--OSalgebra (page 4 of lecture) A3= matrix{{1,0,0},{0,1,0},{0,0,1},{0,1,-1},{1,0,-1},{1,-1,0}} D3= matrix{{1,0,0},{0,1,0},{0,0,1},{0,1,-1},{1,0,-1},{1,-1,0},{1,1,-1}} Stan = matrix{{1,0,0},{0,1,0},{1,1,0},{1,2,0},{1,3,0},{1,1,1},{1,5,17}} ds = (M)->(P:=M**(ZZ/101[z]); c := rank source P; r := rank target P; out={}; scan(c, i->(c1 = subsets(r,i+2); scan(c1,j->(c2 = submatrix(P,j,); if rank c2 < (i+2) then out = append(out,j))))); out) --take a matrix, and returns a list of the dependencies. Rows = H_i --example for A3 osbd = (L) -> (b1:=#L; b2:=reverse subsets(L,b1-1); b3:=0; b4:=vars R; scan(b1, i->( b3=b3+((-1)^i)* (det(diagonalMatrix transpose submatrix(b4,,b2#i),Strategy=>Cofactor)))); b3) --take a list, e.g. {1,2,3} and return the boundary map of the --corresponding monomial e.g. x_2*x_3-x_1*x_3+x_1*x_2. osbds = (M)->(I := ideal (matrix{{0}}**R); scan(M, i->(I = I + ideal osbd(i))); ideal mingens I) --Return the relations of the OS algebra. osideal = (M)->(t0=min mingle M; t1=max mingle M; R=ZZ/31991[x_(t0)..x_(t1),SkewCommutative=>true]; J = transpose mingens osbds M) --return the OS ideal of a hyperplane arrangement, input = dep sets. A3ideal = osideal ds A3 A3ring = (ring A3ideal)/ideal A3ideal reduceHilbert hilbertSeries A3ring ------------------------------------------------------------------------------------------------ --MAGIC TRICK 1 (page 5 of lecture) M=ZZ[t] (1+6*t+25*t^2+90*t^3+301*t^4+966*t^5+3025*t^6)*(1-6*t+11*t^2-6*t^3) --MAGIC TRICK 2 (page 6 of lecture) m=vars A3ring betti res (coker m, DegreeLimit=>2, LengthLimit=>6) -- 0 1 2 3 4 5 -- total: 1 6 25 90 301 966 -- 0: 1 6 25 90 301 966 --------------------------------------------------- --Koszul Section (page 11 of lecture) --Example 6: --Symmetric Alg S=ZZ/31991[x_1..x_3] resS = res coker vars S resS.dd betti resS HSS = hilbertSeries S --Exterior Alg E=ZZ/31991[e_1..e_3, SkewCommutative=>true] betti res(coker vars E, DegreeLimit=>3, LengthLimit=>4) scan(5, i->print hilbertFunction(i,S)) --LOOK AHEAD!!! reduceHilbert hilbertSeries E --Example 8: have seen Tors for A3 is it Koszul? transpose leadTerm ideal A3ideal --CERTIFIED KOSZUL: QGB --how about D3 D3ideal = osideal ds D3 transpose leadTerm ideal D3ideal --Potentially not Koszul D3ring = (ring D3ideal)/ideal D3ideal m=vars D3ring betti res (coker m, DegreeLimit=>3, LengthLimit=>5) -- 0 1 2 3 4 -- total: 1 7 35 156 663 -- 0: 1 7 34 143 560 -- 1: . . 1 13 103 --NOT KOSZUL --Example 9: Pinched Veronese T1=ZZ/31991[x_1..x_3] T2=ZZ/31991[a_1..a_9] PV = matrix{{x_1^3, x_1^2*x_2, x_1*x_2^2,x_2^3,x_1^2*x_3,x_2^2*x_3,x_1*x_3^2,x_2*x_3^2,x_3^3}} --missing xyz PVmap=map(T1,T2,PV) I=ker PVmap transpose leadTerm I --potentially not Koszul, cubic in GB PVring = T2/I m=vars PVring betti res(coker m, DegreeLimit=>3, LengthLimit=>3) -- 0 1 2 3 4 5 6 -- total: 1 9 53 280 1440 7352 37459 -- 0: 1 9 53 280 1440 7352 37459 --KOSZUL, BUT NO QGB ---------------------------------------------------------------------- --E^2 page of the spectral sequence (page 16 of lecture) E2square = (ARR, n,k)->( OSI=osideal ds ARR; mE=coker vars R; ROSI = res(coker gens ideal OSI, DegreeLimit=>k, LengthLimit=>n); TorEA=ROSI**mE; TE = apply(n, i->HH_i(TorEA)); --this gives the Tor^E(A,C). A=R/ideal OSI; mA=coker vars A; TEnew = apply(TE, i->(i**A)); --tensor Tor^E(A,C) with A RA = res(mA, DegreeLimit=>k, LengthLimit=>n); for j from 0 to n-1 do (for i from 0 to n-1 do <<" " <(e0 := min mingle G; if e0 =!= 0 then print "Start with vertex 0!" else (H:=apply(G,sort); --must have edge pairs {i(E=E||matrix{join(toList(i#0:0),{1},toList((i#1-i#0-1):0),{-1},toList((e1-i#1):0))}))); transpose compress transpose E) egypt= {{0,1},{1,2},{2,3},{0,2},{3,4},{0,5},{1,4},{1,3},{2,5},{4,5},{2,4}} --paulo graphic example. Eideal = ideal osideal ds graphic egypt Ering = R/Eideal v=vars Ering betti res (coker v, DegreeLimit=>2, LengthLimit=>5) -- 0 1 2 3 4 5 -- total: 1 11 73 379 1697 6883 -- 0: 1 11 73 379 1697 6883 --CHECK IT (1-2*t)^4*(1-3*t) * (1+11*t+73*t^2+379*t^3+1697*t^4+6883*t^5) -- -- 10 9 8 7 6 -- = - 330384t + 689440t - 543960t + 192840t - 26025t + 1 --------------------------------------------------------------------------