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pushForward(RingMap,Module)

Synopsis

Description

Currently, R and S must both be polynomial rings over the same base field.

This function first checks to see whether M will be a finitely generated R-module via F. If not, an error message describing the codimension of M/(vars of S)M is given (this is equal to the dimension of R iff M is a finitely generated R-module.

Assuming that it is, the push forward F_*(M) is computed. This is done by first finding a presentation for M in terms of a set of elements which generate M as an S-module, and then calling the routine pushForward1.

All optional arguments are passed to pushForward1.

Example: The Auslander-Buchsbaum formula

Let's illustrate the Auslander-Buchsbaum formula. First construct some rings and make a module of projective dimension 2.
i1 : R4 = ZZ/32003[a..d];
i2 : R5 = ZZ/32003[a..e];
i3 : R6 = ZZ/32003[a..f];
i4 : M = coker genericMatrix(R6,a,2,3)

o4 = cokernel | a c e |
              | b d f |

                              2
o4 : R6-module, quotient of R6
i5 : pdim M

o5 = 2
Create ring maps.
i6 : G = map(R6,R5,{a+b+c+d+e+f,b,c,d,e})

o6 = map(R6,R5,{a + b + c + d + e + f, b, c, d, e})

o6 : RingMap R6 <--- R5
i7 : F = map(R5,R4,random(R5^1, R5^{4:-1}))

o7 = map(R5,R4,{- 2136a + 9349b + 8735c - 5609d - 9489e, 13529a - 15802b - 371c - 545d - 2519e, - 11250a - 14212b - 1270c - 1415d + 626e, 1414a - 3327b - 4035c + 11874d - 13874e})

o7 : RingMap R5 <--- R4
The module M, when thought of as an R5 or R4 module, has the same depth, but since depth M + pdim M = dim ring, the projective dimension will drop to 1, respectively 0, for these two rings.
i8 : P = pushForward(G,M)

o8 = cokernel | c -de               |
              | d bc-ad+bd+cd+d2+de |

                              2
o8 : R5-module, quotient of R5
i9 : pdim P

o9 = 1
i10 : Q = pushForward(F,P)

        3
o10 = R4

o10 : R4-module, free, degrees {0, 1, 0}
i11 : pdim Q

o11 = 0

Example: generic projection of a homogeneous coordinate ring

We compute the pushforward N of the homogeneous coordinate ring M of the twisted cubic curve in P^3.
i12 : P3 = QQ[a..d];
i13 : M = comodule monomialCurveIdeal(P3,{1,2,3})

o13 = cokernel | c2-bd bc-ad b2-ac |

                               1
o13 : P3-module, quotient of P3
The result is a module with the same codimension, degree and genus as the twisted cubic, but the support is a cubic in the plane, necessarily having one node.
i14 : P2 = QQ[a,b,c];
i15 : F = map(P3,P2,random(P3^1, P3^{-1,-1,-1}))

                 5     9     2     7    7     6     3     1    10     7     5
o15 = map(P3,P2,{-*a + -*b + -*c + -*d, -*a + -*b + -*c + -*d, --*a + -*b + -*c + 8d})
                 4     2     5     6    6     5     4     4     3     4     6

o15 : RingMap P3 <--- P2
i16 : N = pushForward(F,M)

o16 = cokernel {0} | 821503652433360ab-2378892593212200b2-180967843812504ac+659996719611660bc+20456077852224c2 1643007304866720a2-21543463552689000b2+502985806533000ac+3727516078503900bc+914776078031280c2 4618110968829401670197894687164320000b3-3163611576589491698363463849180648000b2c+59947856690696232009186202915649040ac2+125160281453177557124256750158202600bc2+123254365576028904442854898163719860c3                                       0                                                                                                                                                         |
               {1} | 1418396941154817a-5265014690542360b-171202052055577c                                      -6579999349220375a-29839829515772016b-7056057635649837c                                       1686927222636153255177394681421200005a2+2536043553776579117535040486969212200ab-388195627484498156269005416257189380b2-828317411634297959485490273243340420ac-1183265507822376924552857453091129228bc-952957461124128652576034889254455386c2 287958450810a3-400961287925a2b-2680942780560ab2+5244174813700b3-93738370860a2c+911678082564abc-1609862603460b2c+8536409703ac2-23117264574bc2-1516413743c3 |

                               2
o16 : P2-module, quotient of P2
i17 : hilbertPolynomial M

o17 = - 2*P  + 3*P
           0      1

o17 : ProjectiveHilbertPolynomial
i18 : hilbertPolynomial N

o18 = - 2*P  + 3*P
           0      1

o18 : ProjectiveHilbertPolynomial
i19 : ann N

                         3                2                    2  
o19 = ideal(287958450810a  - 400961287925a b - 2680942780560a*b  +
      -----------------------------------------------------------------------
                    3               2                                      2 
      5244174813700b  - 93738370860a c + 911678082564a*b*c - 1609862603460b c
      -----------------------------------------------------------------------
                     2                 2              3
      + 8536409703a*c  - 23117264574b*c  - 1516413743c )

o19 : Ideal of P2
Note: these examples are from the original Macaulay script by David Eisenbud.

Caveat

The module M must be homogeneous, as must R, S, and f. If you need this function in more general situations, please write it and send it to the Macaulay2 authors, or ask them to write it!

See also