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pureResES1 -- computes the first map of the Eisenbud--Schreyer pure resolution of a given type

Synopsis

Description

Given a degree sequence $d\in \mathbb Z^{n+1}$ and a field $k$ of arbitrary characteristic, this produces the first map of pure resolution of type d as constructed by Eisenbud and Schreyer in Section 5 of ``Betti numbers of graded modules and cohomology of vector bundles''. The cokernel of this map is a module of finite of length over a polynomial ring in $n$ variables.

The code gives an error if d is not strictly increasing with $d_0=0$.

There is an OPTION, MonSize => n (where n is 8,16, or 32). This sets the MonomialSize option when the base ring of flattenedESTensor is created.

i1 : d={0,2,4,5};
 
i2 : p=pureResES1(d,ZZ/32003)

o2 = | x_0^2 x_0x_1 x_1^2-x_0x_2 x_0x_2 x_1x_2 x_2^2        0      0      0      0     |
     | 0     x_0^2  x_0x_1       x_0x_1 x_1^2  x_1x_2       x_0x_2 x_1x_2 x_2^2  0     |
     | 0     0      x_0^2        0      x_0x_1 x_1^2-x_0x_2 0      x_0x_2 x_1x_2 x_2^2 |

               ZZ          3         ZZ          10
o2 : Matrix (-----[x ..x ])  <--- (-----[x ..x ])
             32003  0   2          32003  0   2
 
i3 : betti res coker p

            0  1  2 3
o3 = total: 3 10 15 8
         0: 3  .  . .
         1: . 10  . .
         2: .  . 15 8

o3 : BettiTally
 
i4 : dim coker p

o4 = 0

See also

Ways to use pureResES1 :

For the programmer

The object pureResES1 is a method function with options.