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-- Copyright 1999-2002 by Anton Leykin and Harrison Tsai
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
-- Computes deRham cohomology of a smooth affine variety. For more options on localCohom strategy use ICcohom instead.
-- Special algorithm for the de Rham cohomology of K^n - V(f) using the algorithm of Oaku-Takayama.
--------------------------------------------------------------------------------
--------------------------------------------------------------------------------
deRham = method( Options => {Strategy => Schreyer})
deRham Ideal := options -> I -> (
R:= ring I;
if class R =!= PolynomialRing
then error "expected an ideal in a polynomial ring";
primes := minimalPrimes I;
if #primes > 1 then error "The variety defined by the ideal must be irreducible";
P := primes#0;
S := ideal (singularLocus P);
if R != S then error "expected an ideal defining a smooth affine variety";
n:= numgens R;
d:= dim I;
c:= n-d;
H:=localCohom(c,I);
M:=minimalPresentation H;
integrateTable := Dintegration(M, apply(n,i->1), options);
new HashTable from (for i from 0 to d list i=>integrateTable#(d-i))
)
deRham (ZZ, Ideal) := options -> (k, I) -> (
R:= ring I;
if class R =!= PolynomialRing
then error "expected an ideal in a polynomial ring";
primes := minimalPrimes I;
if #primes > 1 then error "The variety defined by the ideal must be irreducible";
P := primes#0;
S := ideal (singularLocus P);
if R != S then error "expected an ideal defining a smooth affine variety";
n:= numgens R;
d:= dim I;
if k>d then 0
else (
c:= n-d;
H:=localCohom(c,I);
M:=minimalPresentation H;
Dintegration(d-k, M, apply(n,i->1),options)
)
)
deRham RingElement := options -> f -> (
pInfo(1, "ENTERING deRham ...");
R := ring f;
if class R =!= PolynomialRing
then error "expected element of a polynomial ring";
if R.monoid.Options.WeylAlgebra =!= {}
then error "expected element of a commutative polynomial ring";
outputList := {};
n := numgens R;
x := symbol x;
D := symbol D;
WA := (coefficientRing R)[x_1 .. x_n, D_1 .. D_n,
WeylAlgebra => apply(toList(1..n), i -> x_i => D_i)];
RtoWA := map(WA, R, (vars WA)_(toList(0..n-1)));
M := cokernel matrix{{D_1 .. D_n}};
locMod := Dlocalize(M, RtoWA f, Strategy => Oaku);
w := toList(n:1);
integrateTable := Dintegration(locMod, w, options);
homologyTable := hashTable apply(toList(0..n),
i -> (-i+n) => integrateTable#i);
homologyTable
)
deRham (ZZ, RingElement) := options -> (k, f) -> (
pInfo(1, "ENTERING deRham ...");
R := ring f;
if class R =!= PolynomialRing
then error "expected element of a polynomial ring";
if R.monoid.Options.WeylAlgebra =!= {}
then error "expected element of a commutative polynomial ring";
outputList := {};
n := numgens R;
x := symbol x;
D := symbol D;
WA := (coefficientRing R)[x_1 .. x_n, D_1 .. D_n,
WeylAlgebra => apply(toList(1..n), i -> x_i => D_i)];
RtoWA := map(WA, R, (vars WA)_(toList(0..n-1)));
M := cokernel matrix{{D_1 .. D_n}};
locMod := Dlocalize(M, RtoWA f, Strategy => Oaku);
w := toList(n:1);
homologyModule := Dintegration(n-k, locMod, w, options);
homologyModule
)
deRhamAll = method( Options => {Strategy => Schreyer})
deRhamAll RingElement := options -> f -> (
pInfo(1, "ENTERING deRhamAll ...");
R := ring f;
if class R =!= PolynomialRing
then error "expected element of a polynomial ring";
if R.monoid.Options.WeylAlgebra =!= {}
then error "expected element of a commutative polynomial ring";
n := numgens R;
x := symbol x;
D := symbol D;
WA := (coefficientRing R)[x_1 .. x_n, D_1 .. D_n,
WeylAlgebra => apply(toList(1..n), i -> x_i => D_i)];
RtoWA := map(WA, R, (vars WA)_(toList(0..n-1)));
M := cokernel matrix{{D_1 .. D_n}};
WAf := RtoWA f;
LocMap := DlocalizeMap(M, WAf, Strategy => Oaku);
Mf := target LocMap;
w := toList(n:1);
