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ResLengthThree.m2
-* Copyright 2020 Lars Winther Christensen, Luigi Ferraro, Francesca Gandini, Frank Moore, and Oana Veliche. You may redistribute this file under the terms of the GNU General Public License as published by the Free Software Foundation, either version 2 of the License, or any later version. *- newPackage ( "ResLengthThree", Version => "1.0", Date => "3 December 2020", Authors => { { Name => "Lars Winther Christensen", Email => "lars.w.christensen@ttu.edu", HomePage => "http://www.math.ttu.edu/~lchriste/index.html" }, { Name => "Luigi Ferraro", Email => "lferraro@ttu.edu", HomePage => "http://www.math.ttu.edu/~lferraro" }, { Name => "Francesca Gandini", Email => "fra.gandi.phd@gmail.com", HomePage => "https://github.com/fragandi" }, { Name => "Frank Moore", Email => "moorewf@wfu.edu", HomePage => "http://users.wfu.edu/moorewf/" }, { Name => "Oana Veliche", Email => "o.veliche@northeastern.edu", HomePage => "https://web.northeastern.edu/oveliche/index.html" } }, Headline => "Multiplication in free resolutions of length three", Reload => false, DebuggingMode => false, Keywords => { "Homological Algebra" }, PackageExports => { "Complexes" } ) export { "resLengthThreeAlg", "resLengthThreeTorAlg", "multTableOneOne", "multTableOneTwo", "resLengthThreeTorAlgClass", "makeRes", "Labels", "Compact" } --========================================================================== -- EXPORTED FUNCTIONS --========================================================================== resLengthThreeAlg = method() resLengthThreeAlg(Complex, List) := (F, sym) -> ( if F.cache#?"Algebra Structure" then return F.cache#"Algebra Structure"; if length F != 3 then error "Expected a chain complex of length three which is free of rank one in degree zero."; if #sym != 3 or any(sym, s -> (class baseName s =!= Symbol)) then error "Expected a list of three symbols."; mult := multTables(F); Q := ring F; m := numcols F.dd_1; l := numcols F.dd_2; n := numcols F.dd_3; degreesP := if isHomogeneous F then flatten apply(3, j -> apply(degrees source F.dd_(j+1), d -> {j+1} | d)) else flatten apply(3, j -> apply(degrees source F.dd_(j+1), d -> {0} | d)); skewList := toList((0..(m-1)) | ((m+l)..(m+l+n-1))); e := baseName (sym#0); f := baseName (sym#1); g := baseName (sym#2); -- use this line if you want to ensure that 'basis' works properly on the returned ring. --P := first flattenRing (Q[e_1..e_m,f_1..f_l,g_1..g_n,SkewCommutative=>skewList, Degrees => degreesP, Join => false]); P := Q[e_1..e_m,f_1..f_l,g_1..g_n,SkewCommutative=>skewList, Degrees => degreesP, Join => false]; eVector := matrix {apply(m, i -> P_(i))}; fVector := matrix {apply(l, i -> P_(m+i))}; gVector := matrix {apply(n, i -> P_(m+l+i))}; eeGens := flatten entries (matrix {flatten entries (-((transpose eVector) * eVector))} - fVector*(mult#0)); efGens := flatten entries (matrix {flatten entries (((transpose eVector) * fVector))} - gVector*(mult#1)); I := (ideal eeGens) + (ideal efGens) + (ideal apply(m..(m+l-1), i -> P_i))^2 + (ideal apply(0..(m-1), i -> P_i))*(ideal apply((m+l)..(m+l+n-1), i -> P_i)) + (ideal apply(m..