next | previous | forward | backward | up | top | index | toc | Macaulay2 website
EagonResolution :: horizontalStrand

horizontalStrand -- extracts one horizontal strand from an Eagon double complex

Synopsis

Description

The 0-th strand is a possibly non-minimal resolution of the residuce field. More generally, the i-th strand resolves the i-th boundary module in the Koszul complex of R. These resolutions are all minimal iff R is Golod.

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 = horizontalStrand(E,2)

      3      6      17      41      104
o4 = R  <-- R  <-- R   <-- R   <-- R
                                    
     0      1      2       3       4

o4 : ChainComplex
i5 : picture F

     +-------------------------------------------------------------------+
     |+-------+-------+--------+                                         |
o5 = ||       |(3, {})|(0, {2})|                                         |
     |+-------+-------+--------+                                         |
     ||(2, {})|   *   |    *   |                                         |
     |+-------+-------+--------+                                         |
     +-------------------------------------------------------------------+
     |+--------+--------+--------+                                       |
     ||        |(0, {3})|(1, {2})|                                       |
     |+--------+--------+--------+                                       |
     || (3, {})|    *   |    *   |                                       |
     |+--------+--------+--------+                                       |
     ||(0, {2})|    .   |    *   |                                       |
     |+--------+--------+--------+                                       |
     +-------------------------------------------------------------------+
     |+--------+--------+--------+-----------+                           |
     ||        |(1, {3})|(2, {2})|(0, {1, 2})|                           |
     |+--------+--------+--------+-----------+                           |
     ||(0, {3})|    *   |    .   |    2,2    |                           |
     |+--------+--------+--------+-----------+                           |
     ||(1, {2})|    .   |    *   |     *     |                           |
     |+--------+--------+--------+-----------+                           |
     +-------------------------------------------------------------------+
     |+-----------+--------+-----------+--------+-----------+-----------+|
     ||           |(2, {3})|(0, {1, 3})|(3, {2})|(0, {2, 2})|(1, {1, 2})||
     |+-----------+--------+-----------+--------+-----------+-----------+|
     ||  (1, {3}) |    *   |     *     |    .   |     .     |    6,6    ||
     |+-----------+--------+-----------+--------+-----------+-----------+|
     ||  (2, {2}) |    .   |     .     |    *   |     *     |     *     ||
     |+-----------+--------+-----------+--------+-----------+-----------+|
     ||(0, {1, 2})|    .   |     .     |    .   |     .     |     *     ||
     |+-----------+--------+-----------+--------+-----------+-----------+|
     +-------------------------------------------------------------------+

See also

Ways to use horizontalStrand :

For the programmer

The object horizontalStrand is a method function.