outputRequest := {GenCycles, HomologyModules,
VResolution, BFunction, Explicit};
MF := cokernel Fourier relations Mf;
outputTable := computeRestriction (MF, w, -1, n+1,
outputRequest, options);
outputList := {BFunction => outputTable#BFunction,
LocalizeMap => LocMap};
if outputTable#VResolution =!= null then
outputList = append(outputList,
VResolution => FourierInverse outputTable#VResolution)
else outputList = append(outputList, VResolution => null);
outputList = outputList | {
PreCycles => hashTable apply(toList(0..n),
i -> (n-i => FourierInverse outputTable#GenCycles#i)),
CohomologyGroups => hashTable apply(toList(0..n),
i -> (n-i => outputTable#HomologyModules#i)) };
intTable := hashTable outputList;
--P := DintegrationDeRham(Mf, w, Strategy => options.Strategy,
-- Info => options.Info, ExplicitHomology => options.ExplicitHomology);
if intTable#VResolution =!= null then (
pInfo(1, "Making the Koszul complex... ");
createDpairs WA;
Omega := Dres ideal WA.dpairVars#1;
pInfo(1, "Transferring cohomology classes to deRham complex... ");
dbl := Omega**(intTable#VResolution);
transfers := {};
i := 0;
while i <= n do (
pull := intTable#PreCycles#i;
j := 0;
pInfo(2, "\t Degree " | i | "...");
tInfo := toString first timing (
while j < n-i do (
vertMap := zeroize dbl#(n-i-1)^[(j,-j+n-i-1)]*
(dbl.dd#(n-i))*(dbl#(n-i)_[(j,-j+n-i)]);
push := vertMap*pull;
horMap := zeroize dbl#(n-i-1)^[(j,-j+n-i-1)]*
(dbl.dd#(n-i))*(dbl#(n-i)_[(j+1,-j+n-i-1)]);
if ((Dtransposition push) % (Dtransposition horMap) != 0)
then error "syzygy should be produced but wasn't!";
pull = (Dtransposition push) // (Dtransposition horMap);
pull = Dtransposition pull;
j = j+1;
);
);
pInfo(2, "\t\t\t time = " | tInfo | " seconds" );
transfers = append(transfers, i => pull);
i = i+1;
);
outputList = outputList | {TransferCycles =>
hashTable transfers, OmegaRes => Omega};
);
hashTable outputList
)
-- iAllt is by Nobuki Takayama, 2007.
-- iAllt(I) computes integration cohomology groups of D/I with transfer's
-- logCohomology(I) computes DR(tilde M).
--
-- Example:
-- M2
-- load "D-modules.m2"
-- load "trans.m2"
-- use S; f=x*y*(x-y); logCohomology(f);
protect Input -- not exported?
iAllt = method( Options => {Strategy => Schreyer})
iAllt Module:= options -> I -> (
pInfo(2, "iAllt (integrationAllWithTransfer) in trans.m2");
WA := ring gens I;
Mf := I;
pInfo(2,Mf);
n := numgens(WA) // 2; -- do not use numgens(WA)/2
w := toList(n:1);
outputRequest := {GenCycles, HomologyModules,
VResolution, BFunction, Explicit};
MF := cokernel Fourier relations Mf;
outputTable := computeRestriction (MF, w, -1, n+1,
outputRequest, options);
outputList := {BFunction => outputTable#BFunction,
LocalizeMap => LocMap};
outputList = outputList | { Input=> I };
if outputTable#VResolution =!= null then
outputList = append(outputList,
VResolution => FourierInverse outputTable#VResolution)
else outputList = append(outputList, VResolution => null);
outputList = outputList | {
PreCycles => hashTable apply(toList(0..n),
i -> (n-i => FourierInverse outputTable#GenCycles#i)),
CohomologyGroups => hashTable apply(toList(0..n),
i -> (n-i => outputTable#HomologyModules#i)) };
intTable := hashTable outputList;
--P := DintegrationDeRham(Mf, w, Strategy => options.Strategy,
-- Info => options.Info, ExplicitHomology => options.ExplicitHomology);
if intTable#VResolution =!= null then (
pInfo(1, "Making the Koszul complex... ");
createDpairs WA;
Omega := Dres ideal WA.dpairVars#1;
pInfo(1, "Transferring cohomology classes to deRham complex... ");
dbl := Omega**(intTable#VResolution);
pInfo(2,dbl); -- print double complex
transfers := {};
i := 0;
while i <= n do (
pull := intTable#PreCycles#i;
j := 0;
pInfo(2,{i,j,pull});
pInfo(2, "\t Degree " | i | "...");
tInfo := toString first timing (
