Computing Betti numbers using Macaulay2












2












$begingroup$


Let $k$ be a field and $R=k[x,y,z]$, let $M=R/langle x^2,xy,yz^2,y^4rangle$ be $R$-module, how can we compute the left free resolution of $M$, and also the Betti numbers of this resolution?










share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    Let $k$ be a field and $R=k[x,y,z]$, let $M=R/langle x^2,xy,yz^2,y^4rangle$ be $R$-module, how can we compute the left free resolution of $M$, and also the Betti numbers of this resolution?










    share|cite|improve this question











    $endgroup$















      2












      2








      2


      1



      $begingroup$


      Let $k$ be a field and $R=k[x,y,z]$, let $M=R/langle x^2,xy,yz^2,y^4rangle$ be $R$-module, how can we compute the left free resolution of $M$, and also the Betti numbers of this resolution?










      share|cite|improve this question











      $endgroup$




      Let $k$ be a field and $R=k[x,y,z]$, let $M=R/langle x^2,xy,yz^2,y^4rangle$ be $R$-module, how can we compute the left free resolution of $M$, and also the Betti numbers of this resolution?







      homological-algebra betti-numbers macaulay2






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 8 '18 at 17:19









      Rodrigo de Azevedo

      12.9k41857




      12.9k41857










      asked May 28 '12 at 23:39









      kiranovalobaskiranovalobas

      414213




      414213






















          1 Answer
          1






          active

          oldest

          votes


















          8












          $begingroup$

          The first step is to plug your module into Macaulay2. As far as I understand from the official tutorial on modules in Macaulay2, the way to make modules is as kernels or cokernels of linear maps given by matrices. Thus, for example you get:



          R=QQ[x,y,z]
          m=matrix{{x^2,x*y,y*z^2,y^4}}
          M=cokernel m
          C=resolution M
          B=betti C


          The first three commands are self-explanatory. The fourth computes a free resolution of the R-module M, and its output looks like:



                 1      4      4      1
          o19 = R <-- R <-- R <-- R <-- 0

          0 1 2 3 4


          So in partiuclar, we have that the resolution is $Mleftarrow Rleftarrow R^{oplus 4}leftarrow R^{oplus 4}leftarrow Rleftarrow 0$. If you want to see what the individual maps (differentials) are in terms of matrices, you call the .dd method on the stored resolution to get:



          i20 : C.dd

          1 4
          o20 = 0 : R <-------------------- R : 1
          | x2 xy yz2 y4 |

          4 4
          1 : R <-------------------------- R : 2
          {2} | -y 0 0 0 |
          {2} | x -z2 -y3 0 |
          {3} | 0 x 0 -y3 |
          {4} | 0 0 x z2 |

          4 1
          2 : R <--------------- R : 3
          {3} | 0 |
          {4} | y3 |
          {5} | -z2 |
          {6} | x |

          1
          3 : R <----- 0 : 4
          0


          The last command, betti, outputs something called a betti talli, which looks something like this:



                   0 1 2 3
          total: 1 4 4 1
          0: 1 . . .
          1: . 2 1 .
          2: . 1 1 .
          3: . 1 1 .
          4: . . 1 1


          The first row are the indices of a free resolution $Mleftarrow F_0leftarrow F_1leftarrow F_2leftarrow F_3leftarrow 0$, where the $F_i$ are free modules. The second row are the total betti numbers, that is, the ranks of the free modules. Further, we have matrix $(gamma_{ij})$ with a column for each module in the resolution, and as many rows are necessary to encode the graded betti numbers according to the scheme $gamma_{ik}=beta_{i,i+k}$ where $beta_{ij}$ is the degree $j$ graded betti number for the $i^text{th}$ free module.






          share|cite|improve this answer









          $endgroup$













            Your Answer





            StackExchange.ifUsing("editor", function () {
            return StackExchange.using("mathjaxEditing", function () {
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            });
            });
            }, "mathjax-editing");

            StackExchange.ready(function() {
            var channelOptions = {
            tags: "".split(" "),
            id: "69"
            };
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function() {
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled) {
            StackExchange.using("snippets", function() {
            createEditor();
            });
            }
            else {
            createEditor();
            }
            });

            function createEditor() {
            StackExchange.prepareEditor({
            heartbeatType: 'answer',
            autoActivateHeartbeat: false,
            convertImagesToLinks: true,
            noModals: true,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            imageUploader: {
            brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
            contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
            allowUrls: true
            },
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            });


            }
            });














            draft saved

            draft discarded


















            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f150972%2fcomputing-betti-numbers-using-macaulay2%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown

























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            8












            $begingroup$

            The first step is to plug your module into Macaulay2. As far as I understand from the official tutorial on modules in Macaulay2, the way to make modules is as kernels or cokernels of linear maps given by matrices. Thus, for example you get:



            R=QQ[x,y,z]
            m=matrix{{x^2,x*y,y*z^2,y^4}}
            M=cokernel m
            C=resolution M
            B=betti C


