degree(ComplexMap) -- get the degree of a map of chain complexes

Synopsis

Function: degree
Usage:

degree f
Inputs:
- f, a map of complexes,
Outputs:
- an integer,

Description

A complex map $f : C \to D$ of degree $d$ is a sequence of of maps $f_i : C_i \to D_{i+d}$. This method returns $d$.

The degree of the differential of a complex is always -1.

i1 : R = ZZ/101[a..d];

i2 : I = ideal(a^2, b^2, c^2)

             2   2   2
o2 = ideal (a , b , c )

o2 : Ideal of R

i3 : FI = freeResolution I

      1      3      3      1
o3 = R  <-- R  <-- R  <-- R
                           
     0      1      2      3

o3 : Complex

i4 : assert(degree dd^FI == -1)

i5 : FJ = freeResolution (I + ideal(a*b*c))

      1      4      6      3
o5 = R  <-- R  <-- R  <-- R
                           
     0      1      2      3

o5 : Complex

i6 : f = randomComplexMap(FJ, FI, Cycle=>true, Degree => -2)

                    1
o6 = -2 : 0 <----- R  : 0
               0

                    3
     -1 : 0 <----- R  : 1
               0

          1                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             3
     0 : R  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R  : 2
               | 18a4+44a3b-29a2b2+8ab3+6b4+38a3c+50a2bc-48ab2c+50b3c-32a2c2+4abc2-24b2c2+17ac3+22bc3+43c4+31a3d+49a2bd+4ab2d-23b3d+12a2cd-3abcd+7b2cd-20ac2d-49bc2d-8c3d+37a2d2+45abd2+9b2d2+44acd2-11bcd2+36c2d2-39ad3-8bd3-3cd3-22d4 -32a4+20a3b+50a2b2-36ab3+3b4+23a3c+22a2bc+12ab2c-22b3c-18a2c2-11abc2-7b2c2-29ac3+12bc3-15c4+39a3d-6a2bd-11ab2d+47b3d-20a2cd-5abcd+7b2cd-38bc2d+4c3d+14a2d2-40abd2-2b2d2+29acd2-15bcd2+5c2d2-ad3+37bd3+18cd3-39d4 24a4-36a3b+6a2b2-25ab3+9b4-30a3c-42a2bc-16ab2c-23b3c-20a2c2+29abc2-40b2c2+28ac3-45bc3+47c4-29a3d-10a2bd+16ab2d-6b3d-8a2cd+46abcd+25b2cd-21ac2d-20bc2d+7c3d-22a2d2+21abd2+15b2d2-47acd2-25bcd2+c2d2-39ad3+22bd3-47cd3+47d4 |

          4                                                                                                                                                                                                                                              1
     1 : R  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ R  : 3
               {2} | 24a4-36a3b+19a2b2-29ab3-18b4-30a3c+19a2bc-24ab2c-13b3c-29a2c2-16abc2-15b2c2+34ac3+38bc3+45c4-29a3d-10a2bd-38ab2d-43b3d-8a2cd+39abcd-28b2cd+19ac2d+2bc2d-34c3d-22a2d2+21abd2-47b2d2-47acd2+16bcd2-48c2d2-39ad3+22bd3-47cd3+47d4 |
               {2} | 19a4-16a3b-23a2b2+36ab3-3b4+7a3c+39a2bc+35ab2c+22b3c-17a2c2-38abc2-23b2c2+11ac3+29bc3-13c4+15a3d+43a2bd+11ab2d-47b3d-11a2cd+33abcd-7b2cd+46ac2d-47bc2d-10c3d+48a2d2+40abd2+2b2d2-28acd2+15bcd2+30c2d2+ad3-37bd3-18cd3+39d4     |
               {2} | 27a4-22a3b-32a2b2+39ab3+36b4+32a3c-20a2bc+9b3c-30a2c2-49abc2+4b2c2+17ac3+22bc3+43c4-9a3d+24a2bd+33ab2d-39b3d-48a2cd-33abcd+13b2cd-20ac2d-49bc2d-8c3d-15a2d2-19abd2-26b2d2+44acd2-11bcd2+36c2d2-39ad3-8bd3-3cd3-22d4            |
               {3} | 40a3-22a2b+30ab2-47b3+10a2c+13abc+18b2c-13ac2-30bc2-48c3+7a2d-17abd-28b2d+3acd+26bcd+30c2d-41ad2-bd2-37cd2                                                                                                                     |

o6 : ComplexMap

i7 : assert(degree f == -2)

Ways to use this method: