We illustrate some aspects of the paper "A case study in bigraded commutative algebra" by Cox-Dickenstein-Schenck. In that paper, an appropriate term on the E_2 page of a suitable spectral sequence corresponds to non-koszul syzygies.
Using our indexing conventions, the E^2_{3,-1} term will be what the $E^{0,1}_2$ term is in their paper.
We illustrate an instance of the non-generic case for non-Koszul syzygies. To do this we look at the three polynomials used in their Example 4.3. The behaviour that we expect to exhibit is predicted by their Proposition 5.2.
i1 : R = QQ[x,y,z,w, Degrees => {{1,0},{1,0},{0,1},{0,1}}];
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i2 : B = ideal(x*z, x*w, y*z, y*w); o2 : Ideal of R |
i3 : p_0 = x^2*z; |
i4 : p_1 = y^2*w; |
i5 : p_2 = y^2*z+x^2*w; |
i6 : I = ideal(p_0,p_1,p_2); o6 : Ideal of R |
i7 : B = B_*/(x -> x^2)//ideal; o7 : Ideal of R |
i8 : G = complete res image gens B; |
i9 : F = koszul gens I; |
i10 : K = Hom(G, filteredComplex(F)); |
i11 : E = prune spectralSequence K; |
i12 : E^1
+---------------------------------+---------------------------------------------------------+--------------------------------------------------------+--------------------------------+
| 1 | 3 | 3 | 1 |
o12 = |R |R |R |R |
| | | | |
|{0, 0} |{1, 0} |{2, 0} |{3, 0} |
+---------------------------------+---------------------------------------------------------+--------------------------------------------------------+--------------------------------+
|cokernel {-4, 0} | y2 x2 0 0 | |cokernel {-2, 1} | y2 x2 0 0 0 0 0 0 0 0 0 0 | |cokernel {0, 2} | y2 x2 0 0 0 0 0 0 0 0 0 0 ||cokernel {2, 3} | y2 x2 0 0 ||
| {0, -4} | 0 0 w2 z2 | | {-2, 1} | 0 0 y2 x2 0 0 0 0 0 0 0 0 | | {0, 2} | 0 0 y2 x2 0 0 0 0 0 0 0 0 || {6, -1} | 0 0 w2 z2 ||
| | {-2, 1} | 0 0 0 0 y2 x2 0 0 0 0 0 0 | | {0, 2} | 0 0 0 0 y2 x2 0 0 0 0 0 0 || |
|{0, -1} | {2, -3} | 0 0 0 0 0 0 w2 z2 0 0 0 0 | | {4, -2} | 0 0 0 0 0 0 w2 z2 0 0 0 0 ||{3, -1} |
| | {2, -3} | 0 0 0 0 0 0 0 0 w2 z2 0 0 | | {4, -2} | 0 0 0 0 0 0 0 0 w2 z2 0 0 || |
| | {2, -3} | 0 0 0 0 0 0 0 0 0 0 w2 z2 | | {4, -2} | 0 0 0 0 0 0 0 0 0 0 w2 z2 || |
| | | | |
| |{1, -1} |{2, -1} | |
+---------------------------------+---------------------------------------------------------+--------------------------------------------------------+--------------------------------+
|cokernel {-4, -4} | w2 z2 y2 x2 ||cokernel {-2, -3} | w2 z2 y2 x2 0 0 0 0 0 0 0 0 ||cokernel {0, -2} | w2 z2 y2 x2 0 0 0 0 0 0 0 0 ||cokernel {2, -1} | w2 z2 y2 x2 ||
| | {-2, -3} | 0 0 0 0 w2 z2 y2 x2 0 0 0 0 || {0, -2} | 0 0 0 0 w2 z2 y2 x2 0 0 0 0 || |
|{0, -2} | {-2, -3} | 0 0 0 0 0 0 0 0 w2 z2 y2 x2 || {0, -2} | 0 0 0 0 0 0 0 0 w2 z2 y2 x2 ||{3, -2} |
| | | | |
| |{1, -2} |{2, -2} | |
+---------------------------------+---------------------------------------------------------+--------------------------------------------------------+--------------------------------+
o12 : SpectralSequencePage
|
i13 : E^2
+------------------------------------------------+--------------------------------------------------------------------------+------------------------------------------------------------------+--------------------------------+
o13 = |cokernel | y2w y2z+x2w x2z | |cokernel {2, 3} | y2 x2 0 0 | |0 |0 |
| | {6, 1} | 0 0 w z | | | |
|{0, 0} | |{2, 0} |{3, 0} |
| |{1, 0} | | |
