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TensorComplexes :: pureResES1

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

Synopsis

Description

Given a degree sequence d∈ℤn+1 and a field k of arbtirary 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 d0=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 , x , x ])
             32003  0   1   2          32003  0   1   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

  • pureResES -- constructs the Eisenbud--Schreyer pure resolution of a given type

Ways to use pureResES1 :

  • pureResES1(List,Ring) (missing documentation)