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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 if and only if 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 applying the routine coimage to a map whose target is M and whose source is a free module over R.

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 Macaulay 2 authors, or ask them to write it!