+------------------------------------------------+--------------------------------------------------------------------------+------------------------------------------------------------------+--------------------------------+
|cokernel {-4, 0} | y2 x2 0 0 0 0 0 ||cokernel {-2, 1} | y2 x2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||cokernel {0, 2} | y2 x2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ||cokernel {2, 3} | y2 x2 0 0 | |
| {0, -4} | 0 0 w2 z2 y2w y2z+x2w x2z || {-2, 1} | 0 0 y2 x2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {0, 2} | 0 0 y2 x2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 || {6, 1} | 0 0 w z | |
| | {-2, 1} | 0 0 0 0 y2 x2 0 0 0 0 0 0 0 0 0 0 0 0 0 || {0, 2} | 0 0 0 0 y2 x2 0 0 0 0 0 0 0 0 0 0 0 0 || |
|{0, -1} | {2, -2} | 0 0 0 0 0 0 z 0 0 0 w w2 0 0 -y2 y2w 0 0 0 || {4, 0} | 0 0 0 0 0 0 w z 0 0 0 0 x2 -y2 0 0 0 0 ||{3, -1} |
| | {2, -2} | 0 0 0 0 0 0 0 w z 0 0 0 x2 0 0 0 0 0 0 || {4, 0} | 0 0 0 0 0 0 0 0 w z 0 0 -y2 0 0 0 0 0 || |
| | {2, -2} | 0 0 0 0 0 0 0 0 w z 0 0 -y2 0 0 0 0 0 0 || {4, 0} | 0 0 0 0 0 0 0 0 0 0 w z 0 x2 0 0 0 0 || |
| | {2, -2} | 0 0 0 0 0 0 0 0 0 w z 0 0 w2 x2 0 y2w 0 0 || {6, -1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 w z 0 0 || |
| | {6, -3} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 w z || {6, -1} | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 w z || |
| | | | |
| |{1, -1} |{2, -1} | |
+------------------------------------------------+--------------------------------------------------------------------------+------------------------------------------------------------------+--------------------------------+
|cokernel {-4, -4} | w2 z2 y2 x2 | |cokernel {-2, -3} | w2 z2 y2 x2 0 0 0 0 0 0 0 0 | |cokernel {0, -2} | w2 z2 y2 x2 0 0 0 0 0 0 0 0 | |cokernel {2, -1} | w2 z2 y2 x2 ||
| | {-2, -3} | 0 0 0 0 w2 z2 y2 x2 0 0 0 0 | | {0, -2} | 0 0 0 0 w2 z2 y2 x2 0 0 0 0 | | |
|{0, -2} | {-2, -3} | 0 0 0 0 0 0 0 0 w2 z2 y2 x2 | | {0, -2} | 0 0 0 0 0 0 0 0 w2 z2 y2 x2 | |{3, -2} |
| | | | |
| |{1, -2} |{2, -2} | |
+------------------------------------------------+--------------------------------------------------------------------------+------------------------------------------------------------------+--------------------------------+
o13 : SpectralSequencePage
|
The degree zero piece of the module $E^2_{3,-1}$ twisted by $R((2,3))$ below shows that there is a $1$-dimensional space of non-Koszul syzygies of bi-degree $(2,3)$. This is what is predicted by the paper.
i14 : E^2_{3,-1}
o14 = cokernel {2, 3} | y2 x2 0 0 |
{6, 1} | 0 0 w z |
2
o14 : R-module, quotient of R
|
i15 : basis({0,0}, E^2_{3, -1} ** R^{{2, 3}})
o15 = {0, 0} | 1 |
{4, -2} | 0 |
o15 : Matrix
|
i16 : E^2 .dd_{3, -1}
o16 = {2, 3} | -1 0 |
{6, 1} | 0 1 |
o16 : Matrix
|
i17 : basis({0,0}, image E^2 .dd_{3,-1} ** R^{{2,3}})
o17 = {0, 0} | 1 |
{4, -2} | 0 |
o17 : Matrix
|
i18 : basis({0,0}, E^2_{1,0} ** R^{{2,3}})
o18 = {0, 0} | 1 |
{4, -2} | 0 |
o18 : Matrix
|
The degree zero piece of the module $E^2_{3,-1}$ twisted by $R((6,1))$ below shows that there is a $1$-dimensional space of non-Koszul syzygies of bi-degree $(6,1)$. This is also what is predicted by the paper.
i19 : basis({0,0}, E^2 _{3, -1} ** R^{{6,1}})
o19 = {-4, 2} | 0 |
{0, 0} | 1 |
o19 : Matrix
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i20 : isIsomorphism(E^2 .dd_{3, -1})
o20 = true
|