restart
-- Burch's Construction Applied to a Polynomial Ring:

-- pd = 4 (degrees: 2,2,2)
S = QQ[a,b,x,y]
I = ideal(a*x,b*y,a*b+x*y)
F = res (S^1/I)
pdim (S^1/I)
F.dd
degrees  I

-- pd = 5 (degrees: 3,3,4)
S = QQ[a,b,c,x,y,z]
I = ideal(a*b*c,x*y*z,a*b*x*y+a*c*x*z+b*c*y*z)
F = res (S^1/I)
pdim (S^1/I)
degrees I

-- pd = 6 (degrees: 4,4,6)
S = QQ[a,b,c,d,w,x,y,z]
I = ideal(a*b*c*d,w*x*y*z,a*b*c*w*x*y+a*b*d*w*x*z+a*c*d*w*y*z+b*c*d*x*y*z)
F = res (S^1/I)
pdim (S^1/I)
degrees I
-- Note: There was a typo in this one during the talk.

-- In general:
-- pd = n (degrees: n-1, n-1, 2n-2)

loadPackage "Bruns"
S = ZZ/5[a..e]
m = ideal vars S
res comodule m
brunsIdeal m
res oo

restart
-- Three cubics of projective dimension 5
-- I_{2,1,3}. pd = 5 (degrees: 3,3,3)
loadPackage "BigIdeal"
I = jasonIdeal(2,1,3)
S = ring I
res (S^1/I)
betti oo
pdim (S^1/I)

-- Smallest Counterexample to Zhang's Question
I = jasonIdeal(3,4,2)
S = ring I
Q = S/I
m = ideal vars Q
soc = Q_0*Q_1*Q_2
-- Check that x_1*x_2*x_3 is in socle(S/I)
zero soc
zero (soc * m)
-- Then pd(S/I) = dim S = 15
dim S
-- Sum of the degrees of the generators is 14
first sum flatten degrees gens I

-- Growth of pd in 3-generated case is only linear
netList for i from 1 to 10 list(
      if i == 1 then {"degree d", "pd(S/I_{2,1,d})"} else {i, dim ring jasonIdeal(2,1,i)}
     )

-- Examples from talk
I = bigIdeal(2,{3,1})
betti res I
degrees I

netList for i from 1 to 5 list(
     A = for j from 1 to i-1 list i+1;
     A = append(A,0);
     if i == 1 then {"i","degree d", "pd(S/I(2,{i+1,i+1,..,i+1,0}))","i^(i-1)"} else {i, i^2, dim ring bigIdeal(2,A),i^(i-1)}
	  )
-- pd(S/I(2,{i,i,..i})) > i^(i-1)
-- degree = i^2

-- Regularity Examples

-- Caviglia's Family
I = bigIdeal(2,{1,5})
S = ring I
betti res (S^1/I)
regularity (S^1/I)

-- This family shows quadratic regularity growth with 3 generators in degree d
netList for i from 1 to 10 list(
      if i == 1 then {"degree d", "regularity(S/I(2,{1,d-2})"} else {i, regularity bigIdeal(2,{1,i-2})}
     )
-- reg(S/I(2,{1,d-2})) = d^2 - 1

netList for i from 3 to 9 list(
      if i == 3 then {"degree d", "regularity(S/I(2,{2,1,d-4})"} else {i, regularity bigIdeal(2,{2,1,i-4})}
     )
-- rate of growth of reg(S/I(2,{2,1,d-4}) = 1/2 d^3 - 3d^2 + 8d - 7 (d even)
--                                        = 1/2 d^3 - 5/2d^2 + 5d - 3 (d odd)
-- 11    | 415
-- 12    | 521
-- 13    | 738
-- 14    | 889
-- 15    | 1197
-- 16    | 1401   (computed by Mike Stillman)

I = bigIdeal(2,{2,1,0},BaseField=>ZZ/2)
betti res comodule I
betti res comodule bigIdeal(2,{2,2,1,0},BaseField=>ZZ/2)


R = ZZ/2[a..d]
I = ideal(a^2,b^2,a*b)
F = res I
betti res I
F.dd_2
I = bigIdeal(2,{0})

I = bigIdeal(2,{2,1,0})
S = ring I
gens I
gens I * transpose(matrix{{ -S_2^2,S_3^2,S_0*S_2-S_1*S_3}})
flatten entries (F.dd_2)_11
F.dd_2

I = bigIdeal(2,{2,2,2,2,1,0},BaseField=>ZZ/2)
ideal gens I
S = ring I
F = res(I,DegreeLimit=>70)
betti F
flatten entries (F.dd_2)_(5)
F.dd_2
gens I
first flatten entries (gens I * transpose(matrix{{0,0,S_3*S_5+S_0*S_1*(S_4^2-S_5^2)+S_2*S_4-S_3*S_5 }}))
first flatten entries (gens I * transpose(matrix{{0,0,S_3*S_5+S_2*S_4-S_3*S_5 }}))
S_0*S_1*(S_4^3-S_5^2)
regularity bigIdeal(2,{2,2,1,0},BaseField=>ZZ/2)
S = ring I
F = res I
(F.dd_2)_4
I = bigIdeal(2,{2,1,0})
S = ring I
F = res I
(F.dd_2)_6
I = bigIdeal(2,{2,2,1,0})
R = QQ[a,b,c]
res ideal(a^2,a*b,b^2)
oo.dd







loadPackage "BigIdeal"
I = bigIdeal(2,{3})
degree I
pdim comodule I







I = bigIdeal(2,{2,1,2},BaseField=>ZZ/101)
J = monomialSubideal I
S = ring I
R = S/J
socle = ideal syz transpose vars R
betti res I
betti res J
flatten entries gens socle
select(socle_*, m->first degree m == 41)
L = apply(oo,m->lift(m,S))
apply(oo,m->m%I)
mingens ideal oo
unique oo

I = bigIdeal(2,{2,2,2,1,0},BaseField=>ZZ/101)
L = vars ring I
R = ring I
R_{5,4,20,3,20,2,20,1,0,0}%I
socleCheck(I,oo)

gbTrace = 1
gb I
M = L/exponents/first
select(M, e -> e#0 == 5 and e#1 == 2)
L/sum
m = ideal vars ring I
(I:m)
monsInI = {};
for i from 0 to 5 do(
     B = flatten entries basis(i,ring I);
     for m in B do(
	  if m%I== 0 then(
	       minimal = true;
	       for n in monsInI do(
		    if m%n==0 then minimal = false
		    )
	       if minimal == true then monsInI = append(monsInI,m);
	  );
     print i;
     )

I = bigIdeal(2,{2,1,4},BaseField=>ZZ/2)
S = ring I
Q = S/ideal(I_0,I_1)
sub(I,Q)
res oo
betti oo