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D.15.14.15 grtranspose1
Procedure from library gradedModules.lib (see gradedModules_lib).
- Usage:
- grtranspose1(M), graded object or list M
- Return:
- same as input
- Purpose:
- graded transpose of graded object or chain complex M
- Assume:
- M must be a graded object or a list of graded objects
Example:
| | LIB "gradedModules.lib";
"Surface Name: 'k3.d10.g9.quart2' in P^4";
==> Surface Name: 'k3.d10.g9.quart2' in P^4
int @p=31991; ring R = (@p),(x,y,z,u,v), dp;
ideal J = 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5. 104y2v3+32/113xzv3+15544yzv3+31/18z2v3+5909xuv3-67/58yuv3+27/35zuv3+115/9u2v3+79/13xv4+6722yv4-37/114zv4-71/124uv4+4657v5,-77/61x4y-88/101x3y2+93/88x2y3-11/70xy4+9806y5+7896x4z-4699x3yz+55/122x2y2z-63/122xy3z-125/74y4z+47/45x3z2+101/17x2yz2+92/47xy2z2+69/82y3z2+12402x2z3+113/98xyz3-101/33y2z3-15376xz4+47/71yz4-73/10z5+65/74x4u-14409x3yu-14478x2y2u+13593xy3u+102/97y4u+39/62x3zu-34/125x2yzu-83/9xy2zu+45/113y3zu+14484x2z2u-15293xyz2u-26/55y2z2u-958xz3u+67/35yz3u-93/19z4u+25/16x3u2+107/52x2yu2-4599xy2u2-86/51y3u2-9885x2zu2-77/47xyzu2+33/65y2zu2+90/109xz2u2-61/26yz2u2+6198z3u2-38/37x2u3-13935xyu3-142y2u3-64/5xzu3-7228yzu3+1251z2u3+1556xu4+117/121yu4-92/35zu4+99/92u5+13493x4v+12654x3yv+32/101x2y2v-11118xy3v+43/51y4v-575x3zv+103/21x2yzv+85/24xy2zv+1788y3zv+85/3x2z2v-64/25xyz2v+57/35y2z2v+37/120xz3v-69/110yz3v+48/49z4v+55/114x3uv-6439x2yuv+31/51xy2uv-90/49y3uv-45/104x2zuv-12018xyzuv+6/119y2zuv+40/63xz2uv+20/91yz2uv+50/43z3uv+1/26x2u2v-109/47xyu2v+99/7y2u2v+72/83xzu2v+61/118yzu2v+3530z2u2v+6146xu3v+117yu3v-9921zu3v-8708u4v-10/47x3v2-15294x2yv2-7336xy2v2+1/66y3v2-3057x2zv2+74/123xyzv2+146y2zv2-103/34xz2v2-117/76yz2v2+8472z3v2-7/92x2uv2+10033xyuv2+43/53y2uv2+4694xzuv2-49/2yzuv2-71/73z2uv2-125/17xu2v2-9817yu2v2+7218zu2v2+6897u3v2-19/90x2v3+11899xyv3-11779y2v3-5456xzv3+17/42yzv3+15340z2v3+12/7xuv3+9580yuv3-502zuv3-14069u2v3-4371xv4+14452yv4-9423zv4-117/122uv4+1126v5,49/108x4-39/4x3y-67/21x2y2-8/69xy3-9779y4+57/14x3z-11145x2yz+6928xy2z-7824y3z+1/79x2z2+5173xyz2-62/15y2z2-123/112xz3+88/79yz3+1/125z4+57/23x3u-11856x2yu-7444xy2u+115/8y3u-11133x2zu+71/73xyzu-7941y2zu+69/65xz2u+22/75yz2u+65/121z3u+9471x2u2+9167xyu2+51/59y2u2+12835xzu2+15047yzu2+11102z2u2-10059xu3+19/28yu3+65/21zu3-39/28u4-3/73x3v+94/61x2yv+8778xy2v-12922y3v-8711x2zv-37/97xyzv+14270y2zv+4487xz2v-59/112yz2v-14183z3v+15553x2uv+3579xyuv+114/91y2uv-4/97xzuv+13/85yzuv-89/15z2uv+58/75xu2v-34/7yu2v-90/61zu2v+90/101u3v-14673x2v2+90/19xyv2-45/37y2v2+23/49xzv2-71/11yzv2+119/8z2v2+89/10xuv2+109/91yuv2+36/49zuv2-7/31u2v2-40/113xv3-21/121yv3+9910zv3+33/14uv3-23/79v4,93/70x4-43/125x3y+9582x2y2+7565xy3-11511y4-3/79x3z-36/107x2yz-2038xy2z+879y3z-4700x2z2+103/14xyz2+102/79y2z2-67/68xz3-44/25yz3+105/79z4-29/24x3u-74/83x2yu+67/43xy2u+49/12y3u-115/11x2zu+23/67xyzu-61/27y2zu+12257xz2u+14068yz2u+23/15z3u+607x2u2+73/8xyu2+14237y2u2-13/33xzu2+110/71yzu2+41/101z2u2+5708xu3+88/67yu3+1460zu3-2472u4-1629x3v-51/70x2yv-88/73xy2v-36/97y3v+38/11x2zv+15899xyzv+54/19y2zv+9460xz2v-5150yz2v+3462z3v+5522x2uv-19/123xyuv+14871y2uv+53/5xzuv-7535yzuv-13430z2uv+107/47xu2v-8307yu2v-55/79zu2v-11945u3v-16/83x2v2+115/48xyv2+12389y2v2+11545xzv2-25/26yzv2-3755z2v2+4724xuv2-31/21yuv2+7872zuv2+89/45u2v2+87/47xv3+7625yv3+13494zv3-15376uv3-25/126v4;
def I = grobj( groebner(J), intvec(0) ); // ASSUME: no zero entries in J!
