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Macaulay2Doc :: pushForward(RingMap,Module)

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 that generates 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,{15518a + 11673b - 2780c - 15637d + 4010e, 15889a + 1771b - 13085c - 4796d - 3566e, - 1127a - 10588b - 1507c + 2113d + 81e, - 8195a + 1161b + 6680c + 11051d - 5034e})

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

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

o16 = cokernel {0} | 24312912970878ab-53558417980368b2+11035767679344ac-208332755100360bc+108392666235840c2 4052152161813a2-38625686802360b2-2919735355128ac-117096169265820bc+92114562082032c2 2435938617634788214728183914585712b3-384290528615247451220104789394568b2c+4109195066488308651105245818994688ac2-30440791323378185934086788025411712bc2+10522249813613678477603918549958144c3                                  0                                                                                                                                                          |
               {1} | 15433754268565a+113739461997426b+396218376009121c                                      52224662825987a+80538179318172b+207167633670926c                                    330449348561425599845250340874015a2-3545842063890162762779841278514936ab-4981757290597360529004206215380456b2+7701131063978961709869293192952655ac+4055323340826415842573611121695460bc+46017069495501479964492760999389548c2 8875074335a3-114783504864a2b+34457756400ab2-11319830352b3+139659196075a2c+399497485128abc-106198855056b2c-1090185713176ac2-211239154644bc2+1992712156176c3 |

                               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               3
o19 = ideal(8875074335a  - 114783504864a b + 34457756400a*b  - 11319830352b 
      -----------------------------------------------------------------------
                     2                                     2   
      + 139659196075a c + 399497485128a*b*c - 106198855056b c -
      -----------------------------------------------------------------------
                      2                  2                 3
      1090185713176a*c  - 211239154644b*c  + 1992712156176c )

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!