2022


with respect to an ideal

Let $ A $ be a commutative ring with unit. A module $ M $ over $ A $ is said to be of finite length $ n $ if there is a sequence of submodules (a Jordan–Hölder sequence) $ M _ {0} \subset \cdots \subset M _ {n} $ such that each of the quotients $ M _ {i} / M _ {i+ 1} $, $ i = 0, \dots, n - 1 $, is a simple $ A $-module. (The number $ n $ does not depend on the sequence chosen, by the Jordan–Hölder theorem.) Now let $ M $ be an $ A $-module of finite type and $ \mathfrak a $ an ideal contained in the radical of $ A $ and such that $ M / \mathfrak a M $ is of finite length, and let $ M \neq 0 $ be of Krull dimension $ d $. (The Krull dimension of a module $ M $ is equal to the dimension of the ring $ A / \mathfrak q ( M) $ where $ \mathfrak q ( M) $ is the annihilator of $ M $, i.e. $ \mathfrak q ( M) = \{ {a \in A } : {a M = 0 } \} $.) Then there exists a unique integer $ e _ {A} ( \mathfrak a ; M ) $ such that

$$ \textrm{length} _ {A} ( M / \mathfrak a ^ {n+ 1} M ) = \ e _ {A} ( \mathfrak a ; M ) \frac{n ^ {d} }{d!} + \textrm{(lower degree terms)} $$

for $ n $ large enough. The number $ e _ {A} ( \mathfrak a ; M ) $ is called the multiplicity of $ M $ with respect to $ \mathfrak a $. The multiplicity of an ideal $ \mathfrak a $ is $ e ( \mathfrak a ) = e _ {A} ( \mathfrak a ; A ) $. Thus, the multiplicity of the maximal ideal $ \mathfrak m $ of a local ring $ A $ of dimension $ d $ is equal to $ ( d - 1 ) ! $ times the leading coefficient of the Hilbert–Samuel polynomial of $ A $, cf. Local ring.

There are some mild terminological discrepancies in the literature with respect to the Hilbert–Samuel polynomial. Let $ \psi ( n) = \textrm{length} _ {A} ( M / \mathfrak a ^ {n+ 1} M ) $ and $ \chi ( n) = \textrm{length} _ {A} ( \mathfrak a ^ {n} M / \mathfrak a ^ {n+ 1} M ) $. Then both $ \psi ( n) $ and $ \chi ( n) $ are sometimes called Hilbert–Samuel functions. For both $ \psi ( n) $ and $ \chi ( n) $ there are polynomials in $ n $ (of degree $ d $ and $ d - 1 $, respectively) such that $ \psi ( n) $ and $ \chi ( n) $ coincide with these polynomials for large $ n $. Both these polynomials occur in the literature under the name Hilbert–Samuel polynomial.

For a more general set-up cf. [a1].

The multiplicity of a local ring $ A $ is the multiplicity of its maximal ideal $ \mathfrak m $, $ e _ {A} ( \mathfrak m ; A ) $.

References

[a1] N. Bourbaki, "Algèbre commutative" , Masson (1983) pp. Chapt. 8, §4: Dimension MR2333539 MR2284892 MR0260715 MR0194450 MR0217051 MR0171800 Zbl 0579.13001
[a2] M. Nagata, "Local rings" , Interscience (1962) pp. Chapt. III, §23 MR0155856 Zbl 0123.03402
[a3] D. Mumford, "Algebraic geometry" , 1. Complex projective varieties , Springer (1976) pp. Appendix to Chapt. 6 MR0453732 Zbl 0356.14002
[a4] O. Zariski, P. Samuel, "Commutative algebra" , 2 , v. Nostrand (1960) pp. Chapt. VIII, §10 MR0120249 Zbl 0121.27801
How to Cite This Entry:
Multiplicity of a module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplicity_of_a_module&oldid=16483