ASSUME(0, grtest(I));
"Input degrees: "; grview(I);
==> Input degrees:
==> Graded homomorphism: R <- R(-4)^2 + R(-5)^5 + R(-6)^3, given by a matrix,\
with degrees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 ....
==> --- --- --- --- --- --- --- --- --- --- +...
==> 0 : 4 4 5 5 5 5 5 6 6 6 |..1
==> === === === === === === === === === ===
==> 4 4 5 5 5 5 5 6 6 6
def RR = grres(I, 0, 1); list L = RR;
" = Non-minimal betti numbers: "; print(betti(L, 0), "betti");
==> = Non-minimal betti numbers:
==> 0 1 2 3 4
==> ------------------------------------
==> 0: 1 - - - -
==> 1: - - - - -
==> 2: - - - - -
==> 3: - 2 - - -
==> 4: - 4 7 2 -
==> 5: - 3 8 7 2
==> ------------------------------------
==> total: 1 9 15 9 2
==>
"Graded (original) structure of 'res(Input,0)': "; grview(L);
==> Graded (original) structure of 'res(Input,0)':
==> Graded resolution:
==> R <-- d_1 --
==> R(-4)^2 + R(-5)^4 + R(-6)^3 <-- d_2 --
==> R(-7)^2 + R(-6)^7 + R(-7)^6 <-- d_3 --
==> R(-7)^2 + R(-8)^7 <-- d_4 --
==> R(-9)^2 <-- d_5 --
==> 0, given by maps:
==> d_1 :
==> Graded homomorphism: R <- R(-4)^2 + R(-5)^4 + R(-6)^3, given by a matrix,\
with degrees:
==> .1 .2 .3 .4 .5 .6 .7 .8 .9 ...
==> -- -- -- -- -- -- -- -- -- +..
==> 0 : 4 4 5 5 5 5 6 6 6 |.1
==> == == == == == == == == ==
==> 4 4 5 5 5 5 6 6 6
==> d_2 :
==> Graded homomorphism: R(-4)^2 + R(-5)^4 + R(-6)^3 <- R(-7)^2 + R(-6)^7 + R\
(-7)^6
==> , given by a matrix, with degrees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 .11 .12 .13 .14 .15 ....
==> --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- +...
==> 4 : - - 2 2 2 2 2 2 2 3 3 3 3 3 3 |..1
==> 4 : - - 2 2 2 2 2 2 2 3 3 3 3 3 3 |..2
==> 5 : 2 2 1 1 1 1 1 1 1 2 2 2 2 2 2 |..3
==> 5 : 2 2 1 1 1 1 1 1 1 2 2 2 2 2 2 |..4
==> 5 : - - 1 1 1 1 1 1 1 2 2 2 2 2 2 |..5
==> 5 : - - 1 1 1 1 1 1 1 2 2 2 2 2 2 |..6
==> 6 : 1 1 - - - - - - - 1 1 1 1 1 1 |..7
==> 6 : 1 1 - - - - - - - 1 1 1 1 1 1 |..8
==> 6 : 1 1 - - - - - - - 1 1 1 1 1 1 |..9
==> === === === === === === === === === === === === === === ===
==> 7 7 6 6 6 6 6 6 6 7 7 7 7 7 7
==> d_3 :
==> Graded homomorphism: R(-7)^2 + R(-6)^7 + R(-7)^6 <- R(-7)^2 + R(-8)^7, gi\
ven by a matrix, with degrees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 ....
==> --- --- --- --- --- --- --- --- --- +...
==> 7 : - - 1 1 1 1 1 1 1 |..1
==> 7 : - - 1 1 1 1 1 1 1 |..2
==> 6 : 1 1 2 2 2 2 2 2 2 |..3
==> 6 : 1 1 2 2 2 2 2 2 2 |..4
==> 6 : 1 1 2 2 2 2 2 2 2 |..5
==> 6 : 1 1 2 2 2 2 2 2 2 |..6
==> 6 : 1 1 2 2 2 2 2 2 2 |..7
==> 6 : 1 1 2 2 2 2 2 2 2 |..8
==> 6 : 1 1 2 2 2 2 2 2 2 |..9
==> 7 : - - 1 1 1 1 1 1 1 |.10
==> 7 : - - 1 1 1 1 1 1 1 |.11
==> 7 : - - 1 1 1 1 1 1 1 |.12
==> 7 : - - 1 1 1 1 1 1 1 |.13
==> 7 : - - 1 1 1 1 1 1 1 |.14
==> 7 : - - 1 1 1 1 1 1 1 |.15
==> === === === === === === === === ===
==> 7 7 8 8 8 8 8 8 8
==> d_4 :
==> Graded homomorphism: R(-7)^2 + R(-8)^7 <- R(-9)^2, given by a matrix, wit\
h degrees:
==> .1 .2 ...