(m+l-1), i -> P_i))*(ideal apply((m+l)..(m+l+n-1), i -> P_i)) + (ideal apply((m+l)..(m+l+n-1), i -> P_i))^2; I = ideal mingens I; A := P/I; A.cache#"l" = l; A.cache#"m" = m; A.cache#"n" = n; F.cache#"Algebra Structure" = A; A ) resLengthThreeAlg( Complex ) := (F) -> ( resLengthThreeAlg(F, {getSymbol "e", getSymbol "f", getSymbol "g" }) ) resLengthThreeTorAlg = method() resLengthThreeTorAlg(Complex,List) := (F,sym) -> ( if F.cache#?"Tor Algebra Structure" then return F.cache#"Tor Algebra Structure"; A := resLengthThreeAlg(F,sym); P := ambient A; Q := ring F; kk := coefficientRing first flattenRing Q; PP := kk monoid P; I := ideal mingens sub(ideal A, PP); B := PP/I; B.cache#"l" = A.cache#"l"; B.cache#"m" = A.cache#"m"; B.cache#"n" = A.cache#"n"; F.cache#"Tor Algebra Structure" = B; B ) resLengthThreeTorAlg( Complex ) := (F) -> ( resLengthThreeTorAlg(F, {getSymbol "e", getSymbol "f", getSymbol "g" }) ) makeRes = (d1,d2,d3) -> ( -- first check that the differentials are ok if ( (image matrix entries gens ker d1) != image matrix entries gens image d2 or (image matrix entries gens ker d2) != image matrix entries gens image d3 or ker d3 !=0) then error "Expected differentials of resolution of length three"; f2 := map(source d1,, matrix d2); f3 := map(source f2,, matrix d3); complex {d1, f2, f3} ) multTableOneOne = method(Options => {Labels => true, Compact => false}) multTableOneOne(Ring) := opts -> A -> ( if not (A.cache#?"l" and A.cache#?"m" and A.cache#?"n") then error "Expected an algebra created with a resLengthThree routine."; l := A.cache#"l"; m := A.cache#"m"; n := A.cache#"n"; eVector := matrix {apply(m, i -> A_i)}; oneTimesOneA := table(m,m, (i,j) -> if i <= j then (A_i)*(A_j) else if opts.Compact then "." else (A_i)*(A_j)); topLine := {{" "} | flatten entries eVector}; sideLine := entries transpose eVector; result := (topLine | apply(sideLine,oneTimesOneA, (i,j) -> i | j)); if (opts.Labels) then result else oneTimesOneA ) multTableOneTwo = method(Options => { Labels => true} ) multTableOneTwo(Ring) := opts -> A -> ( if not (A.cache#?"l" and A.cache#?"m" and A.cache#?"n") then error "Expected an algebra created with a resLengthThree routine."; l := A.cache#"l"; m := A.cache#"m"; n := A.cache#"n"; eVector := matrix {apply(m, i -> A_i)}; fVector := matrix {apply(l, i -> A_(m+i))}; oneTimesTwoA := matrix table(m,l,(i,j) -> (A_i)*(A_(m+j))); -- put on the row and column labels for fun result := matrix entries ((matrix {{0}} | fVector) || ((transpose eVector) | oneTimesTwoA)); if (opts.Labels) then entries result else entries oneTimesTwoA ) resLengthThreeTorAlgClass = method() resLengthThreeTorAlgClass Complex := F -> ( A := resLengthThreeTorAlg(F); p := rank multMap(A,1,1); q := rank multMap(A,1,2); r := rank homothetyMap(A,2,1); tau := first tauMaps(A,1,1,1); if (p >= 4 or p == 2) then return ("H(" | p | "," | q | ")") else if (p == 3) then ( if (q > 1) then return ("H(" | p | "," | q | ")") else if (q == 1 and r != 1) then return "C(3)" else if (q == 0 and tau == 0) then return ("H(" | p | "," | q | ")") else return "T"; ) else if (p == 1) then ( if (q != r) then return "B" else