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# Spectral sequences and non-Koszul syzygies

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}}]; 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 i20 : isIsomorphism(E^2 .dd_{3, -1}) o20 = true