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localCohom.exa.m2
restart load "Dloadfile.m2" -- set your favorite strategy Str = Walther LocStr = Oaku -- Example (Harry) setInfoLevel 666 W = QQ[X, dX, Y, dY, Z, dZ, WeylAlgebra=>{X=>dX, Y=>dY, Z=>dZ}] I = ideal (X*(Y-Z), X*Y*Z) M = ideal (dX, dY, dZ) time h = localCohom(I, Strategy=>Walther) pruneLocalCohom h time h = localCohom(2, I) pruneLocalCohom h -- Example (Anton) W = QQ[X, dX, Y, dY, WeylAlgebra=>{X=>dX, Y=>dY}] I = ideal (X^2+Y^2, X*Y) time localCohom(I, Strategy=>Str) -- OT "Algs for D-mods" (1998), Example 7.5 W = QQ[x,y,z,dx,dy,dz, WeylAlgebra=>{x=>dx,y=>dy,z=>dz}] R = cokernel matrix{{dx, dy, z^3*dz+z}} I = ideal (x*z, y*z) localCohom(I, R, Strategy=>Str) -------------------------------------------------------------------------------- restart load "Dloadfile.m2" -- b^2-ac, c^2-bd, ad-bc. W = QQ[x_1..x_4, dx_1..dx_4, WeylAlgebra => toList (1..4) / (i -> x_i => dx_i)] I = minors(2, matrix{{x_1, x_2, x_3}, {x_2, x_3, x_4}}) r = localCohom (I, W^1 / ideal {dx_1, dx_2, dx_3, dx_4}, Strategy=>Walther, LocStrategy=>Oaku); restart load "Dloadfile.m2" -- Example (!!! Unable to compute !!!) W = QQ[x_1..x_6, dx_1..dx_6, WeylAlgebra => toList (1..6) / (i -> x_i => dx_i)] I = minors(2, matrix{{x_1, x_2, x_3}, {x_4, x_5, x_6}}) h = localCohom ({0,1,3}, I, W^1 / ideal {dx_1, dx_2, dx_3, dx_4, dx_5, dx_6}, Strategy=>Walther, LocStrategy=>Oaku); pruneLocalCohom oo ----------------------------------------------------------------------------- -- EXAMPLES for the local cohomology paper ----------------------------------------------------------------------------- restart load "Dloadfile.m2" W = QQ[X, dX, Y, dY, Z, dZ, WeylAlgebra=>{X=>dX, Y=>dY, Z=>dZ}]; I = ideal (X*(Y-Z), X*Y*Z); time h = localCohom(I, Strategy=>Walther) pruneLocalCohom h time localCohom(I, Strategy=>OaTa) M = cokernel matrix{{dX, dY, Z^3*dZ+Z}}; I = ideal (X*Z, Y*Z); time h = localCohom(I, M, Strategy=>Walther, LocStrategy=>Oaku) pruneLocalCohom h time localCohom(I, M, Strategy=>OaTa) W = QQ[X, dX, Y, dY, WeylAlgebra=>{X=>dX, Y=>dY}] I = ideal (X^2+Y^2, X*Y) time h = localCohom(I, Strategy=>Walther) pruneLocalCohom h time localCohom(I, Strategy=>OaTa) -- twisted cubic W = QQ[x_1..x_4, dx_1..dx_4, WeylAlgebra => toList (1..4) / (i -> x_i => dx_i)]; I = minors(2, matrix{{x_1, x_2, x_3}, {x_2, x_3, x_4}}); time h = localCohom ({0,1,3}, I, W^1 / ideal {dx_1, dx_2, dx_3, dx_4}, Strategy=>Walther, LocStrategy=>Oaku); pruneLocalCohom h time localCohom (2, I, W^1 / ideal {dx_1, dx_2, dx_3, dx_4}, Strategy=>Walther, LocStrategy=>Oaku); time h = localCohom ({0,1,3}, I, W^1 / ideal {dx_1, dx_2, dx_3, dx_4}, Strategy=>OaTa); -- 2x2 minor W = QQ[x_1..x_6, dx_1..dx_6, WeylAlgebra => toList (1..6) / (i -> x_i => dx_i)]; I = minors(2, matrix{{x_1, x_2, x_3}, {x_4, x_5, x_6}}); time h = localCohom ({0,1,3}, I, W^1 / ideal {dx_1, dx_2, dx_3, dx_4, dx_5, dx_6}, Strategy=>Walther, LocStrategy=>Oaku); pruneLocalCohom h time h = localCohom (2, I, W^1 / ideal {dx_1, dx_2, dx_3, dx_4, dx_5, dx_6}, Strategy=>Walther, LocStrategy=>Oaku); ----------------------------------------------------------------- -- Simple examples restart load "Dloadfile.m2" W = QQ[x_1..x_6, dx_1..dx_6, WeylAlgebra => toList (1..6) / (i -> x_i => dx_i)]; M = W^10/ideal{dx_1^3, dx_2, dx_3, dx_4, dx_5, dx_6} M = makeCyclic presentation M M = makeCyclic(matrix{{dx_1^2, dx_2^2, dx_3^2, dx_4^2, dx_5^2, dx_6^2}}) M = makeCyclic(matrix{{dx_1, dx_2, dx_3, dx_4, dx_5, dx_6}|toList(12:0), toList(6:0)|{dx_1, dx_2, dx_3, dx_4, dx_5, dx_6}|toList(6:0), toList(12:0)|{dx_1^2, dx_2^2, dx_3^2, dx_4^2, dx_5^2, dx_6^2}} ) time h = localCohom (ideal{x_1,x_2,x_3}, W^1/M.AnnG, Strategy=>Walther, LocStrategy=>Oaku); pruneLocalCohom h time h = localCohom (ideal{x_1,x_2,x_3}, W^1/M.AnnG, Strategy=>OaTa); time h = localCohom (ideal{x_1,x_2,x_3}, M, Strategy=>OaTa); pruneLocalCohom h --------------------------------------------------- -- Foolproof check --------------------------------------------------- W = QQ[X, dX, Y, dY, WeylAlgebra=>{X=>dX, Y=>dY}] I = ideal (X^2+Y^2, X*Y) M = cokernel dX Dtrace 1 time localCohom(I,M)