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Minor changes to KustinMiller to get it working with Complexes
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@mikestillman
mikestillman committed Oct 7, 2024
1 parent cc90018 commit 4f17a11
Showing 1 changed file with 39 additions and 34 deletions.
73 changes: 39 additions & 34 deletions M2/Macaulay2/packages/KustinMiller.m2
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Original file line number Diff line number Diff line change
Expand Up @@ -13,7 +13,7 @@ newPackage(
},
Headline => "unprojection and the Kustin-Miller complex construction",
Keywords => {"Commutative Algebra"},
PackageExports => {"SimplicialComplexes"},
PackageExports => {"SimplicialComplexes", "Complexes"},
Certification => {
"journal name" => "The Journal of Software for Algebra and Geometry: Macaulay2",
"journal URI" => "https://msp.org/jsag/",
Expand Down Expand Up @@ -149,7 +149,10 @@ resBE Matrix := A -> (
p:=gens pfaffians(-1+rank source A,A);
n:=rank source p;
g:=matrix {apply(n,j-> (-1)^(j)*p_(n-j-1)_0)};
chainComplex {g,A,transpose g,map(R^1,R^0,0)})
g2 := map(source g,, A);
complex {g, g2, map(source g2,, transpose g)}
)
--complex {g,A,transpose g,map(R^1,R^0,0)}) -- previously a zero was added on.



Expand Down Expand Up @@ -311,7 +314,7 @@ kustinMillerComplex(Ideal,Ideal,PolynomialRing):=opt->(I,J,T0)->(
kustinMillerComplex(res I,res J,T0,opt))


kustinMillerComplex(ChainComplex,ChainComplex,PolynomialRing):=opt->(cI0,cJ0,T0)->(
kustinMillerComplex(Complex,Complex,PolynomialRing):=opt->(cI0,cJ0,T0)->(
if opt.Verbose>1 then (
<<"------------------------------------------------------------------------------------------------------------------------"<<endl;
<<"res(I):"<<endl;
Expand Down Expand Up @@ -348,7 +351,7 @@ kustinMillerComplex(ChainComplex,ChainComplex,PolynomialRing):=opt->(cI0,cJ0,T0)
);
gJs:=sub(gens source phi,S);
IJmap:=sub(matrix entries phi,S)*((gens ideal (dualcJ.dd_0))//gJs);
alphaDual:=extend(dualcI,dualcJ, map(dualcI#0,dualcJ#0,IJmap));
alphaDual:=extend(dualcI,dualcJ, map(dualcI_0,dualcJ_0,IJmap));
w:=(alphaDual_(length cI))_0_0;
alpha:=toList apply(g-1,j->sub(w^(-1),S)*((transpose alphaDual_(g-2-j))**S^{-degshift}));
if opt.Verbose>1 then (
Expand All @@ -359,7 +362,7 @@ kustinMillerComplex(ChainComplex,ChainComplex,PolynomialRing):=opt->(cI0,cJ0,T0)
<<endl<<"------------------------------------------------------------------------------------------------------------------------"<<endl;
);
cJ1:=cJ[1];
betaMap:=map((cI#0,(cJ1#0)**S^{-degshift},-sub(matrix entries phi,S)*((gens ideal (cJ1.dd_0))//gJs)));
betaMap:=map((cI_0,(cJ1_0)**S^{-degshift},-sub(matrix entries phi,S)*((gens ideal (cJ1.dd_0))//gJs)));
beta1:=extend(cI,cJ1**S^{-degshift},betaMap);
beta:=toList apply(g, j-> (-1)^(j + codim I)*beta1_j);
if opt.Verbose>1 then (
Expand All @@ -376,10 +379,10 @@ kustinMillerComplex(ChainComplex,ChainComplex,PolynomialRing):=opt->(cI0,cJ0,T0)
-- construct the homotopy h
h:={0_S};
for j from 1 to g-1 do (
tC1:= chainComplex { id_(S^(rank (cI#j)))};
tC2:= chainComplex { cI.dd_j };
hi:= (extend ( tC2, tC1, map (tC2#0, tC1#0, beta#(j-1)*alpha#(j-1) - h#(j-1)*cI.dd_j )));
h=append(h,map(cI#j,cI#j**S^{-degshift},hi_1));
tC1:= complex { id_(S^(rank (cI_j)))};
tC2:= complex { cI.dd_j };
hi:= (extend ( tC2, tC1, map (tC2_0, tC1_0, beta#(j-1)*alpha#(j-1) - h#(j-1)*cI.dd_j )));
h=append(h,map(cI_j,cI_j**S^{-degshift},hi_1));
);
if opt.Verbose>1 then (
for j from 1 to #h-1 do (
Expand Down Expand Up @@ -422,7 +425,7 @@ kustinMillerComplex(ChainComplex,ChainComplex,PolynomialRing):=opt->(cI0,cJ0,T0)
<<endl<<"------------------------------------------------------------------------------------------------------------------------"<<endl;
);
);
chainComplex L)
complex L)

--kustinMillerComplex(I,J,QQ[t])