return ("H(" | p | "," | q | ")"); ) else if (p == 0) then ( if (q != r) then return ("G(" | r | ")") else return ("H(" | p | "," | q | ")"); ); ) resLengthThreeTorAlgClass Ideal := I -> ( resLengthThreeTorAlgClass freeResolution I ) --====================================================================== -- INTERNAL FUNCTIONS --====================================================================== multTables = F -> ( Q := ring F; d1:= matrix entries F.dd_1; d2:= matrix entries F.dd_2; d3:= matrix entries F.dd_3; m := numcols d1; l := numcols d2; n := numcols d3; multEE := (matrix entries (d1**(id_(source d1)) - (id_(source d1))**d1)) // d2; multEF := (matrix entries (d1**(id_(source d2)) - multEE * (id_(source d1)**d2))) // d3; {multEE,multEF} ) multMap = method() multMap(Ring, ZZ, ZZ) := (A,m,n) -> ( Abasism := basis(m,A); Abasisn := basis(n,A); AbasismPlusn := basis(m+n,A); AmTimesAn := matrix {flatten entries ((transpose Abasism) * Abasisn)}; sub(last coefficients(AmTimesAn, Monomials=>AbasismPlusn), coefficientRing A) ) multMap(RingElement,ZZ) := (f,m) -> ( -- returns the matrix of left multiplication by f A := ring f; n := first degree f; Abasism := basis(m,A); AbasismPlusn := basis(m+n,A); fTimesAbasism := f*Abasism; sub(last coefficients(fTimesAbasism, Monomials=>AbasismPlusn), coefficientRing A) ) homothetyMap = method() homothetyMap(Ring,ZZ,ZZ) := (A,m,n) -> ( Abasism := basis(m,A); homothetyList := apply(flatten entries Abasism, f -> transpose matrix {flatten entries multMap(f,n)}); matrix {homothetyList} ) tauMaps = method() tauMaps(Ring,ZZ,ZZ,ZZ) := (A,l,m,n) -> ( kk := coefficientRing A; multMaplm := multMap(A,l,m); multMapmn := multMap(A,m,n); Al := kk^(numcols basis(l,A)); An := kk^(numcols basis(n,A)); lTensmn := (id_Al) ** multMapmn; lmTensn := multMaplm ** (id_An); psi := matrix {{lTensmn},{lmTensn}}; {rank lTensmn + rank lmTensn - rank psi,lTensmn, lmTensn, psi} ) --=================================================================================================== -- TESTS --=================================================================================================== TEST /// Q = QQ[x,y,z]; F = freeResolution ideal (x*y, y*z, x^3, y^3-x*z^2,x^2*z,z^3); G = resLengthThreeAlg F; assert ( e_1*e_2 == y*f_1 ) assert ( e_1*f_4 == -x*g_1 ) /// TEST /// Q = QQ[x,y,z]; F = freeResolution ideal (x*y, y*z, x^3, y^3-x*z^2,x^2*z,z^3); G = resLengthThreeAlg F assert ( e_1*e_2 == y*f_1 ) assert ( e_1*f_4 == -x*g_1 ) /// TEST /// Q = QQ[x,y,z] I = ideal(x^2,x*y,z^2,y*z,z^2) assert( resLengthThreeTorAlgClass I === "B" ) /// TEST /// Q = QQ[u,v,w,x,y,z] I = ideal (u*v,w*x,y*z) assert( resLengthThreeTorAlgClass I === "C(3)" ) /// TEST /// Q = QQ[x,y,z] I = ideal(x^3,x^2*z,x*(z^2+x*y),z^3-2*x*y*z,y*(z^2+x*y),y^2*z,y^3) assert( resLengthThreeTorAlgClass I === "G(7)" ) /// TEST /// Q = QQ[u,v,w,x,y,z] I = ideal(x*y^2,x*y*z,y*z^2,x^4-y^3*z,x*z^3-y^4) assert( resLengthThreeTorAlgClass I === "G(2)" ) /// TEST /// Q = QQ[x,y,z] I = ideal(x^2,x*y^2)*ideal(y*z,x*z,z^2) assert( resLengthThreeTorAlgClass freeResolution I === "H(0,0)" ) /// TEST /// Q = QQ[x,y,z] I = ideal(x^2,x*y^2)*ideal(y*z,x*z,z^2) assert( resLengthThreeTorAlgClass I === "H(0,0)" ) /// TEST /// Q = QQ[x,y,z] I = ideal(x^3,y^3,z^3,x*y*z) assert( resLengthThreeTorAlgClass I === "T" ) /// TEST /// Q = QQ[x,y,z] I = ideal(x^5,y^5,x*y^4,x^2*y^3,x^3*y^2,x^4*y,z^3) assert( resLengthThreeTorAlgClass I === "H(6,5)" ) /// TEST /// Q = QQ[x,y,z] d1 = matrix{{-x^2,z^2-x*y,-y^2,-x*z,-y*z}} d2 = matrix{{0,0,z,0,-y},{0,0,0,-y,x},{-z,0,0,x,0},{0,y,-x,0,z},{y,-x,0,-z,0}} d3 = transpose d1 F = makeRes(d1,d2,d3) A = resLengthThreeAlg F assert(e_2*e_4===y*f_3+z*f_5) assert(e_1*e_2===-z*f_3-x*f_5) assert(e_3*e_5===-y*f_1) T = resLengthThreeTorAlg F assert(e_2*e_4===sub(0,T)) assert(e_1*e_2===sub(0,T)) assert(e_3*e_5===sub(0,T)) /// TEST /// Q = QQ[x,y,z] I = ideal(x*y, y*z, x^3, y^3-x*z^2,x^2*z, z^3) G = resLengthThreeAlg freeResolution I assert( e_1*e_2 == y*f_1 ) assert( e_1*e_3 == x*f_2 ) assert( e_1*e_4 == -x*z*f_1 + y*f_3 - z*f_5 ) assert( e_1*e_5 == x^2*f_1 + x*f_5 ) assert( e_1*e_6 == z^2*f_1 + x*f_7 ) assert( e_1*f_1 == 0 ) assert( e_1*f_2 == 0 ) assert( e_1*f_3 == 0 ) assert( e_1*f_4 == -x*g_1 ) assert( e_1*f_5 == 0 ) assert( e_1*f_6 == g_2 ) assert( e_1*f_7 == 0 ) /// --========================================================================== -- DOCUMENTATION --========================================================================== beginDocumentation() document{ Key => {ResLengthThree}, Headline => "Computation of multiplicative structures on free resolutions of length three", PARA { "Let ", EM "I ", " be a homogeneous ideal contained in the irrelevant maximal ideal of a graded ring ", EM "Q ", " (obtained as a quotient of a polynomial ring). If the length of the minimal free resolution ", EM "F ", " of ", TEX /// $R=Q/I$ ///, " is 3, then the resolution admits the structure of a differential graded algebra. The induced algebra structure on ", TEX /// $A = Tor^Q(R,k)$ ///, " is unique and provides for a classification of such quotient rings. The package determines a multiplicative structure on the free resolution ", EM "F ", " as well as the unique induced structure on ", EM "A ", "and the class of the quotient ", EM "R ", "according to the classification scheme of ", HREF{"https://doi.org/10.1016/0021-8693(88)90056-7","Avramov, Kustin, and Miller"},"." } } document{ Key => { resLengthThreeAlg, (resLengthThreeAlg, Complex ), (resLengthThreeAlg, Complex, List ) }, Headline => "the minimal free resolution presented as a graded-commutative ring", Usage => "resLengthThreeAlg F, resLengthThreeAlg(F,L)", Inputs =>{ "F" => Complex => "a length three free resolution of a cyclic module", "L" => List => "of three symbols" }, Outputs => { QuotientRing => "the resolution presented as a quotient of a graded-commutative free algebra over the ambient ring"}, PARA { "For a free resolution ", TT "F", " over a ring ", TT "Q", ", the function returns the resolution ", TT "F", " as a quotient of a graded-commutative free algebra over ", TT "Q", ". The basis vectors in degrees 1, 2, and 3 are named with the symbols from the list ", TT "L", ". The default symbols are ", TT "e", ", ", TT "f", ", and ", TT "g", "." }, EXAMPLE { "Q = QQ[x,y,z];", "A = resLengthThreeAlg freeResolution ideal (x^2,y^2,z^2)", "describe A", "e_1*e_2", "e_1*f_2", "e_1*f_3", "f_1*f_2", }, PARA { "The ambient ring ", TT "Q", " does not need to be a polynomial algebra." }, EXAMPLE { "P = QQ[u,v,x,y,z];", "Q = P/ideal(u^2,u*v);", "F = resLengthThreeAlg(freeResolution(ideal (x^2,x*y,y^2,z^2), LengthLimit => 3), {a,b,c} )", "describe F", }, PARA { }, EXAMPLE { "P = QQ[u,v];", "Q = (P/ideal(u^2,u*v))[x,y,z];", "A = resLengthThreeAlg freeResolution(ideal (x^2,x*y,y^2,z^2), LengthLimit => 3)", "describe A", }, PARA { }, EXAMPLE { "P = ZZ[x,y,z];", "Q = P/ideal(4_P);", "A = resLengthThreeAlg freeResolution(ideal (x^2,y^2,z^2), LengthLimit => 4)", "describe A" }, Caveat => { "The ambient ring ", TT "Q ", "must be homogeneous." } } document{ Key => { resLengthThreeTorAlg, (resLengthThreeTorAlg, Complex ), (resLengthThreeTorAlg, Complex, List ) }, Headline => "the Tor algebra presented as a graded-commutative ring", Usage => "resLengthThreeTorAlg F, resLengthThreeTorAlg(F,L)", Inputs =>{ "F" => Complex => "a length three free resolution of a cyclic module", "L" => List => "of three symbols" }, Outputs => { QuotientRing => "the Tor algebra presented as a quotient of a graded-commutative free algebra over the residue field of the ambient ring"}, PARA { "For a free resolution ", TT "F", " over a ring ", TT "Q", ", the function returns the algebra ", TEX /// $Tor^Q(R,k)$ ///, " as a quotient of a graded-commutative free algebra over the residue field of ", TT "Q", ". The basis vectors in degrees 1, 2, and 3 are named with the symbols from the list ", TT "L", ". The default symbols are ", TT "e", ", ", TT "f", ", and ", TT "g", "." }, EXAMPLE { "Q = QQ[x,y,z];", "A = resLengthThreeTorAlg freeResolution ideal (x^2,y^2,z^2)", "describe A", "e_1*e_2", "e_1*f_2", "e_1*f_3", "f_1*f_2", }, PARA { "The ambient ring ", TT "Q", " does not need to be a polynomial algebra." }, EXAMPLE { "P = QQ[u,v,x,y,z];", "Q = P/ideal(u^2,u*v);", "A = resLengthThreeTorAlg (freeResolution(ideal (x^2,x*y,y^2,z^2), LengthLimit => 4), {a,b,c} )", "describe A", }, PARA { }, EXAMPLE { "P = QQ[u,v];", "Q = (P/ideal(u^2,u*v))[x,y,z];", "A = resLengthThreeTorAlg ( freeResolution(ideal (x^2,x*y,y^2,z^2), LengthLimit => 4), {a,b,c} )", "describe A", }, Caveat => { "For the function to return an algebra over the residue field of the ambient ring ", TT "Q ", "that ring must a homogeneous quotient of a polynomial algebra over a field."}, } document{ Key => { multTableOneOne, (multTableOneOne, Ring) }, Headline => "the multiplication table for products of elements in degree one", Usage => "multTableOneOne A", Inputs =>{ "A" => Ring => { "created with ", TO resLengthThreeAlg, " or ", TO resLengthThreeTorAlg } }, Outputs => { List => { "of the rows in the multiplication table; use ", TO netList, " to display it as a table" } }, PARA { "For a free resolution of length three described as a graded-commutative ring ", TT "A", ", the function returns a list of the rows of the multiplication table of elements in degree one. It also computes the multiplication table for products of elements in degree one in the graded-commutative homology algebra obtained from ", TT "A", "." }, EXAMPLE { "Q = QQ[x,y,z];", "A = resLengthThreeAlg freeResolution ideal (x^2,y^2,z^2)", "multTableOneOne A", "netList multTableOneOne A" }, } document{ Key => { [multTableOneOne, Labels], }, Headline => "an optional argument for multTableOneOne determining whether to label rows and columns", Usage => "multTableOneOne A", Inputs =>{ "A" => Ring => { "created with ", TO resLengthThreeAlg, " or ", TO resLengthThreeTorAlg } }, Outputs => { List => { "of the rows in the multiplication table; use ", TO netList, " to display it as a table" } }, PARA { "The default value of ", TO Labels, " is ", TO true, ". Changing the value to ", TO false, " removes the row and column labels." }, EXAMPLE { "Q = QQ[x,y,z];", "A = resLengthThreeAlg freeResolution ideal (x^2,y^2,z^2)", "netList multTableOneOne (A, Labels => false)", }, } document{ Key => { [multTableOneOne, Compact], Compact }, Headline => "an optional argument for multTableOneOne that prints dots below the diagonal" , Usage => "multTableOneOne A", Inputs =>{ "A" => Ring => { "created with ", TO resLengthThreeAlg, " or ", TO resLengthThreeTorAlg } }, Outputs => { List => { "of the rows in the multiplication table; use ", TO netList, " to display it as a table" } }, PARA { "The default value of ", TO Compact, " is ", TO false, ". Changing the value to ", TO true, " saves space by printing dots for the products ", TEX /// $e_ie_j$ ///, " for ", TEX /// $i>j$ /// }, EXAMPLE { "Q = QQ[x,y,z];", "A = resLengthThreeAlg freeResolution ideal (x^2,y^2,z^2)", "multTableOneOne (A, Compact => true)", "netList multTableOneOne(A, Compact => true)", }, } document{ Key => { multTableOneTwo, (multTableOneTwo, Ring) }, Headline => "the multiplication table for products of elements in degree one with elements in degree two", Usage => "multTableOneTwo A", Inputs =>{ "A" => Ring => { "created with ", TO resLengthThreeAlg, " or ", TO resLengthThreeTorAlg } }, Outputs => { List => { "of the rows in the multiplication table; use ", TO netList, " to display it as a table" } }, PARA { "For a free resolution of length three described as a graded-commutative ring ", TT "A", ", the function returns a list of the rows of the multiplication table of elements in degree one with elements in degree two. It also computes the multiplication table for products of elements in degree one with elements in degree two in the graded-commutative homology algebra obtained from ", TT "A", "." }, EXAMPLE { "Q = QQ[x,y,z];", "A = resLengthThreeAlg freeResolution ideal (x^2,y^2,z^2)", "multTableOneTwo A", "netList multTableOneTwo A" }, } document{ Key => { Labels }, Headline => "an optional argument for multTableOneOne and MultTableOneTwo determining whether to label rows and columns", Usage => "multTableOneOne A", "multTableOneTwo A", Inputs =>{ "A" => Ring => { "created with ", TO resLengthThreeAlg, " or ", TO resLengthThreeTorAlg } }, Outputs => { List => { "of the rows in the multiplication table, use ", TO netList, " to display it as a table" } }, PARA { "The default value of ", TO Labels, " is ", TO true, ". Changing the value to ", TO