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verticalStrand -- extracts one vertical strand from an Eagon double complex

Synopsis

Description

The 0-th vertical strand is the Koszul complex of R. The vertical strands are never resolutions unless R is regular. The key lemma in Eagon's treatment identifies the i-th homology H_i of the n-th vertical strand with H_0**X_i.

i1 : S = ZZ/101[x,y,z]

o1 = S

o1 : PolynomialRing
 
i2 : R = S/((ideal(x,y))^2+ideal(z^3))

o2 = R

o2 : QuotientRing
 
i3 : E = eagon(R,5);
 
i4 : F = verticalStrand(E,3)

      18      45      41      14
o4 = R   <-- R   <-- R   <-- R
                              
     0       1       2       3

o4 : ChainComplex
 
i5 : picture F

     +-------------------------------------------------+
     |+--------+--------+--------+--------+-----------+|
o5 = ||        |(0, {3})|(1, {2})|(2, {1})|(0, {1, 1})||
     |+--------+--------+--------+--------+-----------+|
     || (3, {})|    *   |    *   |    *   |     *     ||
     |+--------+--------+--------+--------+-----------+|
     ||(0, {2})|    .   |    *   |    *   |    5,3    ||
     |+--------+--------+--------+--------+-----------+|
     ||(1, {1})|    .   |    .   |    *   |     *     ||
     |+--------+--------+--------+--------+-----------+|
     +-------------------------------------------------+
     |+-----------+--------+--------+-----------+      |
     ||           |(1, {3})|(2, {2})|(0, {1, 2})|      |
     |+-----------+--------+--------+-----------+      |
     ||  (0, {3}) |    *   |    .   |    2,2    |      |
     |+-----------+--------+--------+-----------+      |
     ||  (1, {2}) |    .   |    *   |     *     |      |
     |+-----------+--------+--------+-----------+      |
     ||  (2, {1}) |    .   |    .   |     .     |      |
     |+-----------+--------+--------+-----------+      |
     ||(0, {1, 1})|    .   |    .   |     .     |      |
     |+-----------+--------+--------+-----------+      |
     +-------------------------------------------------+
     |+-----------+--------+-----------+               |
     ||           |(2, {3})|(0, {1, 3})|               |
     |+-----------+--------+-----------+               |
     ||  (1, {3}) |    *   |     *     |               |
     |+-----------+--------+-----------+               |
     ||  (2, {2}) |    .   |     .     |               |
     |+-----------+--------+-----------+               |
     ||(0, {1, 2})|    .   |     .     |               |
     |+-----------+--------+-----------+               |
     +-------------------------------------------------+

See also

Ways to use verticalStrand :

For the programmer

The object verticalStrand is a method function.