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frobeniusDirectImage(ZZ,Matrix) -- compute the pushforward of map of modules under the $p$th toric Frobenius map

Description

The $p$th toric Frobenius is a toric morphism $F_p \colon X \rightarrow X$ which is the extension of the natural group homomorphism $T_X \rightarrow T_X$ given by raising all coordinates to the $p$th power. This allows one to view the Cox ring $R$ of $X$ as a module over itself, with the module action being $r \cdot m :\!= r^p m$. The extension of this action to modules also allows one to compute the pushforward by $F_p$. Note that $p$ need not be prime, nor related to the characteristic of the ground field in any way. The pushforward is an endofunctor on the category of $R$-modules, so we may apply it to maps between modules.

i1 : X = hirzebruchSurface 1;
i2 : R = ring X;
i3 : M = koszul_1 vars R

o3 = | x_0 x_1 x_2 x_3 |

             1      4
o3 : Matrix R  <-- R
i4 : frobeniusDirectImage_2 M

o4 = {0, 0} | x_0 0   0 0 0 0   0 x_1 0 0   x_2 0 x_3 0 0   0 |
     {0, 1} | 0   x_0 0 0 0 0   1 0   0 0   0   1 0   1 0   0 |
     {1, 0} | 0   0   1 0 0 x_1 0 0   1 0   0   0 0   0 x_3 0 |
     {0, 1} | 0   0   0 1 1 0   0 0   0 x_2 0   0 0   0 0   1 |

             4      16
o4 : Matrix R  <-- R

See also

Ways to use this method:


The source of this document is in /build/reproducible-path/macaulay2-1.25.05+ds/M2/Macaulay2/packages/ToricHigherDirectImages.m2:670:0.