isShortExactSequence(SimplicialModuleMap,SimplicialModuleMap) -- whether a pair of simplicial module maps forms a short exact sequence

Function: isShortExactSequence
Usage:

isShortExactSequence(g, f)
Inputs:
- f, a Map of Simplicial Modules,
- g, a Map of Simplicial Modules,
Outputs:
- a Boolean value, that is true if these form a short exact sequence

Description

A short exact sequence of simplicial modules \[ 0 \to B \xrightarrow{f} C \xrightarrow{g} D \to 0\] consists of two morphisms of simplicial modules $f \colon B \to C$ and $g \colon C \to D$ such that $g f = 0$, $\operatorname{image} f = \operatorname{ker} g$, $\operatorname{ker} f = 0$, and $\operatorname{coker} g = 0$.

From a simplicial morphism $h \colon B \to C$, one obtains a short exact sequence \[ 0 \to \operatorname{image} h \to C \to \operatorname{coker} h \to 0. \]

i1 : R = ZZ/101[a,b,c];

i2 : B = simplicialModule(freeResolution coker matrix{{a^2*b, a*b*c, c^3}},  Degeneracy => true)

      1      4      9
o2 = R  <-- R  <-- R <-- ...
                    
     0      1      2

o2 : SimplicialModule

i3 : C = simplicialModule(freeResolution coker vars R, 2, Degeneracy => true)

      1      4      10
o3 = R  <-- R  <-- R  <-- ...
                    
     0      1      2

o3 : SimplicialModule

i4 : h = randomSimplicialMap(C, B, Cycle => true)

          1               1
o4 = 0 : R  <----------- R  : 0
               | -47 |

          4                                                                                                  4
     1 : R  <---------------------------------------------------------------------------------------------- R  : 1
               {0} | -47 0                        0                        0                            |
               {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2       |
               {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2    |
               {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2 |

          10                                                                                                                                                                                                                                                             9
     2 : R   <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R  : 2
                {0} | -47 0                        0                        0                            0                        0                        0                            0                            0                                              |
                {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2       0                        0                        0                            0                            0                                              |
                {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2    0                        0                        0                            0                            0                                              |
                {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2 0                        0                        0                            0                            0                                              |
                {1} | 0   0                        0                        0                            35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2       0                            0                                              |
                {1} | 0   0                        0                        0                            19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2    0                            0                                              |
                {1} | 0   0                        0                        0                            -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2 0                            0                                              |
                {2} | 0   0                        0                        0                            0                        0                        0                            -34a2-19ab-44ac+15bc+2c2     -39a2b-18ab2+24a2c-49abc+18b2c+4ac2-47c3       |
                {2} | 0   0                        0                        0                            0                        0                        0                            43a2+40ab-5b2+36ac+49bc+29c2 -22a2b-49ab2-18b3-49abc+19b2c-43ac2-15bc2-28c3 |
                {2} | 0   0                        0                        0                            0                        0                        0                            -16a2+39ab+21ac+24bc+38c2    24a3-36a2b-29ab2-30a2c+19abc+45ac2-34bc2-48c3  |

o4 : SimplicialModuleMap

i5 : f = inducedMap(C, image h)

          1
o5 = 0 : R  <----------- image | -47 | : 0
               | -47 |

          4
     1 : R  <---------------------------------------------------------------------------------------------- image {0} | -47 0                        0                        0                            | : 1
               {0} | -47 0                        0                        0                            |         {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2       |
               {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2       |         {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2    |
               {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2    |         {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2 |
               {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2 |

          10
     2 : R   <--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- image {0} | -47 0                        0                        0                            0                        0                        0                            0                            0                                              | : 2
                {0} | -47 0                        0                        0                            0                        0                        0                            0                            0                                              |         {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2       0                        0                        0                            0                            0                                              |
                {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2       0                        0                        0                            0                            0                                              |         {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2    0                        0                        0                            0                            0                                              |
                {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2    0                        0                        0                            0                            0                                              |         {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2 0                        0                        0                            0                            0                                              |
                {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2 0                        0                        0                            0                            0                                              |         {1} | 0   0                        0                        0                            35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2       0                            0                                              |
                {1} | 0   0                        0                        0                            35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2       0                            0                                              |         {1} | 0   0                        0                        0                            19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2    0                            0                                              |
                {1} | 0   0                        0                        0                            19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2    0                            0                                              |         {1} | 0   0                        0                        0                            -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2 0                            0                                              |
                {1} | 0   0                        0                        0                            -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2 0                            0                                              |         {2} | 0   0                        0                        0                            0                        0                        0                            -34a2-19ab-44ac+15bc+2c2     -39a2b-18ab2+24a2c-49abc+18b2c+4ac2-47c3       |
                {2} | 0   0                        0                        0                            0                        0                        0                            -34a2-19ab-44ac+15bc+2c2     -39a2b-18ab2+24a2c-49abc+18b2c+4ac2-47c3       |         {2} | 0   0                        0                        0                            0                        0                        0                            43a2+40ab-5b2+36ac+49bc+29c2 -22a2b-49ab2-18b3-49abc+19b2c-43ac2-15bc2-28c3 |
                {2} | 0   0                        0                        0                            0                        0                        0                            43a2+40ab-5b2+36ac+49bc+29c2 -22a2b-49ab2-18b3-49abc+19b2c-43ac2-15bc2-28c3 |         {2} | 0   0                        0                        0                            0                        0                        0                            -16a2+39ab+21ac+24bc+38c2    24a3-36a2b-29ab2-30a2c+19abc+45ac2-34bc2-48c3  |
                {2} | 0   0                        0                        0                            0                        0                        0                            -16a2+39ab+21ac+24bc+38c2    24a3-36a2b-29ab2-30a2c+19abc+45ac2-34bc2-48c3  |

o5 : SimplicialModuleMap

i6 : g = inducedMap(coker h, C)

