symExt -- the first differential of the complex R(M)
Synopsis
-
- Usage:
symExt(m,E)
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Inputs:
-
m, a matrix, a presentation matrix for a positively graded module M over a polynomial ring
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E, a polynomial ring, exterior algebra
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Outputs:
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a matrix, a matrix representing the map M_1 ** omega_E <-- M_0 ** omega_E
Description
This function takes as input a matrix
m with linear entries, which we think of as a presentation matrix for a positively graded
S-module
M matrix representing the map
M_1 ** omega_E <-- M_0 ** omega_E which is the first differential of the complex
R(M).
i1 : S = ZZ/32003[x_0..x_2];
|
i2 : E = ZZ/32003[e_0..e_2, SkewCommutative=>true];
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i3 : M = coker matrix {{x_0^2, x_1^2}};
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i4 : m = presentation truncate(regularity M,M);
4 8
o4 : Matrix S <--- S
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i5 : symExt(m,E)
o5 = {-1} | e_2 0 0 0 |
{-1} | e_1 e_2 0 0 |
{-1} | e_0 0 e_2 0 |
{-1} | 0 e_0 e_1 e_2 |
4 4
o5 : Matrix E <--- E
|
Caveat
This function is a quick-and-dirty tool which requires little computation. However if it is called on two successive truncations of a module, then the maps it produces may NOT compose to zero because the choice of bases is not consistent.
See also
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bgg -- the ith differential of the complex R(M)
Ways to use symExt :
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"symExt(Matrix,PolynomialRing)"