flattenedESTensor -- make a flattened tensor from elementary symmetric functions

Synopsis

• Usage:
flattenedESTensor(L,kk)
• Inputs:
• L, a list, of positive ZZ
• kk, a ring, Name of ground field (or ring)
• Optional inputs:
• MonSize (missing documentation) => ..., default value 32,
• Outputs:

Description

Given a list $L = \{a, b_1,\dots, b_n\}$ of positive integers with $a= sum_i b_i,$ and a field (or ring of integers) kk, the script creates a ring $S = kk[x_1,\dots,x_n]$ and a map $$f: A \to B_1\otimes\cdots \otimes B_n$$ of LabeledModules over $S$, where $A$ is a free LabeledModule of rank $a$ and $B_i$ is a free LabeledModule of rank $b_i$. The map $f$ is constructed from symmetric functions, and corresponds to collection of linear forms on $P^{b_1-1}\times\cdots\timesß P^{b_n-1}$ as used in the construction of pure resolutions in the paper Betti numbers of graded modules and cohomology of vector bundles'' of Eisenbud and Schreyer.

The format of $F$ is the one required by tensorComplex1, namely $f: A \to B_1\otimes \cdots \otimes B_n$, with $a = rank A, b_i = rank B_i$.

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 : kk = ZZ/101 o1 = kk o1 : QuotientRing i2 : f = flattenedESTensor({5,2,1,2},kk) o2 = | x_0 x_1 x_2 0 0 | | 0 x_0 x_1 x_2 0 | | 0 x_0 x_1 x_2 0 | | 0 0 x_0 x_1 x_2 | 4 5 o2 : Matrix (kk[x ..x ]) <--- (kk[x ..x ]) 0 2 0 2 i3 : numgens ring f o3 = 3 i4 : betti matrix f 0 1 o4 = total: 4 5 -1: . 5 0: 4 . o4 : BettiTally i5 : S = ring f o5 = S o5 : PolynomialRing i6 : g = tensorComplex1 f o6 = | x_0^2 0 x_0x_1 0 x_1^2-x_0x_2 0 x_0x_2 0 x_1x_2 0 x_2^2 0 0 0 0 0 0 0 0 0 | | 0 x_0^2 x_0^2 x_0x_1 x_0x_1 x_1^2-x_0x_2 x_0x_1 x_0x_2 x_1^2 x_1x_2 x_1x_2 x_2^2 x_0x_2 0 x_1x_2 0 x_2^2 0 0 0 | | 0 0 0 x_0^2 x_0^2 x_0x_1 0 x_0x_1 x_0x_1 x_1^2 x_1^2-x_0x_2 x_1x_2 0 x_0x_2 x_0x_2 x_1x_2 x_1x_2 x_2^2 x_2^2 0 | | 0 0 0 0 0 x_0^2 0 0 0 x_0x_1 0 x_1^2-x_0x_2 0 0 0 x_0x_2 0 x_1x_2 0 x_2^2 | 4 20 o6 : Matrix S <--- S i7 : betti res coker g 0 1 2 3 o7 = total: 4 20 20 4 0: 4 . . . 1: . 20 20 . 2: . . . 4 o7 : BettiTally