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

## Synopsis

• Usage:
pureResES1(d,kk)
• Inputs:
• Optional inputs:
• MonSize (missing documentation) => ..., default value 32,
• Outputs:

## 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