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D.15.8.27 grlifting
Procedure from library gradedModules.lib (see gradedModules_lib).
- Usage:
- grlifting(M,N), graded objects M and N
- Return:
- map of chain complexes (as a list)
- Purpose:
- construct a map of chain complexes between free resolutions of Img(M) and Img(N).
Example:
| LIB "gradedModules.lib";
/*
ring r=32003,(x,y,z),dp;
module P=grobj(module([xy,0,xz]),intvec(0,1,0));
grview(P);
module D=grobj(module([y,0,z],[x2+y2,z,0]),intvec(0,1,0));
grview(D);
def G=grlifting(D,P);
grview(G);
kill r;
ring r=32003,(x,y,z),dp;
module D=grobj(module([y,0,z],[x2+y2,z,0], [z3, xy, xy2]),intvec(0,1,0));
D = grgroebner(D);
grview( grres(D, 0));
def G=grlifting(D, D);
grview(G);
*/
ring S = 0, (x(0..3)), dp;
list kos = grres(grobj(maxideal(1), intvec(0)), 0);
print( betti(kos), "betti");
==> 0 1 2 3 4
==> ------------------------------------
==> 0: 1 4 6 4 1
==> ------------------------------------
==> total: 1 4 6 4 1
==>
grview(kos);
==> Graded resolution:
==> S <-- d_1 --
==> S(-1)^4 <-- d_2 --
==> S(-2)^6 <-- d_3 --
==> S(-3)^4 <-- d_4 --
==> S(-4) <-- d_5 --
==> 0, given by maps:
==> d_1 :
==> Graded homomorphism: S <- S(-1)^4, given by a matrix, with degrees:
==> .1 .2 .3 .4 ...
==> -- -- -- -- +..
==> 0 : 1 1 1 1 |.1
==> == == == ==
==> 1 1 1 1
==> d_2 :
==> Graded homomorphism: S(-1)^4 <- S(-2)^6, given by a matrix, with degrees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ....
==> --- --- --- --- --- --- +...
==> 1 : 1 1 - 1 - - |..1
==> 1 : 1 - 1 - 1 - |..2
==> 1 : - 1 1 - - 1 |..3
==> 1 : - - - 1 1 1 |..4
==> === === === === === ===
==> 2 2 2 2 2 2
==> d_3 :
==> Graded homomorphism: S(-2)^6 <- S(-3)^4, given by a matrix, with degrees:
==> ..1 ..2 ..3 ..4 ....
==> --- --- --- --- +...
==> 2 : 1 1 - - |..1
==> 2 : 1 - 1 - |..2
==> 2 : 1 - - 1 |..3
==> 2 : - 1 1 - |..4
==> 2 : - 1 - 1 |..5
==> 2 : - - 1 1 |..6
==> === === === ===
==> 3 3 3 3
==> d_4 :
==> Graded homomorphism: S(-3)^4 <- S(-4), given by a matrix, with degrees:
==> .1 ...
==> -- +..
==> 3 : 1 |.1
==> 3 : 1 |.2
==> 3 : 1 |.3
==> 3 : 1 |.4
==> ==
==> 4
==> d_5 :
==> Graded homomorphism: S(-4) <- 0, given by zero (1 x 0) matrix.
// module M = grshift(kos[4], 2); // phi, Syz_3(K(2))
def M = KeneshlouMatrixPresentation(intvec(0,0,1,0));
grview( grres(M, 0) );
==> Graded resolution:
==> S(-1)^4 <-- d_1 --
==> S(-2) <-- d_2 --
==> 0, given by maps:
==> d_1 :
==> Graded homomorphism: S(-1)^4 <- S(-2), given by a matrix, with degrees:
==> .1 ...
==> -- +..
==> 1 : 1 |.1
==> 1 : 1 |.2
==> 1 : 1 |.3
==> 1 : 1 |.4
==> ==
==> 2
==> d_2 :
==> Graded homomorphism: S(-2) <- 0, given by zero (1 x 0) matrix.
// module N = grshift(kos[3], 1); // psi, Syz_2(K(1))
def N = KeneshlouMatrixPresentation(intvec(0,1,0,0));
grview( grres(N, 0) );
==> Graded resolution:
==> S(-1)^6 <-- d_1 --
==> S(-2)^4 <-- d_2 --
==> S(-3) <-- d_3 --
==> 0, given by maps:
==> d_1 :
==> Graded homomorphism: S(-1)^6 <- S(-2)^4, given by a matrix, with degrees:
==> ..1 ..2 ..3 ..4 ....
==> --- --- --- --- +...
==> 1 : 1 1 - - |..1
==> 1 : 1 - 1 - |..2
==> 1 : 1 - - 1 |..3
==> 1 : - 1 1 - |..4
==> 1 : - 1 - 1 |..5
==> 1 : - - 1 1 |..6
==> === === === ===
==> 2 2 2 2
==> d_2 :
==> Graded homomorphism: S(-2)^4 <- S(-3), given by a matrix, with degrees:
==> .1 ...
==> -- +..
==> 2 : 1 |.1
==> 2 : 1 |.2
==> 2 : 1 |.3
==> 2 : 1 |.4
==> ==
==> 3
==> d_3 :
==> Graded homomorphism: S(-3) <- 0, given by zero (1 x 0) matrix.
grlifting(M, N); // grview(G);
==> t: 2
==> _[1]=26642*gen(4)+24263*gen(3)+5664*gen(2)+24170*gen(1)
// def G=grlifting( grgens(M), grgens(N) ); grview(G);
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