eagon(R,b) computes the first b columns of the Eagon double complex Y^*_* of R, and caches them in a HashTable of class EagonData in of R.cache.EagonData. (The command eagon(R,-1) removes this.)
Following Gulliksen-Levin we think of Y^n_* as the n-th column, and Y^*_i as the i-th row. The columns Y^n are not acyclic. The i-th row is a resolution of the i-th module of boundaries in the Koszul complex K of the variables of R; in particular, the "Eagon Resolution" is the 0-th row,
Y^b_0 \to...\to Y^1_0 \to Y^0_0.
Let X_i be the free module R**H_i(K), which is also the R**F_i, where F is a minimal free resolution of R as a module over the polynomial ring on the same set of variables.
We count X_i as having homological degree i+1. With this convention, Y^*_0 has the form K\otimes T(F'), where T denotes the tensor algebra and F' is the F_1++F_2++... .
The module Y^n_i = Eagon#{0,n,i} is described in Gulliksen-Levin as: Y^0 = koszul vars R Y^{n+1}_0 = Y^n_1; and for i>0, Y^{n+1}_i = Y^n_{i+1} ++ Y^n_0**X_i
Note that Y^n_i == 0 for i>1+length koszul vars R - n,
The i-th homology of Y^n_* is H_i(Y^n) = H_0(Y^n_*)**X_i (proved in Gulliksen-Levin). Part of the inductive construction will be a map inducing this isomorphism
alpha^n_i = eagonBeta^n_i + dHor^n_0**1: Y^n_0**X_i \to Y^{n-1}_{i+1} ++ Y^{n-1}_0**X_i = Y^n
Assume that the differential of Y^n and the maps dVert^n and alpha^n are known. We take
dHor^{n+1}_0: Y^{n+1}_0 = Y^n_1 -> Y^n_0 to be dVert^n_1.
The remaining horizontal differentials dHor^{n+1}_i: Y^{n+1} \to Y^n have source and target as follows:
Y^{n+1}_i = Y^n_{i+1} ++ Y^n_0**X_i -> Y^n_i = Y^{n-1}_{i+1} ++ Y^{n-1}_0**X_i.
We take dHor^{n+1}_i to be the sum of two maps:
dVert^n_{i+1} Y^n_{i+1} -> Y^n_i ++ Y^{n-1}_0**X_i.
and alpha^{n+1}_i = eagonBeta^{n+1}_i + dHor^n_0**1: Y^n_0**X_i \to Y^n_i ++ Y^{n-1}_0**X(i).
It remains to define eagonBeta^{n+1}_i; we take this to be the negative of
a lifting along the map from Y^{n+1}_{i-1} \subset Y^n_i to Y^n_{i-1} of the composite
dVert^{n+1}_{i-1} * (dHor^n_0 ** X_i): Y^n_0**X_i -> Y^{n-1}_0.
i1 : S = ZZ/101[a,b,c] o1 = S o1 : PolynomialRing |
i2 : I = ideal(a,b)*ideal"a3,b3,c3"
4 3 3 3 4 3
o2 = ideal (a , a*b , a*c , a b, b , b*c )
o2 : Ideal of S
|
i3 : R = S/I o3 = R o3 : QuotientRing |
i4 : needsPackage "DGAlgebras"; isGolod R o5 = true |
i6 : E = eagon(R,6) o6 = EagonData in <ring>.cache computed to length 6 o6 : EagonData |
We can see the vertical and horizontal strands, and the eagonBeta maps
i7 : verticalStrand(E,3)
27 81 81 27
o7 = R <-- R <-- R <-- R
0 1 2 3
o7 : ChainComplex
|
i8 : horizontalStrand(E,2)
3 9 27 81 243 729
o8 = R <-- R <-- R <-- R <-- R <-- R
0 1 2 3 4 5
o8 : ChainComplex
|
i9 : horizontalStrand (E,0)
1 3 9 27 81 243
o9 = R <-- R <-- R <-- R <-- R <-- R
0 1 2 3 4 5
o9 : ChainComplex
|
i10 : F = eagonResolution E
1 3 9 27 81 243 729
o10 = R <-- R <-- R <-- R <-- R <-- R <-- R
0 1 2 3 4 5 6
o10 : ChainComplex
|
i11 : eagonBeta E
+----------------------------------------------------------------+
|+-------+--------+ |
o11 = || |(0, {1})| |
|+-------+--------+ |
||(1, {})| * | |
|+-------+--------+ |
+----------------------------------------------------------------+
|+-------+--------+ |
|| |(1, {1})| |
|+-------+--------+ |
||(2, {})| * | |
|+-------+--------+ |
+----------------------------------------------------------------+
|+--------+--------+-----------+ |
|| |(2, {1})|(0, {1, 1})| |
|+--------+--------+-----------+ |
|| (3, {})| * | . | |
|+--------+--------+-----------+ |
||(0, {2})| . | . | |
|+--------+--------+-----------+ |
