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TestIdeals :: ascendModule

ascendModule -- find the smallest submodule of free module containing a given submodule which is compatible with a given Cartier linear map

Synopsis

Description

Given an $n\times n$ matrix $U$ and a submodule $M$ of a free module $R^n$, ascendModule finds the smallest submodule $V$ of $R^n$ containing $M$ and which satisfies $U^{1 + p + \cdots + p^{e-1}} V\subseteq V^{[p^e]}$.

i1 : R = ZZ/2[a,b,c,d];
i2 : A = matrix {{b*c, a, 0}, {a^2* d, d^2 , c + d}};

             2       3
o2 : Matrix R  <--- R
i3 : M = image A;
i4 : U = matrix {{a^4  + a*b*c^2  + a*b*c*d, a^2* b}, {a^2*c*d^3 , a^3* c*d + a^3 *d^2  + b*c*d^3}};

             2       2
o4 : Matrix R  <--- R
i5 : ascendModule(1, M, U)

o5 = image | 0   a  bc  |
           | c+d d2 a2d |

                             2
o5 : R-module, submodule of R

For ease of use, instead of passing the module $M$, one can instead pass a matrix $A$ whose image is $M$, and ascendModule will return a matrix whose image is $V$.

i6 : ascendModule(1, A, U)

o6 = | 0   a  bc  |
     | c+d d2 a2d |

             2       3
o6 : Matrix R  <--- R

This method is described in M. Katzman and W. Zhang's "Annihilators of Artinian modules compatible with a Frobenius map", under the name "star-closure".

Ways to use ascendModule :

For the programmer

The object ascendModule is a method function.