Macaulay2 » Documentation
Packages » ToricHigherDirectImages :: frobeniusDirectImage(ZZ,Complex)
next | previous | forward | backward | up | index | toc

frobeniusDirectImage(ZZ,Complex) -- compute the pushforward of a complex 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 complexes of $R$-modules.

i1 : X = hirzebruchSurface 1;
i2 : R = ring X;
i3 : C = complex koszul vars R

      1      4      6      4      1
o3 = R  <-- R  <-- R  <-- R  <-- R
                                  
     0      1      2      3      4

o3 : Complex
i4 : frobeniusDirectImage_2 C

      4      16      24      16      4
o4 = R  <-- R   <-- R   <-- R   <-- R
                                     
     0      1       2       3       4

o4 : Complex

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.m2:706:0.