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Dlocalize.exa.m2
restart needsPackage "Dmodules" ------------------------- EXAMPLES for Dlocalization -------------------------------- -- Example 1: Simple example W = QQ[x,dx,WeylAlgebra=>{x=>dx}] M = cokernel matrix{{dx}} Dlocalize(M, x) DlocalizeMap(M, x) DlocalizeAll(M, x) Dlocalize(M, x, Strategy => Oaku) DlocalizeMap(M, x, Strategy => Oaku) DlocalizeAll(M, x, Strategy => Oaku) -- Example 2: Cusp W = QQ[x,y,dx,dy, WeylAlgebra=>{x=>dx,y=>dy}] M = cokernel matrix{{dx,dy}} f = x^2-y^3 Dlocalize(M, f) DlocalizeAll(M, f) Dlocalize(M, f, Strategy => Oaku) DlocalizeAll(M, f, Strategy => Oaku) -- Example 3: from Coutinho "Primer..." n = 4 W = QQ[u_1..u_n, Du_1..Du_n, WeylAlgebra => apply(toList(1..n), i -> u_i => Du_i)] M = W^1/ideal(Du_1..Du_n) f = sum(toList(1..n), i -> u_i^2) Dlocalize(M, f) DlocalizeAll(M, f) Dlocalize(M, f, Strategy => Oaku) DlocalizeAll(M, f, Strategy => Oaku) -- Example 4: from OTW, Example 8 W = QQ[x,y,z,Dx,Dy,Dz,WeylAlgebra => {x=>Dx,y=>Dy,z=>Dz}] I = ideal((x^3-y^2*z^2)^2*Dx+3*x^2, (x^3-y^2*z^2)^2*Dy-2*y*z^2, (x^3-y^2*z^2)^2*Dz-2*y^2*z) -- some annihilators of e^(1/x3-y2z2) nh = x -- non-holonomic locus M = W^1/I Dlocalize(M,nh) DlocalizeAll(M,nh) Dlocalize(M, nh, Strategy => Oaku) -- NOTE THAT I IS NOT SATURATED DlocalizeAll(M, nh, Strategy => Oaku) -- LEADS TO EXTRA FACTOR IN BFUNCTION? -- Example 5a: Simple example -- pure torsion module W = QQ[x,dx,WeylAlgebra=>{x=>dx}] M = cokernel matrix{{x}} Dlocalize(M, x) DlocalizeMap(M, x) DlocalizeAll(M, x) Dlocalize(M, x, Strategy => Oaku) DlocalizeMap(M, x, Strategy => Oaku) DlocalizeAll(M, x, Strategy => Oaku) -- Example 5b: Simple example -- already localized W = QQ[x,dx,WeylAlgebra=>{x=>dx}] M = cokernel matrix{{x*dx+7}} Dlocalize(M, x) DlocalizeMap(M, x) DlocalizeAll(M, x) Dlocalize(M, x, Strategy => Oaku) DlocalizeMap(M, x, Strategy => Oaku) DlocalizeAll(M, x, Strategy => Oaku) N = cokernel matrix{{x*dx+1/2}} Dlocalize(N, x) Dlocalize(N, x, Strategy => Oaku) -- Example 6: Boundary cases W = QQ[x,dx,WeylAlgebra=>{x=>dx}] M = cokernel matrix{{dx}} Dlocalize(M, 0_W) Dlocalize(M, 0_W, Strategy => Oaku) Dlocalize(M, 1_W) Dlocalize(M, 1_W, Strategy => Oaku) -- Example 7: Compare OTW and OTWcyclic W = makeWA (QQ[x,y,z]) I = PolyAnn(x*y-z^2+2) fs = {x*y+z^2-4,x*y*z,x*y+z^3,x*y*z+z^2} for f in fs do ( elapsedTime resOTW := Dlocalize(I,f,Strategy => OTW); elapsedTime resOTWcyclic := Dlocalize(I,f,Strategy => OTW); assert(resOTW == resOTWcyclic) ) -- Example 8: W = makeWA(QQ[x,y]); M = coker matrix{{dx,dy,0,0},{0,0,x,y}} f=x^2+y^2; Mf=Dlocalize(M,f); assert(Mf == Dlocalize(W^1/ideal(dx,dy),f))
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