+----------------------------------------------------------------+
|+--------+--------+-----------+-----------+ |
|| |(3, {1})|(0, {2, 1})|(1, {1, 1})| |
|+--------+--------+-----------+-----------+ |
||(0, {3})| . | . | . | |
|+--------+--------+-----------+-----------+ |
||(1, {2})| . | . | . | |
|+--------+--------+-----------+-----------+ |
+----------------------------------------------------------------+
|+-----------+-----------+-----------+-----------+--------------+|
|| |(0, {3, 1})|(1, {2, 1})|(2, {1, 1})|(0, {1, 1, 1})||
|+-----------+-----------+-----------+-----------+--------------+|
|| (1, {3}) | . | . | . | . ||
|+-----------+-----------+-----------+-----------+--------------+|
|| (2, {2}) | . | . | . | . ||
|+-----------+-----------+-----------+-----------+--------------+|
||(0, {1, 2})| . | . | . | . ||
|+-----------+-----------+-----------+-----------+--------------+|
+----------------------------------------------------------------+
|
With the default option CompressBeta => true, only a subset of the components of Y^{n+1}_{i-1} are used. To see the effect of CompressBeta => true, consider:
i12 : eagon(R,-1) EagonData removed from R.cache |
i13 : E = eagon(R,6, Verbose =>true) Used 1 of 1 blocks of eagonBeta (2, 1) Used 1 of 1 blocks of eagonBeta (2, 2) Used 1 of 1 blocks of eagonBeta (2, 3) Used 1 of 1 blocks of eagonBeta (2, 4) Used 1 of 2 blocks of eagonBeta (3, 1) Used 1 of 2 blocks of eagonBeta (3, 2) Used 1 of 2 blocks of eagonBeta (3, 3) Used 1 of 1 blocks of eagonBeta (3, 4) Used 1 of 3 blocks of eagonBeta (4, 1) Used 1 of 3 blocks of eagonBeta (4, 2) Used 1 of 2 blocks of eagonBeta (4, 3) Used 1 of 1 blocks of eagonBeta (4, 4) Used 1 of 5 blocks of eagonBeta (5, 1) Used 1 of 4 blocks of eagonBeta (5, 2) Used 1 of 2 blocks of eagonBeta (5, 3) Used 1 of 1 blocks of eagonBeta (5, 4) Used 1 of 7 blocks of eagonBeta (6, 1) Used 1 of 5 blocks of eagonBeta (6, 2) Used 1 of 2 blocks of eagonBeta (6, 3) Used 1 of 1 blocks of eagonBeta (6, 4) o13 = EagonData in <ring>.cache computed to length 6 o13 : EagonData |
i14 : eagon(R,-1) EagonData removed from R.cache |
i15 : En = eagon(R,6,CompressBeta => false) o15 = EagonData in <ring>.cache computed to length 6 o15 : EagonData |
i16 : eagonBeta (E,4), eagonBeta(E,5)
+--------+--------+-----------+
o16 = (| |(2, {1})|(0, {1, 1})|,
+--------+--------+-----------+
| (3, {})| * | . |
+--------+--------+-----------+
|(0, {2})| . | . |
+--------+--------+-----------+
-----------------------------------------------------------------------
+--------+--------+-----------+-----------+
| |(3, {1})|(0, {2, 1})|(1, {1, 1})|)
+--------+--------+-----------+-----------+
|(0, {3})| . | . | . |
+--------+--------+-----------+-----------+
|(1, {2})| . | . | . |
+--------+--------+-----------+-----------+
o16 : Sequence
|
i17 : eagonBeta (En,4), eagonBeta(En,5)
+--------+--------+-----------+
o17 = (| |(2, {1})|(0, {1, 1})|,
+--------+--------+-----------+
| (3, {})| * | . |
+--------+--------+-----------+
|(0, {2})| * | . |
+--------+--------+-----------+
-----------------------------------------------------------------------
+--------+--------+-----------+-----------+
| |(3, {1})|(0, {2, 1})|(1, {1, 1})|)
+--------+--------+-----------+-----------+
|(0, {3})| . | . | * |
+--------+--------+-----------+-----------+
|(1, {2})| * | . | * |
+--------+--------+-----------+-----------+
o17 : Sequence
|
There are also ways to investigate the components of dVert, dHor, and eagonBeta; see picture, DisplayBlocks, and mapComponent.
The object eagon is a method function with options.