restart -- Example 1 R = QQ[x] I = ideal(x^2) S = R/I m = ideal vars S F = res comodule m -- by default, LengthLimit = #vars in over-ring R. -- Hence no ending 0 => truly infinite free resolution F = res(comodule m, LengthLimit=>5) F = res(comodule m, LengthLimit=>10) F.dd betti F -- Example 2 R = QQ[x,y] I = ideal(x^2,y^2) S = R/I m = ideal vars S F = res comodule m F = res(comodule m, LengthLimit=>8) betti F -- Example 2b - Module where Betti Numbers decrease for an arbitrary -- number of steps. (Here arbitrary means 8.) G = dual F M = coker G.dd_(-6) H = res(M**S^{-7},LengthLimit=>16) H.dd_8 betti H -- Example 3 R = QQ[x,y] I = ideal(x^2,x*y,y^2) S = R/I m = ideal vars S F = res comodule m F = res(comodule m, LengthLimit=>8) betti F -- Exponential growth of Betti numbers. Note that S is not a complete -- intersection as in Examples 1 and 2 -- Example 4 R = QQ[w..z] I = ideal(x*y-w*z) S = R/I V = matrix{{w,x},{y,z}} F = res coker V F.dd --periodic of period 2 (up to isomorphism) restart R = QQ[a..i] V = matrix{{a,b,c},{d,e,f},{g,h,i}} isInjective V f = det V f*coker V == 0 -- f annihilates coker M C = chainComplex V s = nullhomotopy(-f*id_C) U = s_0 U*V V*U -f*id_(R^3) S = R/ideal(f) P = map(S,R) M = coker P(V) F = res M F.dd -- A periodic resolution over R/(f) -- Example 5 R = QQ[x,y,z] f = x^3 + y^3 + z^3 S = R/ideal(f) m = ideal vars S F = res(comodule m^2,LengthLimit=>8) F.dd L = coker F.dd_3 V = lift(F.dd_3,R) LR = coker V res LR C = chainComplex V U = (nullhomotopy(-f*id_C))_0 U*V V*U -- We get a matrix factorization from the infinite free resolution above -- Example 6 -- Examples of Gasharov and Peeva -- Periodic resolutions of period n -- Pick a in the coefficient field K with order n in the group K* -- Period 2: p = 3 a = 2 R = ZZ/p[v..z]/ideal(a*w*y+x*y,w*z+x*z,y^2-x*v+a*w*v,z^2-x*v+w*v,w^2,x^2,y*z,y*v,z*v,v^2) M = coker matrix{{w,a*y+z},{0,x}} F = res(M,LengthLimit=>8) F.dd -- Period 3: p = 7 a = 2 R = ZZ/p[v..z]/ideal(a*w*y+x*y,w*z+x*z,y^2-x*v+a*w*v,z^2-x*v+w*v,w^2,x^2,y*z,y*v,z*v,v^2) M = coker matrix{{w,a*y+z},{0,x}} F = res(M,LengthLimit=>8) F.dd -- Period 4: p = 5 a = 2 R = ZZ/p[v..z]/ideal(a*w*y+x*y,w*z+x*z,y^2-x*v+a*w*v,z^2-x*v+w*v,w^2,x^2,y*z,y*v,z*v,v^2) M = coker matrix{{w,a*y+z},{0,x}} F = res(M,LengthLimit=>8) F.dd -- Example 6b -- Another Gasharov-Peeva example of how to get periodic resolutions (of period n) -- over an arbitrary field restart n = 5 R = QQ[x_1,x_2,x_3,y_1..y_n] I = ideal mingens( ideal(x_1*y_1 + x_2*y_n) + ideal(for i from 2 to n list (x_1*y_i + x_2*y_(i-1))) + ideal(x_2*x_3-y_1^2) + ideal(for i from 2 to 1 + floor(n/2) list (x_2*x_3-y_i*y_(n+2-i))) + ideal(for i from 1 to floor((n+1)/2) list (x_1*x_3 + y_i*y_(n+1-i))) + ideal(x_1^2,x_2^2,x_3^3) + ideal(for i from 1 to n list (y_i*y_3)) + ideal(flatten for i from 1 to n list for j from 1 to n list(if(not(i == 1 and j == 1)) and i + j != n+1 and i+j != n+2 and i <= j then y_i*y_j else 0))) I = I + ideal(x_3) ideal flatten entries mingens I S = R/(I) M = coker matrix {{x_2,0},{y_1,x_1}} F = res(M,LengthLimit=>n+2) F.dd -- Example 7 -- Example of Anick of an Artinian ring for with P_R(k) is not rational restart R = QQ[a..e] m = ideal vars R I = ideal(a^2,b^2,d^2,e^2,a*b,d*e,a*c+c*d+b*e) + m^3 S = R/I betti res(comodule ideal vars S,LengthLimit=>7) poincareN res(comodule ideal vars S,LengthLimit=>7) (map(ZZ[Z],ring oo,{1,Z}))(oo) -- Example 8 -- Tate Resolution R = QQ[x,y]/ideal(x^2,y^2) F = res(comodule ideal vars R,LengthLimit=>5) F.dd_1 F.dd_2 F.dd_3 F.dd_4 -- Example 9 -- Lech's Example of a quadratic algebra that is not Koszul restart R = QQ[w,x,y,z] I = ideal(w^2,x^2,y^2,z^2,w*x+y*z) S = R/I betti res comodule ideal vars S hilbertFunction(3,S) hilbertSeries S reduceHilbert oo -- If S was Koszul, then the Poincare Series would be 1/Hilb_S(-z) T = QQ[Z] loadPackage "PowerSeries" s = series(1 - 4*Z + 5*Z^2) t = inverse s setDegree(8,t) -- Poincare Series can't have negative coefficients -- Hence S is not Koszul -- Example 10 -- Quadratic Complete Intersections (and hence Koszul algebras) -- that do not have a quadratic GB restart R = QQ[x,y,z] I = ideal(random(2,R),random(2,R),random(2,R)) -- 3 general quadrics codim I loadPackage "gfanInterface" (M,L) = gfan I; for i from 0 to length M - 1 do print M_i for i from 0 to length M - 1 do print max apply(M_i,x->degree x) -- Every GB has a quartic element I = ideal(x^2+y*z,y^2+x*z,z^2+x*y) -- a cleaner example codim I (M,L) = gfan I; for i from 0 to length M - 1 do print M_i for i from 0 to length M - 1 do print max apply(M_i,x->degree x) -- Example 11 -- Roos' Example of quadratic algebras whose resolutions of K -- become nonlinear at an arbitrary step in the resolution restart n = 3 R = QQ[x_1..x_6] I = ideal(x_1^2,x_1*x_2,x_2*x_3,x_3^2,x_3*x_4,x_4^2,x_4*x_5,x_5*x_6,x_6^2,x_1*x_3+n*x_3*x_6-x_4*x_6, x_3*x_6 + x_1*x_4 + (n - 2)*x_4*x_6) S = R/I F = res(comodule ideal vars S,LengthLimit=>4) betti F n = 4 R = QQ[x_1..x_6] I = ideal(x_1^2,x_1*x_2,x_2*x_3,x_3^2,x_3*x_4,x_4^2,x_4*x_5,x_5*x_6,x_6^2,x_1*x_3+n*x_3*x_6-x_4*x_6, x_3*x_6 + x_1*x_4 + (n - 2)*x_4*x_6) S = R/I F = res(comodule ideal vars S,LengthLimit=>5) betti F n = 5 R = QQ[x_1..x_6] I = ideal(x_1^2,x_1*x_2,x_2*x_3,x_3^2,x_3*x_4,x_4^2,x_4*x_5,x_5*x_6,x_6^2,x_1*x_3+n*x_3*x_6-x_4*x_6, x_3*x_6 + x_1*x_4 + (n - 2)*x_4*x_6) S = R/I F = res(comodule ideal vars S,LengthLimit=>6) betti F -- Refs: -- 1. Avramov and Grayson, "Resolutions and Cohomology over Complete -- Intersections". (part of Computations in algebraic geometry with Macaulay 2) -- Available for free at: http://www.math.uiuc.edu/Macaulay2/Book/ -- 2. Avramov, "Infinite Free Resolutions." (part of Six Lectures on -- Commutative Algebra) -- 3. Conca, "Koszul Algebras and Grobner Bases of Quadrics" (Nagoya Symposium Proceedings) -- 4. Froberg, "Koszul Algebras"