正則列









曖昧さ回避
 この項目では、可換環論における正則列について説明しています。距離空間における点列については「コーシー列」をご覧ください。


数学、特に可換環論において正則列（せいそくれつ、英: regular sequence）とは、不定元のように振る舞う可環環の元の列のことである[1]。例えば、係数環 R を持つ多項式環 R[X1, ..., Xn] において X1, ..., Xn は正則列である。




目次






  • 1 定義


  • 2 例


  • 3 脚注


  • 4 参考文献


  • 5 関連項目





定義


R を可換環、M を R-加群とする。元 x ∈ R が M-正則元（M-regular element）であるとは、x が加群 M 上の零因子でないことである。列 x1, ..., xn ∈ R が M-正則列（
M-regular sequence）であるとは2条件



  • 各 xi は M/(x1, ..., xi−1)M-正則元

  • M/(x1, ..., xn)M ≠ 0


が成り立つことである。M = R のときには接頭語「R-」はしばしば省略される。


M-正則列を並び変えたものは M-正則列になるとは限らない[2]。ただしネーター局所環の極大イデアルに含まれる正則列は並び変えても正則列であることがわかる[3]。



例


R を可換環とする。



  • 多項式環 R[X1, ..., Xn] において X1, ..., Xn は正則列である。


脚注





  1. ^ Bruns & Herzog 1993.


  2. ^ Eisenbud 1995, Example 17.3.


  3. ^ Eisenbud 1995, Corollary 17.2.




参考文献




  • Bruns, Winfried; Herzog, Jürgen (1993). Cohen-Macaulay rings. Cambridge Studies in Advanced Mathematics. 39 (Rev. ed.). Cambridge University Press. ISBN 0-521-56674-6. MR 1251956. Zbl 0909.13005. https://books.google.com/books?id=LF6CbQk9uScC&pg=PA3. 


  • Eisenbud, David (1995). Commutative algebra. Graduate Texts in Mathematics. 150. Springer-Verlag. ISBN 0-387-94268-8. MR 1322960. Zbl 0819.13001. https://books.google.com/books?id=xDwmBQAAQBAJ&pg=PA417. 


  • Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge Studies in Advanced Mathematics. 8. Cambridge University Press. ISBN 0-521-36764-6. MR 0879273. Zbl 0603.13001. https://books.google.com/books?id=yJwNrABugDEC&pg=PA123. 



関連項目


  • 深さ (環論)



Popular posts from this blog

Can a sorcerer learn a 5th-level spell early by creating spell slots using the Font of Magic feature?

Image
28 6 $begingroup$ Per the Font of Magic feature, sorcerer can use Flexible Casting to create 5th-level spell slots at level 7, even though they are not typically available until level 9. You can transform unexpended sorcery points into one spell slot as a bonus action on your turn. The Creating Spell Slots table shows the cost of creating a spell slot of [5th level is 7] If such a sorcerer levels up while still having this 5th-level spell slot, can they choose a 5th-level spell as the spell they gain upon levelling up? The Spellcasting feature states: Additionally, when you gain a level in this class, you can choose one of the sorcerer spells you know and replace it with another spell from the sorcerer spell list, which also must be of a level for which you have spell slots.
Read more

Does disintegrating a polymorphed enemy still kill it after the 2018 errata?

Image
up vote 13 down vote favorite 1 After the 2018 errata, the disintegrate spell description now reads: A creature targeted by this spell must make a Dexterity saving throw. On a failed save, the target takes 10d6 + 40 force damage. The target is disintegrated if this damage leaves it with 0 hit points. If you were to polymorph an enemy into a rat and then disintegrate it, would the enemy be disintegrated or would it just return to its original form? I know that for Druids, it’s not an instant kill anymore, but is this the case for polymorph as well? dnd-5e spells polymorph errata share | improve this question edited 18 hours ago
Read more

A Topological Invariant for $pi_3(U(n))$

Image
3 3 $begingroup$ Recently, I saw a construction of topological invariant for $pi_3(U(n))$ with $ngeq 2$ : $$ N=frac{1}{24pi^2}int_{S^3} d^3x epsilon^{ijk} Tr[(U^{-1}partial_{x_i}U)(U^{-1}partial_{x_j}U)(U^{-1}partial_{x_k}U)] , $$ where $Uin U(n)$ depends on $boldsymbol{x}=(x_1,x_2,x_3)in S^3$ , $epsilon^{ijk}$ is the Levi-Civita symbol, $i,j,k=1,2,3$ , and the duplicated indexes are summed over. It is claimed that $N$ is an integer, but why? Update 02/02/2019 I think I got an argument for $n=2$ . In this case, $U=e^{i varphi} q$ with $qin SU(2)$ . Due to the trace and the Levi-Civita symbol in $N$ , $varphi$ does not contribute to $N$ . As $Tr[q^{dagger}partial_i q]=0$ and $(q^{dagger}partial_i q)^{dagger}=-q^{dagger}partial_i q$ , $q^{dagger}partial_i q$ in geneeral has the form $q^{dagger}partia
Read more