One Hat Cyber Team
Your IP :
216.73.216.135
Server IP :
194.44.31.54
Server :
Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64
Server Software :
Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k
PHP Version :
5.6.40
Buat File
|
Buat Folder
Eksekusi
Dir :
~
/
usr
/
share
/
Macaulay2
/
Saturation
/
View File Name :
doc.m2
doc /// Node Key Saturation Headline ideal and submodule quotient, saturation, and annihilator routines Description Text This package provides various basic commutative algebra routines for working with ideals and modules. Tree :Main functionalities "ideal quotients and saturation" "module quotients, saturation, and annihilator" Subnodes (quotient, Ideal, Ideal) annihilator saturate Node Key "ideal quotients and saturation" Description Text @SUBSECTION "Ideal Quotients"@ The @TO2 {(quotient, Ideal, Ideal), "quotient"}@ of two ideals $I, J\subset R$ is ideal $I:J$ of elements $f\in R$ such that $f J \subset I$. Example R = QQ[a..d]; I = ideal(a^2*b-c^2, a*b^2-d^3, c^5-d); J = ideal(a^2,b^2,c^2,d^2); I:J P = quotient(I,J) Text The functions @TO ":"@ and @TO "quotient"@ perform the same basic operation, however @TT "quotient"@ takes options. @SUBSECTION "Saturation of Ideals"@ The @TO2 {(saturate, Ideal, Ideal), "saturation"}@ of an ideal $I\subset R$ with respect to another ideal $J\subset R$ is the ideal $I:J^\infty$ of elements $f\in R$ such that $f J^N\subset I$ for some $N$ large enough. If the ideal $J$ is not given, the ideal generated by the variables of the ring $R$ is used. For example, one way to homogenize an ideal is to homogenize the generators and then saturate with respect to the homogenizing variable. Example R = ZZ/32003[a..d]; I = ideal(a^3-b, a^4-c) Ih = homogenize(I, d) saturate(Ih, d) -- TODO Node Key "module quotients, saturation, and annihilator" Description Text @SUBSECTION "Module Quotients"@ There are two notions associated to the colon construction for modules. Similar to the case of ideals, the @TO2 {(quotient, Module, Module), "quotient"}@ of two $R$-modules $M, N$ contained in the same @TO ambient@ module is an ideal $M:N$ of elements $f\in R$ such that $f N \subset M$. This is equivalent to the @TO annihilator@ of the quotient module $(M+N)/M$. -- TODO: -- Example Text -- TODO: what is the ambient module assumed to be? The @TO2 {(quotient, Module, Ideal), "quotient"}@ of an $R$-module $M\subset F$ with respect to an ideal $J\subset R$ is the module $M:_F J$ of elements $f\in F$ such that $J f\subset M$. -- TODO: -- Example Text @SUBSECTION "Saturation of Modules"@ The @TO2 {(saturate, Module, Ideal), "saturation"}@ of an $R$-module $M\subset F$ with respect to an ideal $J\subset R$ is an $R$-module $M:_F J^\infty$ of elements $f\in F$ such that $J^N f\subset M$ for some $N$ large enough. If the ideal $J$ is not given, the ideal generated by the variables of the ring $R$ is used. If $M=M:_F J^\infty$ (or, equivalently, $M=M:_F J$), we say that $M$ is saturated with respect to $J$. We can use this command to remove graded submodules of finite length. Example R = ZZ/32003[a..d]; m = ideal vars R M = R^1 / (a * m^2) M / saturate 0_M Text @SUBSECTION "Module Annihilators"@ The @TO annihilator@ of an $R$-module $M$ is the ideal $\mathrm{ann}(M) = \{ f \in R | f M = 0 \}$. Example R = QQ[a..i]; M = cokernel genericMatrix(R,a,3,3) annihilator M Text You may also use the abbreviation @TT "ann"@: Example ann (M/(a*M)) ///