false, " removes the row and column labels." }, EXAMPLE { "Q = QQ[x,y,z];", "A = resLengthThreeAlg freeResolution ideal (x^2,y^2,z^2)", "netList multTableOneOne (A, Labels => false)", "netList multTableOneTwo (A, Labels => false)", }, } document{ Key => { [multTableOneTwo, Labels], }, Headline => "an optional argument for multTableOneTwo determining whether to label rows and columns", Usage => "multTableOneTwo A", Inputs =>{ "A" => Ring => { "created with ", TO resLengthThreeAlg, " or ", TO resLengthThreeTorAlg } }, Outputs => { List => { "of the rows in the multiplication table. Use ", TO netList, " to display it as a table" } }, PARA { "The default value of ", TO Labels, " is ", TO true, ". Changing the value to ", TO false, " removes the row and column labels." }, EXAMPLE { "Q = QQ[x,y,z];", "A = resLengthThreeAlg freeResolution ideal (x^2,y^2,z^2)", "netList multTableOneTwo (A, Labels => false)", }, } document{ Key => { resLengthThreeTorAlgClass, (resLengthThreeTorAlgClass, Complex),(resLengthThreeTorAlgClass, Ideal), }, Headline => "the class (w.r.t. multiplication in homology) of an ideal", Usage => "resLengthThreeTorAlgClass F", Inputs =>{ "F" => Complex => { "a length three free resolution of a cyclic module" } , "I" => Ideal => {"an ideal of codepth 3"}, }, Outputs => { String => { "the (parametrized) class of the ideal I"} }, PARA { "Classifies the ideal ", TEX /// $I$ ///, " as belonging to one of the (parametrized) classes ", BOLD /// B ///,", ",BOLD /// C///,"(c), ",BOLD /// G///,"(r), ",BOLD ///H///,"(p,q) , ", BOLD ///T///,", provided that it is codepth 3." }, EXAMPLE { "Q = QQ[x,y,z];", "resLengthThreeTorAlgClass ideal (x*y,x^2,y*z,z^2)", "resLengthThreeTorAlgClass ideal (x^2,y^2,z^2)", "resLengthThreeTorAlgClass ideal (x*y,y*z,x^3,x^2*z,x*z^2-y^3,z^3)", "resLengthThreeTorAlgClass ideal (x*z+y*z,x*y+y*z,x^2-y*z,y*z^2+z^3,y^3-z^3)", "resLengthThreeTorAlgClass ideal (x^2,y^2,z^2,x*z)", "resLengthThreeTorAlgClass ideal (x^2,y^2,z^2,x*y*z)", }, Caveat => { "The codepth of the ideal ", TEX///I///," must be exactly 3, and the length of the complex ",TEX///F///, " must be exactly 3."}, } document{ Key => { makeRes, }, Headline => "creates a resolution starting from three matrices", Usage => "makeRes(d1,d2,d3)", Inputs =>{ "d1" => Matrix => {"of the differential in degree 1"} , "d2" => Matrix => {"of the differential in degree 2"}, "d3" => Matrix => {"of the differential in degree 3"}, }, Outputs => { Complex => { "the resolution with differentials d1, d2, d3."} }, PARA { "Creates a resolution of length 3 that has the given three matrices as differentials." }, EXAMPLE { "Q = QQ[x,y,z];", "d1=matrix{{-x^2,z^2-x*y,-y^2,-x*z,-y*z}}", "d2=matrix{{0,0,z,0,-y},{0,0,0,-y,x},{-z,0,0,x,0},{0,y,-x,0,z},{y,-x,0,-z,0}}", "d3=transpose d1", "makeRes(d1,d2,d3)", }, } end --========================================================================== -- end of package code --===============================d=========================================== uninstallPackage "ResLengthThree" restart installPackage "ResLengthThree" debug loadPackage "ResLengthThree" check "ResLengthThree" viewHelp "ResLengthThree"