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Difference between revisions of "Multiplicity of a module"

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<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>
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<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>

Revision as of 21:54, 30 March 2012

with respect to an ideal

Let be a commutative ring with unit. A module over is said to be of finite length if there is a sequence of submodules (a Jordan–Hölder sequence) such that each of the quotients , , is a simple -module. (The number does not depend on the sequence chosen, by the Jordan–Hölder theorem.) Now let be an -module of finite type and an ideal contained in the radical of and such that is of finite length, and let be of Krull dimension . (The Krull dimension of a module is equal to the dimension of the ring where is the annihilator of , i.e. .) Then there exists a unique integer such that

for large enough. The number is called the multiplicity of with respect to . The multiplicity of an ideal is . Thus, the multiplicity of the maximal ideal of a local ring of dimension is equal to times the leading coefficient of the Hilbert–Samuel polynomial of , cf. Local ring.

There are some mild terminological discrepancies in the literature with respect to the Hilbert–Samuel polynomial. Let and . Then both and are sometimes called Hilbert–Samuel functions. For both and there are polynomials in (of degree and , respectively) such that and coincide with these polynomials for large . 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 is the multiplicity of its maximal ideal , .

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=23907