            The first three commands are self-explanatory. The fourth computes a free resolution of the R-module M, and its output looks like:



                   1      4      4      1
            o19 = R <-- R <-- R <-- R <-- 0

            0 1 2 3 4


            So in partiuclar, we have that the resolution is $Mleftarrow Rleftarrow R^{oplus 4}leftarrow R^{oplus 4}leftarrow Rleftarrow 0$. If you want to see what the individual maps (differentials) are in terms of matrices, you call the .dd method on the stored resolution to get:



            i20 : C.dd

            1 4
            o20 = 0 : R <-------------------- R : 1
            | x2 xy yz2 y4 |

            4 4
            1 : R <-------------------------- R : 2
            {2} | -y 0 0 0 |
            {2} | x -z2 -y3 0 |
            {3} | 0 x 0 -y3 |
            {4} | 0 0 x z2 |

            4 1
            2 : R <--------------- R : 3
            {3} | 0 |
            {4} | y3 |
            {5} | -z2 |
            {6} | x |

            1
            3 : R <----- 0 : 4
            0


            The last command, betti, outputs something called a betti talli, which looks something like this:



                     0 1 2 3
            total: 1 4 4 1
            0: 1 . . .
            1: . 2 1 .
            2: . 1 1 .
            3: . 1 1 .
            4: . . 1 1


            The first row are the indices of a free resolution $Mleftarrow F_0leftarrow F_1leftarrow F_2leftarrow F_3leftarrow 0$, where the $F_i$ are free modules. The second row are the total betti numbers, that is, the ranks of the free modules. Further, we have matrix $(gamma_{ij})$ with a column for each module in the resolution, and as many rows are necessary to encode the graded betti numbers according to the scheme $gamma_{ik}=beta_{i,i+k}$ where $beta_{ij}$ is the degree $j$ graded betti number for the $i^text{th}$ free module.






            share|cite|improve this answer









            $endgroup$


















              8












              $begingroup$

              The first step is to plug your module into Macaulay2. As far as I understand from the official tutorial on modules in Macaulay2, the way to make modules is as kernels or cokernels of linear maps given by matrices. Thus, for example you get:



              R=QQ[x,y,z]
              m=matrix{{x^2,x*y,y*z^2,y^4}}
              M=cokernel m
              C=resolution M
              B=betti C


              The first three commands are self-explanatory. The fourth computes a free resolution of the R-module M, and its output looks like:



                     1      4      4      1
              o19 = R <-- R <-- R <-- R <-- 0

              0 1 2 3 4


              So in partiuclar, we have that the resolution is $Mleftarrow Rleftarrow R^{oplus 4}leftarrow R^{oplus 4}leftarrow Rleftarrow 0$. If you want to see what the individual maps (differentials) are in terms of matrices, you call the .dd method on the stored resolution to get:



              i20 : C.dd

              1 4
              o20 = 0 : R <-------------------- R : 1
              | x2 xy yz2 y4 |

              4 4
              1 : R <-------------------------- R : 2
              {2} | -y 0 0 0 |
              {2} | x -z2 -y3 0 |
              {3} | 0 x 0 -y3 |
              {4} | 0 0 x z2 |

              4 1
              2 : R <--------------- R : 3
              {3} | 0 |
              {4} | y3 |
              {5} | -z2 |
              {6} | x |

              1
              3 : R <----- 0 : 4
              0


              The last command, betti, outputs something called a betti talli, which looks something like this:



                       0 1 2 3
              total: 1 4 4 1
              0: 1 . . .
              1: . 2 1 .
              2: . 1 1 .
              3: . 1 1 .
              4: . . 1 1


              The first row are the indices of a free resolution $Mleftarrow F_0leftarrow F_1leftarrow F_2leftarrow F_3leftarrow 0$, where the $F_i$ are free modules. The second row are the total betti numbers, that is, the ranks of the free modules. Further, we have matrix $(gamma_{ij})$ with a column for each module in the resolution, and as many rows are necessary to encode the graded betti numbers according to the scheme $gamma_{ik}=beta_{i,i+k}$ where $beta_{ij}$ is the degree $j$ graded betti number for the $i^text{th}$ free module.






              share|cite|improve this answer









              $endgroup$
















                8












                8








                8





                $begingroup$

                The first step is to plug your module into Macaulay2. As far as I understand from the official tutorial on modules in Macaulay2, the way to make modules is as kernels or cokernels of linear maps given by matrices. Thus, for example you get:



                R=QQ[x,y,z]
                m=matrix{{x^2,x*y,y*z^2,y^4}}
                M=cokernel m
                C=resolution M
                B=betti C


                The first three commands are self-explanatory. The fourth computes a free resolution of the R-module M, and its output looks like:



                       1      4      4      1
                o19 = R <-- R <-- R <-- R <-- 0

                0 1 2 3 4


                So in partiuclar, we have that the resolution is $Mleftarrow Rleftarrow R^{oplus 4}leftarrow R^{oplus 4}leftarrow Rleftarrow 0$. If you want to see what the individual maps (differentials) are in terms of matrices, you call the .dd method on the stored resolution to get:



                i20 : C.dd

                1 4
                o20 = 0 : R <-------------------- R : 1
                | x2 xy yz2 y4 |

                4 4
                1 : R <-------------------------- R : 2
                {2} | -y 0 0 0 |
                {2} | x -z2 -y3 0 |
                {3} | 0 x 0 -y3 |
                {4} | 0 0 x z2 |

                4 1
                2 : R <--------------- R : 3
                {3} | 0 |
                {4} | y3 |
                {5} | -z2 |
                {6} | x |

                1
                3 : R <----- 0 : 4
                0


                The last command, betti, outputs something called a betti talli, which looks something like this:



                         0 1 2 3
                total: 1 4 4 1
                0: 1 . . .
                1: . 2 1 .
                2: . 1 1 .
                3: . 1 1 .
                4: . . 1 1


                The first row are the indices of a free resolution $Mleftarrow F_0leftarrow F_1leftarrow F_2leftarrow F_3leftarrow 0$, where the $F_i$ are free modules. The second row are the total betti numbers, that is, the ranks of the free modules. Further, we have matrix $(gamma_{ij})$ with a column for each module in the resolution, and as many rows are necessary to encode the graded betti numbers according to the scheme $gamma_{ik}=beta_{i,i+k}$ where $beta_{ij}$ is the degree $j$ graded betti number for the $i^text{th}$ free module.






                share|cite|improve this answer









                $endgroup$



                The first step is to plug your module into Macaulay2. As far as I understand from the official tutorial on modules in Macaulay2, the way to make modules is as kernels or cokernels of linear maps given by matrices. Thus, for example you get:



                R=QQ[x,y,z]
                m=matrix{{x^2,x*y,y*z^2,y^4}}
                M=cokernel m
                C=resolution M
                B=betti C


                The first three commands are self-explanatory. The fourth computes a free resolution of the R-module M, and its output looks like:



                       1      4      4      1
                o19 = R <-- R <-- R <-- R <-- 0

                0 1 2 3 4


                So in partiuclar, we have that the resolution is $Mleftarrow Rleftarrow R^{oplus 4}leftarrow R^{oplus 4}leftarrow Rleftarrow 0$. If you want to see what the individual maps (differentials) are in terms of matrices, you call the .dd method on the stored resolution to get:



                i20 : C.dd

                1 4
                o20 = 0 : R <-------------------- R : 1
                | x2 xy yz2 y4 |

                4 4
                1 : R <-------------------------- R : 2
                {2} | -y 0 0 0 |
                {2} | x -z2 -y3 0 |
                {3} | 0 x 0 -y3 |
                {4} | 0 0 x z2 |

                4 1
                2 : R <--------------- R : 3
                {3} | 0 |
                {4} | y3 |
                {5} | -z2 |
                {6} | x |

                1
                3 : R <----- 0 : 4
                0


                The last command, betti, outputs something called a betti talli, which looks something like this:



                         0 1 2 3
                total: 1 4 4 1
                0: 1 . . .
                1: . 2 1 .
                2: . 1 1 .
                3: . 1 1 .
                4: . . 1 1


                The first row are the indices of a free resolution $Mleftarrow F_0leftarrow F_1leftarrow F_2leftarrow F_3leftarrow 0$, where the $F_i$ are free modules. The second row are the total betti numbers, that is, the ranks of the free modules. Further, we have matrix $(gamma_{ij})$ with a column for each module in the resolution, and as many rows are necessary to encode the graded betti numbers according to the scheme $gamma_{ik}=beta_{i,i+k}$ where $beta_{ij}$ is the degree $j$ graded betti number for the $i^text{th}$ free module.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered May 29 '12 at 0:24









                Vladimir SotirovVladimir Sotirov

                8,63611948




                8,63611948






























                    draft saved

                    draft discarded




















































                    Thanks for contributing an answer to Mathematics Stack Exchange!


                    • Please be sure to answer the question. Provide details and share your research!

                    But avoid



                    • Asking for help, clarification, or responding to other answers.

                    • Making statements based on opinion; back them up with references or personal experience.


                    Use MathJax to format equations. MathJax reference.


                    To learn more, see our tips on writing great answers.




                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function () {
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f150972%2fcomputing-betti-numbers-using-macaulay2%23new-answer', 'question_page');
                    }
                    );

                    Post as a guest















                    Required, but never shown





















































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown

































                    Required, but never shown














                    Required, but never shown












                    Required, but never shown







                    Required, but never shown







                    Popular posts from this blog

                    Aardman Animations

                    Are they similar matrix

                    “minimization” problem in Euclidean space related to orthonormal basis