                                  1
o6 = 0 : cokernel | -47 | <----- R  : 0
                             0

                                                                                                                                   4
     1 : cokernel {0} | -47 0                        0                        0                            | <------------------- R  : 1
                  {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2       |    {0} | 0 0 0 0 |
                  {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2    |    {1} | 0 1 0 0 |
                  {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2 |    {1} | 0 0 1 0 |
                                                                                                                {1} | 0 0 0 1 |

                                                                                                                                                                                                                                                                                                          10
     2 : cokernel {0} | -47 0                        0                        0                            0                        0                        0                            0                            0                                              | <------------------------------- R   : 2
                  {1} | 0   35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2       0                        0                        0                            0                            0                                              |    {0} | 0 0 0 0 0 0 0 0 0 0 |
                  {1} | 0   19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2    0                        0                        0                            0                            0                                              |    {1} | 0 1 0 0 0 0 0 0 0 0 |
                  {1} | 0   -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2 0                        0                        0                            0                            0                                              |    {1} | 0 0 1 0 0 0 0 0 0 0 |
                  {1} | 0   0                        0                        0                            35ab+10b2+8ac-50bc+29c2  34ab+19b2-43ac+39bc-28c2 39ab+18b2-2ac-3bc-22c2       0                            0                                              |    {1} | 0 0 0 1 0 0 0 0 0 0 |
                  {1} | 0   0                        0                        0                            19a2-10ab-29ac+24bc+38c2 -34a2-19ab-9ac-34bc-48c2 -39a2-18ab-13ac+47bc-38c2    0                            0                                              |    {1} | 0 0 0 0 1 0 0 0 0 0 |
                  {1} | 0   0                        0                        0                            -8a2-22ab-24b2-29ac-38bc 43a2+24ab+34b2+28ac+48bc 2a2+16ab-47b2+22ac+38bc-47c2 0                            0                                              |    {1} | 0 0 0 0 0 1 0 0 0 0 |
                  {2} | 0   0                        0                        0                            0                        0                        0                            -34a2-19ab-44ac+15bc+2c2     -39a2b-18ab2+24a2c-49abc+18b2c+4ac2-47c3       |    {1} | 0 0 0 0 0 0 1 0 0 0 |
                  {2} | 0   0                        0                        0                            0                        0                        0                            43a2+40ab-5b2+36ac+49bc+29c2 -22a2b-49ab2-18b3-49abc+19b2c-43ac2-15bc2-28c3 |    {2} | 0 0 0 0 0 0 0 1 0 0 |
                  {2} | 0   0                        0                        0                            0                        0                        0                            -16a2+39ab+21ac+24bc+38c2    24a3-36a2b-29ab2-30a2c+19abc+45ac2-34bc2-48c3  |    {2} | 0 0 0 0 0 0 0 0 1 0 |
                                                                                                                                                                                                                                                                           {2} | 0 0 0 0 0 0 0 0 0 1 |

o6 : SimplicialModuleMap

i7 : assert isShortExactSequence(g,f)

A short exact sequence of modules gives rise to a short exact sequence of simplicial modules. These simplicial modules arise as free resolutions of the modules.

i8 : I = ideal(a^3, b^3, c^3)

             3   3   3
o8 = ideal (a , b , c )

o8 : Ideal of R

i9 : J = I + ideal(a*b*c)

             3   3   3
o9 = ideal (a , b , c , a*b*c)

o9 : Ideal of R

i10 : K = I : ideal(a*b*c)

              2   2   2
o10 = ideal (c , b , a )

o10 : Ideal of R

i11 : SES = complex{
          map(comodule J, comodule I, 1),
          map(comodule I, (comodule K) ** R^{-3}, {{a*b*c}})
          }

o11 = cokernel | a3 b3 c3 abc | <-- cokernel | a3 b3 c3 | <-- cokernel {3} | c2 b2 a2 |
                                                               
      0                             1                         2

o11 : Complex

i12 : assert isWellDefined SES

i13 : assert isShortExactSequence(dd^SES_1, dd^SES_2)

i14 : (g,f) = (horseshoeResolution SES)/simplicialModule;

i15 : assert isShortExactSequence(g,f)

Ways to use this method:

isShortExactSequence(SimplicialModuleMap,SimplicialModuleMap) -- whether a pair of simplicial module maps forms a short exact sequence

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SimplicialModules/SimplicialModuleDOC.m2:3095:0.

isShortExactSequence(SimplicialModuleMap,SimplicialModuleMap) -- whether a pair of simplicial module maps forms a short exact sequence

Description

See also

Ways to use this method: