sunnyquiver
commented
1 year ago Hopefully this bug is fixed by the suggested change in commit 1dff97e on August 14th, 2023.
Closed Nakayamaalgebra closed 1 year ago
sunnyquiver
commented
1 year ago Hopefully this bug is fixed by the suggested change in commit 1dff97e on August 14th, 2023.
The following is the quiver and relations of the Auslander algebra A of linear oriented path algebra KQ with Q of Dynkin type D_5 :
Quiver( ["1","2","3","4","5","6","7","8","9","10","11","12","13","14","15","16","17","18","19","20"], [["3","2","a_3_2"],["2","1","a_2_1"],["6","3","a_6_3"],["6","4","a_6_4"],["4","2","a_4_2"],["6","5"\ ,"a_6_5"],["5","2","a_5_2"],["7","6","a_7_6"],["8","6","a_8_6"],["11","6","a_11_6"],["11","10","a_11_10"],["10","5","a_10_5"],["12","8","a_12_8"],["12","7","a_12_7"],["12","11","a_12_11"],["13","12","a\ _13_12"],["14","12","a_14_12"],["2","9","a_2_9"],["15","11","a_15_11"],["16","15","a_16_15"],["16","12","a_16_12"],["17","16","a_17_16"],["17","13","a_17_13"],["17","14","a_17_14"],["18","16","a_18_16"\ ],["19","18","a_19_18"],["19","17","a_19_17"],["20","19","a_20_19"]] ) [ (Z(3)^0)*a_3_2*a_2_1, (Z(3)^0)*a_4_2*a_2_9, (Z(3)^0)*a_6_3*a_3_2+(Z(3)^0)*a_6_4*a_4_2+(Z(3)^0)*a_6_5*a_5_2, (Z(3)^0)*a_7_6*a_6_3, (Z(3)^0)*a_8_6*a_6_4, (Z(3)^0)*a_11_6*a_6_5+(Z(3)^0)*a_11_10*a_10_5, (Z(3)^0)*a_12_8*a_8_6+(Z(3)^0)*a_12_7*a_7_6+(Z(3)^0)*a_12_11*a_11_6, (Z(3)^0)*a_13_12*a_12_7, (Z(3)^0)*a_14_12*a_12_8, (Z(3)^0)*a_15_11*a_11_10, (Z(3)^0)*a_16_15*a_15_11+(Z(3)^0)*a_16_12*a_12_11, (Z(3)^0)*a_17_16*a_16_12+(Z(3)^0)*a_17_13*a_13_12+(Z(3)^0)*a_17_14*a_14_12, (Z(3)^0)*a_18_16*a_16_15, (Z(3)^0)*a_19_18*a_18_16+(Z(3)^0)*a_19_17*a_17_16, (Z(3)^0)*a_20_19*a_19_18, (Z(3)^0)*a_15_11*a_11_6*a_6_5, (Z(3))*a_16_15*a_15_11*a_11_6+(Z(3)^0)*a_16_12*a_12_8*a_8_6+(Z(3)^0)*a_16_12*a_12_7*a_7_6, (Z(3)^0)*a_17_16*a_16_12*a_12_8+(Z(3)^0)*a_17_13*a_13_12*a_12_8, (Z(3)^0)*a_15_11*a_11_6*a_6_3*a_3_2+(Z(3)^0)*a_15_11*a_11_6*a_6_4*a_4_2, (Z(3))*a_16_15*a_15_11*a_11_6*a_6_3+(Z(3)^0)*a_16_12*a_12_8*a_8_6*a_6_3, (Z(3)^0)*a_15_11*a_11_6*a_6_3*a_3_2*a_2_9 ]It should be correct as it has global and dominant dimension equal to two.
Now I want the stable Auslander algebra of KQ and it seems there might be a problem with the command QuiverAlgebraOfAmodAeA (note that the stable auslander algebra is A/AeA where e corresponds to the projective-injective A-modules of the Auslander algebra A). Here A=C.
injC:=IndecInjectiveModules(C);n:=Size(injC);W:=Filtered([1..n],x->IsProjectiveModule(injC[x])=true);T:=QuiverAlgebraOfAmodAeA(C,W);GlobalDimensionOfAlgebra(T,33);
gives and algebra of global dimension 3 and
projC:=IndecProjectiveModules(C);n:=Size(projC);W:=Filtered([1..n],x->IsInjectiveModule(projC[x])=true);T:=QuiverAlgebraOfAmodAeA(C,W);GlobalDimensionOfAlgebra(T,33);
gives an algebra of global dimension 4, thus both give the wrong answer since the stable Auslander algebra has global dimension 2.
Someone gave the following guess about the problem in the code of QuiverAlgebraOfAmodAeA: In the line 3160: r[2], I suspect it should be r[2*i+2]
After changing this in the code it seems to give the correct algebra (at least the global dimension is then 2 as it should be).
Thanks for any help.