Definition of an exact sequence











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I am a beginner at homology, and I am trying to learn it from this text: http://www.seas.upenn.edu/~jean/sheaves-cohomology.pdf



I see the following definitions for exact sequences and short exact sequences :



enter image description here



But then later on I see this:



enter image description here



This is where I get confused :




  • How can that sequence be exact ? from the earlier definition I thought that this would require 2 mappings ?


  • why is g surjective ? I see surjectivity appear in the context of short exact sequences, not in the context of (merely) exact sequences (and that sequence is not a short exact sequence either since it does not start with 0 as per the earlier definition ?)







definition homology-cohomology homological-algebra exact-sequence







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    up vote
    0
    down vote

    favorite












    I am a beginner at homology, and I am trying to learn it from this text: http://www.seas.upenn.edu/~jean/sheaves-cohomology.pdf



    I see the following definitions for exact sequences and short exact sequences :



    enter image description here



    But then later on I see this:



    enter image description here



    This is where I get confused :




    • How can that sequence be exact ? from the earlier definition I thought that this would require 2 mappings ?


    • why is g surjective ? I see surjectivity appear in the context of short exact sequences, not in the context of (merely) exact sequences (and that sequence is not a short exact sequence either since it does not start with 0 as per the earlier definition ?)







    definition homology-cohomology homological-algebra exact-sequence







    share|cite|improve this question


























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I am a beginner at homology, and I am trying to learn it from this text: http://www.seas.upenn.edu/~jean/sheaves-cohomology.pdf



      I see the following definitions for exact sequences and short exact sequences :



      enter image description here



      But then later on I see this:



      enter image description here



      This is where I get confused :




      • How can that sequence be exact ? from the earlier definition I thought that this would require 2 mappings ?


      • why is g surjective ? I see surjectivity appear in the context of short exact sequences, not in the context of (merely) exact sequences (and that sequence is not a short exact sequence either since it does not start with 0 as per the earlier definition ?)







      definition homology-cohomology homological-algebra exact-sequence







      share|cite|improve this question















      I am a beginner at homology, and I am trying to learn it from this text: http://www.seas.upenn.edu/~jean/sheaves-cohomology.pdf



      I see the following definitions for exact sequences and short exact sequences :



      enter image description here



      But then later on I see this:



      enter image description here



      This is where I get confused :




      • How can that sequence be exact ? from the earlier definition I thought that this would require 2 mappings ?


      • why is g surjective ? I see surjectivity appear in the context of short exact sequences, not in the context of (merely) exact sequences (and that sequence is not a short exact sequence either since it does not start with 0 as per the earlier definition ?)







      definition homology-cohomology homological-algebra exact-sequence




      definition homology-cohomology homological-algebra exact-sequence






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      share|cite|improve this question













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      edited Nov 17 at 10:49









      José Carlos Santos

      142k20112208




      142k20112208










      asked Nov 17 at 10:08









      user3203476

      672612




      672612






















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          That sequence has two mappings. The one from $C$ to the null module $0$ is the null map (there are no other choices). And the kernel of the null map is, of course, the whole $C$. Therefore, claiming that that sequence is exact is equivalent to the assertion that $g$ is surjective.






          share|cite|improve this answer





















          • many thanks for your time !
            – user3203476
            Nov 17 at 10:26












          • I'm glad I could help.
            – José Carlos Santos
            Nov 17 at 10:34











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          active

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          1 Answer
          1






          active

          oldest

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          active

          oldest

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          active

          oldest

          votes








          up vote
          2
          down vote



          accepted










          That sequence has two mappings. The one from $C$ to the null module $0$ is the null map (there are no other choices). And the kernel of the null map is, of course, the whole $C$. Therefore, claiming that that sequence is exact is equivalent to the assertion that $g$ is surjective.






          share|cite|improve this answer





















          • many thanks for your time !
            – user3203476
            Nov 17 at 10:26












          • I'm glad I could help.
            – José Carlos Santos
            Nov 17 at 10:34















          up vote
          2
          down vote



          accepted










          That sequence has two mappings. The one from $C$ to the null module $0$ is the null map (there are no other choices). And the kernel of the null map is, of course, the whole $C$. Therefore, claiming that that sequence is exact is equivalent to the assertion that $g$ is surjective.






          share|cite|improve this answer





















          • many thanks for your time !
            – user3203476
            Nov 17 at 10:26












          • I'm glad I could help.
            – José Carlos Santos
            Nov 17 at 10:34













          up vote
          2
          down vote



          accepted







          up vote
          2
          down vote



          accepted






          That sequence has two mappings. The one from $C$ to the null module $0$ is the null map (there are no other choices). And the kernel of the null map is, of course, the whole $C$. Therefore, claiming that that sequence is exact is equivalent to the assertion that $g$ is surjective.






          share|cite|improve this answer












          That sequence has two mappings. The one from $C$ to the null module $0$ is the null map (there are no other choices). And the kernel of the null map is, of course, the whole $C$. Therefore, claiming that that sequence is exact is equivalent to the assertion that $g$ is surjective.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 17 at 10:13









          José Carlos Santos

          142k20112208




          142k20112208












          • many thanks for your time !
            – user3203476
            Nov 17 at 10:26












          • I'm glad I could help.
            – José Carlos Santos
            Nov 17 at 10:34


















          • many thanks for your time !
            – user3203476
            Nov 17 at 10:26












          • I'm glad I could help.
            – José Carlos Santos
            Nov 17 at 10:34
















          many thanks for your time !
          – user3203476
          Nov 17 at 10:26






          many thanks for your time !
          – user3203476
          Nov 17 at 10:26














          I'm glad I could help.
          – José Carlos Santos
          Nov 17 at 10:34




          I'm glad I could help.
          – José Carlos Santos
          Nov 17 at 10:34


















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