The source and target of a a map in the Eagon double complex, such as dVert, dHor, and eagonBeta, are direct sums of tensor products of the form K_i**X_{u_1}**..**X_{u_s} where K_i is a term of the Koszul complex and X_i is a term of the S-free resolution of R, all tensored with R. This tensor product is represented by a symbol that is a two element Sequence
(i, \{u_1..u_s\})
The block structure of the matrix, together with the source and target Sequences, can be seen from picture M.
The function mapComponent returns a single block.
i1 : S = ZZ/101[a,b,c,d,e] o1 = S o1 : PolynomialRing |
i2 : R = S/(ideal(e^2,d*e^4)+(ideal"ab,ac")^2) --a non-Golod ring, generators in different degrees o2 = R o2 : QuotientRing |
i3 : E = eagon (R,5); |
i4 : picture E#{"dHor",3,0}
+--------+-------+--------+--------+
o4 = | |(3, {})|(0, {2})|(1, {1})|
+--------+-------+--------+--------+
| (2, {})| * | * | * |
+--------+-------+--------+--------+
|(0, {1})| . | . | * |
+--------+-------+--------+--------+
|
i5 : mapComponent(E#{"dHor",3,0}, (0,{1}),(1,{1}))
o5 = {2} | a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 e 0 0 0 |
{4} | 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 e 0 0 |
{4} | 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 e 0 |
{4} | 0 0 0 a 0 0 0 b 0 0 0 c 0 0 0 d 0 0 0 e |
4 20
o5 : Matrix R <--- R
|
i6 : picture E#{"dVert",3,1}
+--------+-------+--------+--------+--------+-----------+
o6 = | |(4, {})|(0, {3})|(1, {2})|(2, {1})|(0, {1, 1})|
+--------+-------+--------+--------+--------+-----------+
| (3, {})| * | * | * | * | * |
+--------+-------+--------+--------+--------+-----------+
|(0, {2})| . | . | * | * | 5,3 |
+--------+-------+--------+--------+--------+-----------+
|(1, {1})| . | . | . | * | * |
+--------+-------+--------+--------+--------+-----------+
|
i7 : mapComponent(E#{"dVert",3,1}, (0,{2}),(0,{1,1}))
o7 = {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{6} | 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
{6} | 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 |
{6} | 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 |
5 16
o7 : Matrix R <--- R
|
i8 : picture E#{"eagonBeta",3,1}
+--------+--------+-----------+
o8 = | |(2, {1})|(0, {1, 1})|
+--------+--------+-----------+
| (3, {})| * | * |
+--------+--------+-----------+
|(0, {2})| * | 5,3 |
+--------+--------+-----------+
|
i9 : mapComponent(E#{"eagonBeta",3,1}, (0,{2}),(0,{1,1}))
o9 = {5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{5} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
{6} | 0 -1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
{6} | 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 |
{6} | 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 0 0 |
5 16
o9 : Matrix R <--- R
|
The object mapComponent is a method function.