Expand Down Expand Up @@ -587,12 +590,12 @@ assert(not isGorenstein ideal (x_1*x_2, x_1*x_3))

-- we first find the first non-zero module
firstNonzero=method()
firstNonzero(ChainComplex):= cc -> (
firstNonzero(Complex):= cc -> (
for i from min cc to max cc do if cc_i!=0 then return i;
infinity)

isExactRes=method()
isExactRes(ChainComplex):= cc ->(
isExactRes(Complex):= cc ->(
for j from firstNonzero(cc)+1 to max(cc)+1 do (
if cc.dd_(j)*cc.dd_(j+1) !=0 then return false;
if (HH_j cc) !=0 then return false;
Expand All @@ -602,13 +605,15 @@ isExactRes(ChainComplex):= cc ->(
-- with this method also the substituted complexes
-- recognize, when printed, if a name is assigned to
-- the ring of the complex
substitute(ChainComplex,Ring):=(cc,S)->(
dual cc;
cn:= new ChainComplex;
cn.ring = S;
for i from min(cc) to max(cc) do cn#i = S^(degrees (cc#i));
for i from min(cc)+1 to max(cc) do cn.dd_i = sub(cc.dd_i,S);
cn)
substitute(Complex, Ring) := (C, S) -> (
(lo,hi) := concentration C;
moduleHash := hashTable for i from lo to hi list i => S^(degrees C_i);
if lo === hi then
return complex(moduleHash#lo, Base=>lo);
mapHash := hashTable for i from lo+1 to hi list
i => map(moduleHash#(i-1), moduleHash#i, sub(dd^C_i, S));
complex mapHash
)

--------------------------------------------------------------------------
-- Stellar subdivision code
Expand Down Expand Up @@ -779,7 +784,7 @@ doc ///
Key
kustinMillerComplex
(kustinMillerComplex,Ideal,Ideal,PolynomialRing)
(kustinMillerComplex,ChainComplex,ChainComplex,PolynomialRing)
(kustinMillerComplex,Complex,Complex,PolynomialRing)
Headline
Compute Kustin-Miller resolution of the unprojection of I in J
Usage
Expand All @@ -790,15 +795,15 @@ doc ///
in a positively graded polynomial ring R
I:Ideal
contained in J
cI:ChainComplex
cI:Complex
resolution of I
cJ:ChainComplex
cJ:Complex
resolution of J
W:PolynomialRing
over the same @TO coefficientRing@ as R
with one variable T.
Outputs
:ChainComplex
:Complex
Description
Text
Compute Kustin-Miller resolution of the unprojection of I in J (or
Expand Down Expand Up @@ -1004,7 +1009,7 @@ doc ///
A:Matrix
skew-symmetric
Outputs
:ChainComplex
:Complex
Description
Text
Returns the Buchsbaum-Eisenbud resolution of the ideal of submaximal @TO pfaffians@
Expand All @@ -1023,13 +1028,13 @@ doc ///
doc ///
Key
isExactRes
(isExactRes,ChainComplex)
(isExactRes,Complex)
Headline
Test whether a chain complex is an exact resolution.
Usage
isExactRes(cc)
Inputs
cc:ChainComplex
cc:Complex
Outputs
:Boolean
Description
Expand All @@ -1054,16 +1059,16 @@ doc ///

doc ///
Key
(substitute,ChainComplex,Ring)
(substitute,Complex,Ring)
Headline
Substitute a chain complex to a new ring.
Usage
substitute(cc,R)
Inputs
cc:ChainComplex
cc:Complex
R:Ring
Outputs
:ChainComplex
:Complex
Description
Text
Substitute a chain complex cc to a new ring R.
Expand Down Expand Up @@ -1168,7 +1173,7 @@ assert(pfaffians(4,A)==ideal cc.dd_1);
I = ideal(z_2*z_3-z_1*z_4,x_4*z_3-x_3*z_4,x_2*z_2-x_1*z_4,x_4*z_1-x_3*z_2,x_2*z_1-x_1*z_3);
cc= res I;
assert(isExactRes cc);
C=chainComplex presentation QQ^1;
C=complex presentation QQ^1;
assert(isExactRes C);
assert(isExactRes(C[1]));
assert(isExactRes(C[-1]));
Expand Down Expand Up @@ -1202,8 +1207,8 @@ TEST ///
I=sub(ideal C47,R)
c47=res I;
cc=kustinMillerComplex(c47,c26,K[x_8]);
assert(rank(cc#1)==16);
assert(rank(cc#2)==30);
assert(rank(cc_1)==16);
assert(rank(cc_2)==30);
assert(isExactRes(cc));
///

Expand Down Expand Up @@ -1235,8 +1240,8 @@ TEST ///
cJ=res J
betti cJ
cc=kustinMillerComplex(cI,cJ,QQ[T]);
assert(rank(cc#1)==9);
assert(rank(cc#2)==16);
assert(rank(cc_1)==9);
assert(rank(cc_2)==16);
assert(isExactRes cc);
///

Expand Down

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