# symExt -- the first differential of the complex R(M)

## Synopsis

• Usage:
symExt(m,E)
• Inputs:
• m, , a presentation matrix for a positively graded module M over a polynomial ring
• E, , exterior algebra
• Outputs:
• , 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]; i3 : M = coker matrix {{x_0^2, x_1^2}}; i4 : m = presentation truncate(regularity M,M); 4 8 o4 : Matrix S <--- S 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

• bgg -- the ith differential of the complex R(M)

## Ways to use symExt :

• "symExt(Matrix,PolynomialRing)"

## For the programmer

The object symExt is .