while j < n-i do (
vertMap := zeroize dbl#(n-i-1)^[(j,-j+n-i-1)]*
(dbl.dd#(n-i))*(dbl#(n-i)_[(j,-j+n-i)]);
push := vertMap*pull;
horMap := zeroize dbl#(n-i-1)^[(j,-j+n-i-1)]*
(dbl.dd#(n-i))*(dbl#(n-i)_[(j+1,-j+n-i-1)]);
if ((Dtransposition push) % (Dtransposition horMap) != 0)
then error "syzygy should be produced but wasn't!";
pull = (Dtransposition push) // (Dtransposition horMap);
pull = Dtransposition pull;
pInfo(2,{i,j,push,pull});
j = j+1;
);
);
pInfo(2, "\t\t\t time = " | tInfo | " seconds" );
transfers = append(transfers, i => pull);
i = i+1;
);
outputList = outputList | {TransferCycles =>
hashTable transfers, OmegaRes => Omega};
);
hashTable outputList
)
-- derLogF returns generators for widetilde{Der}(-log f)
-- Example: QQ[x,y]
-- f = x*y*(x-y)
-- iAllt cokernel matrix derLogF(f)
derLogF = method();
derLogF RingElement := f -> (
R := ring f;
if class R =!= PolynomialRing
then error "expected element of a polynomial ring";
if R.monoid.Options.WeylAlgebra =!= {}
then error "expected element of a commutative polynomial ring";
v := generators(R);
n := #v;
d := apply(v, i->diff(i,f));
d = join({f},d);
syzf := kernel matrix{d};
m := numgens syzf;
-- msyz = gens syzf;
msyz := gens syzf;
-- note: msyz^{1} get 1-th row
-- note: msyz_{i} get i-th column
-- note: msyz^{i}_{j} get (i,j)-element
-- note: entries
-- note: #List, length
pInfo(2,msyz);
-- see annFs.m2, oxclient.m2
W := makeWeylAlgebra( R, SetVariables=>false); --note: g=map(R,W,matrix{{x,y,0,0}}); g(x) does not work
phi := map(W,R);
vw := generators(W);
i:=0; j:=0; ell:=0; t:=0;
op:={};
while i<m do (
j = 1;
ell=-phi msyz_(0,i);
while j<=n do (
ell = ell + phi msyz_(j,i) * vw_(n+j-1);
j = j+1;
);
op = join(op,{ell});
i = i+1;
);
{op}
)
logCohomology=method();
logCohomology RingElement := f -> (
ii := derLogF f;
iAllt(cokernel(matrix(ii)))
)
-- use S; logCohomology(x^3-y)
-- Warning: do not input like logCohomology x^3-y
-- Example:
-- use S; f=(x^3+y^4+x*y^3)*(x^2+y^2); logCohomology f;
-- the transfer cycles contains dx and dy.
-- use S; f=(x^3+y^4+x*y^3)*(x^2-y^2); logCohomology f;
-- the transfer cycles contains dx and dy.
getTransfer=method();
getTransfer (HashTable,ZZ) := (cohom,k) -> (
t:=cohom#TransferCycles;
tk:=transpose(t)#k;
entries(tk)
)
-- Example:
-- use S; f=x*y*(x-y); cc=logCohomology f; getReducedTransfer(cc,1);
-- use S; f=(x^3+y^4+x*y^3)*(x^2+y^2); cc=logCohomology f; getReducedTransfer(cc,1);
getReducedTransfer=method();
getReducedTransfer (HashTable,ZZ) := (cohom,k) -> (
tmp:=cohom;
tk:=getTransfer(cohom,k);
-- cf. Dbasic.m2, holonomicRank
W := ring(cohom#VResolution);
createDpairs W;
n := #(W.dpairInds#0);
m := numgens W;
-- get weight vectors for the order filtration refined
-- by lex on the derivatives
weightList := { apply ( toList(0..m-1), i -> if member(i, W.dpairInds#1)
then 1 else 0 ) };
-- ring equipped with the new order
tempW := (coefficientRing W)(monoid [(entries vars W)#0,
WeylAlgebra => W.monoid.Options.WeylAlgebra,
Weights => weightList]);
WtotempW := map (tempW, W, vars tempW);
-- map to the lex ring.
tktemp := apply(tk,i->apply(i,j->WtotempW(j)));
iii := WtotempW( ((cohom#VResolution).dd)#1 );
ggg := gb iii;
tkreduced := apply(tktemp,i->apply(i,j-> (j % ggg)))
)
TEST ///
--Boundary cases
x = symbol x; y = symbol y;
R = QQ[x,y]
default := hashTable {0=>QQ^1, 1=>QQ^0, 2=>QQ^0};
F2 = deRham(1_R); -- affine space
assert all (keys default, i -> (F2#i == default#i));
--Small change doesn't affect deRham groups
F1 = deRham(x^2-30*y^3)
F2 = deRham(x^2-31*y^3)
assert all (keys F1, i -> (F1#i == F2#i));
--These lead to problems in factor at the moment
F1 = deRham(x^2-y^10)
F2 = deRham(2*x^2 - 3*y^10)
assert all (keys F1, i -> (F1#i == F2#i));
///
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