==> -- -- +..
==> 7 : 2 2 |.1
==> 7 : 2 2 |.2
==> 8 : 1 1 |.3
==> 8 : 1 1 |.4
==> 8 : 1 1 |.5
==> 8 : 1 1 |.6
==> 8 : 1 1 |.7
==> 8 : 1 1 |.8
==> 8 : 1 1 |.9
==> == ==
==> 9 9
==> d_5 :
==> Graded homomorphism: R(-9)^2 <- 0, given by zero (2 x 0) matrix.
"Graded transpose of the previous resolution "; list LLL = grtranspose1( L ); grview( LLL );
==> Graded transpose of the previous resolution
==> Graded resolution:
==> R(9)^2 <-- d_1 --
==> R(8)^7 + R(7)^2 <-- d_2 --
==> R(7)^8 + R(6)^7 <-- d_3 --
==> R(6)^3 + R(5)^4 + R(4)^2 <-- d_4 --
==> R, given by maps:
==> d_1 :
==> Graded homomorphism: R(9)^2 <- R(8)^7 + R(7)^2, given by a matrix, with d\
egrees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 ....
==> --- --- --- --- --- --- --- --- --- +...
==> -9 : 1 1 1 1 1 1 1 2 2 |..1
==> -9 : 1 1 1 1 1 1 1 2 2 |..2
==> === === === === === === === === ===
==> -8 -8 -8 -8 -8 -8 -8 -7 -7
==> d_2 :
==> Graded homomorphism: R(8)^7 + R(7)^2 <- R(7)^8 + R(6)^7, given by a matri\
x, with degrees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 .10 .11 .12 .13 .14 .15 ....
==> --- --- --- --- --- --- --- --- --- --- --- --- --- --- --- +...
==> -8 : 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 |..1
==> -8 : 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 |..2
==> -8 : 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 |..3
==> -8 : 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 |..4
==> -8 : 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 |..5
==> -8 : 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 |..6
==> -8 : 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 |..7
==> -7 : - - - - - - - - 1 1 1 1 1 1 1 |..8
==> -7 : - - - - - - - - 1 1 1 1 1 1 1 |..9
==> === === === === === === === === === === === === === === ===
==> -7 -7 -7 -7 -7 -7 -7 -7 -6 -6 -6 -6 -6 -6 -6
==> d_3 :
==> Graded homomorphism: R(7)^8 + R(6)^7 <- R(6)^3 + R(5)^4 + R(4)^2, given b\
y a matrix, with degrees:
==> ..1 ..2 ..3 ..4 ..5 ..6 ..7 ..8 ..9 ....
==> --- --- --- --- --- --- --- --- --- +...
==> -7 : 1 1 1 2 2 - - - - |..1
==> -7 : 1 1 1 2 2 - - - - |..2
==> -7 : 1 1 1 2 2 2 2 3 3 |..3
==> -7 : 1 1 1 2 2 2 2 3 3 |..4
==> -7 : 1 1 1 2 2 2 2 3 3 |..5
==> -7 : 1 1 1 2 2 2 2 3 3 |..6
==> -7 : 1 1 1 2 2 2 2 3 3 |..7
==> -7 : 1 1 1 2 2 2 2 3 3 |..8
==> -6 : - - - 1 1 1 1 2 2 |..9
==> -6 : - - - 1 1 1 1 2 2 |.10
==> -6 : - - - 1 1 1 1 2 2 |.11
==> -6 : - - - 1 1 1 1 2 2 |.12
==> -6 : - - - 1 1 1 1 2 2 |.13
==> -6 : - - - 1 1 1 1 2 2 |.14
==> -6 : - - - 1 1 1 1 2 2 |.15
==> === === === === === === === === ===
==> -6 -6 -6 -5 -5 -5 -5 -4 -4
==> d_4 :
==> Graded homomorphism: R(6)^3 + R(5)^4 + R(4)^2 <- R, given by a matrix, wi\
th degrees:
==> ..1 ....
==> --- +...
==> -6 : 6 |..1
==> -6 : 6 |..2
==> -6 : 6 |..3
==> -5 : 5 |..4
==> -5 : 5 |..5
==> -5 : 5 |..6
==> -5 : 5 |..7
==> -4 : 4 |..8
==> -4 : 4 |..9
==> ===
==> 0
"Its non-minimal betti numbers: "; print(betti(LLL, 0), "betti");
==> Its non-minimal betti numbers:
==> 0 1 2 3 4
==> ------------------------------------
==> -9: 2 7 8 3 -
==> -8: - 2 7 4 -
==> -7: - - - 2 -
==> -6: - - - - -
==> -5: - - - - -
==> -4: - - - - 1
==> ------------------------------------
==> total: 2 9 15 9 1
==>
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