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nLab > Latest Changes: salamander lemma

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  1.  
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 14th 2016

    Under definition 1 of salamander lemma, I fixed a mistake in the definition of A □A_\Box where there was a direct sum of two submodules, where there needed to be a sum (i.e., join) instead.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 14th 2016

    Thanks for catching this.

    And thanks for looking into writing out a proof of the braid lemma, if I am guessing correctly that this is what you are doing?! :-)

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 14th 2016

    I hope to be doing that, yes; it may take a little while because I’ve never looked carefully at the salamander lemma – I’ve only had an idea what it was good for.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeApr 14th 2016
    • (edited Apr 14th 2016)

    Also in salamander lemma, looking at the zig-zags after remark 3, they look off. The \swarrows should pertain to vertical differentials according to 2. under remark 3, but in the zig-zags below the \swarrows go from X k+1,l□X_{k+1,l}_\Box to □X k,l^\Box X_{k, l}, i.e., an extramural map corresponding to a horizontal differential X k+1,l→X k,lX_{k+1, l} \to X_{k, l}. So I assume the arrows in both zig-zags should be switched to the opposite directions.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 15th 2016
    • (edited Apr 15th 2016)

    Thanks, Todd. Right now I don’t have the leisure to check. I could check later today. But if you looked at it and think it’s a typo, I suppose you should feel free to fix it.

  6. Morever -> Moreover

    Anonymous

    diff, v29, current

  7. add missing “be” in first definition

    Anonymous

    diff, v30, current

  8. Reworded statement of Lemma 2.5 (extramural maps).

    Anonymous

    diff, v31, current

  9. Delete extraneous “all” in last sentence of “far diagonal identifications” paragraph.

    Mark S Davis

    diff, v32, current

  10. Replace “Dually” by “Similarly” in the Salamander Lemma (this is not a “dual” statement in the category-theoretic sense, which context is appropriate here).

    Mark S Davis

    diff, v33, current

  11. Replace \simeq with = in the second statement of 2.10 (extramural isomorphisms) for consistency with the first statement.

    Mark S Davis

    diff, v34, current

  12. hence => that is in #IntraExtramuralIsomorphisms

    Mark S Davis

    diff, v35, current

  13. In the proof of #IntramuralIsos, replace 0 objects in the zigzag diagrams with bullets to represent an “unknown” object, since from the double complexes in the statement, we don’t know that the corresponding objects are actually 0 (and what they are is immaterial to the proof).

    Mark S Davis

    diff, v36, current

  14. Correct superscript vert -> hor in 2nd of fourth pair of isomorphisms in statement for #IntramuralIsos.

    Mark S Davis

    diff, v36, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeOct 25th 2020

    Thanks for all your work on this entry!

    Re-reading it now, I made some minor cosmetic edits to the text.

    diff, v37, current

  16. Add some missing primes in the proof that A’^{hor} = 0 in #ShortSharp3x3.

    Mark S Davis

    diff, v38, current

  17. Add braces to correct formatting in previous edit

    Mark S Davis

    diff, v38, current

  18. Change \partial to \delta in the long exact sequence display in the statement of the snake lemma, to match the rest of the statement.

    Mark S Davis

    diff, v39, current

  19. Move double quote at the start of “nxn-lemmas” inside the TeX delimiters to avoid line break after the opening quote.

    Mark S Davis

    diff, v39, current

  20. Another attempt at fixing the quotes and line breaks in Remark 3.2 (why is there no preview mode?)

    Mark S Davis

    diff, v39, current

  21. Take 3 of fixing the quotes and line breaks in Remark 3.2

    Mark S Davis

    diff, v39, current

  22. Giving up on 3.2; reverting to what it was before.

    Mark S Davis

    diff, v39, current

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2020
    • (edited Oct 26th 2020)

    It would be good to have a preview mode, but it needs somebody to go and program it.

    For the time being, one can use the Sandbox for testing/preview purposes.

  24. Replace \oplus with + in proof of #ExtramuralIso (this sum needn’t be direct).

    Mark S Davis

    diff, v40, current

  25. Insert a paragraph at the beginning of the proof of #Sharp3x3 explaining how the diagram can be augmented with coker (B -> B”), which is needed to get the extramural isomorphism from B -> B”.

    Mark S Davis

    diff, v41, current

  26. Fix a few TeX objects from the previous edit.

    Mark S Davis

    diff, v41, current

  27. Correct indices in displayed equation for the isomorphisms between homology groups (these appear to have been adapted from Bergman’s paper but the indexing scheme is different and what is currently there doesn’t match up).

    Mark S Davis

    diff, v42, current

  28. Remove period at the end of the array in the proof of #FourLemma (the period showed up as a superscript period for ν, and anyway that display doesn’t end a sentence).

    Mark S Davis

    diff, v42, current

  29. Fix a couple of typos in proof of #TheSnakeLemma:

    • ker(f){\Box} => ker(g){\Box} before 2nd display

    • as => at after 2nd display

    Mark S Davis

    diff, v43, current

  30. Fix the end of the chain of isomorphisms giving exactness at ker(g) in #TheSnakeLemma by directly using the extramural iso {}^{\Box}Y_1 \equiv 0_{\Box} = 0 (since there’s no context for an intramural iso at Y_1).

    Mark S Davis

    diff, v43, current

  31. Rewrite Remark 3.8 to clarify that one obtains in turn the connecting homomorphism between homology groups leading to the long exact sequence in homology.

    Mark S Davis

    diff, v43, current

  32. Delete an extraneous preposition in Remark 3.8.

    Mark S Davis

    diff, v43, current

  33. In the proof of #TheSnakeLemma, point out that the squares involving ker(l) and coker(r) in the first display commute; also remove the unused arrow labels p and i.

    Mark S Davis

    diff, v44, current

  34. Reword start of last paragraph of proof of #TheSnakeLemma: “to build” => “building”

    Mark S Davis

    diff, v44, current

  35. Delete reference to Jonathan Wise’s paper. The link is broken, and he isn’t posting it at his current web page; further, https://mathoverflow.net/a/7531 suggests the proof is incomplete.

    Mark S Davis

    diff, v45, current

    • CommentRowNumber36.
    • CommentAuthorIsky Mathews
    • CommentTimeSep 5th 2023

    OK so I made an nForum account just to make some edits here! There were a few times in the proof of the Sharp 3x3 Lemma (Prop 3.1) and the next that we refer to the zigzag of morphisms as being made up of intramural rather than extramural morphisms, which is fixed, along with some minor misspellings here and there.

    diff, v46, current

    • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeSep 5th 2023

    Looks good. Thanks!

    • CommentRowNumber38.
    • CommentAuthorIsky Mathews
    • CommentTimeSep 5th 2023

    I have a few other remarks which I want to make about the page but are more seriously mathematical so wanted to ask on here if they make sense before sending the edits through:

    In the proof of the Salamander Lemma, I think there are two really minor issues. In point 3, it is said that bb has to satisfy ∂ horb=0\partial^{\text{hor}} b = 0 which is true but I think what’s relevant is that it satisfies ∂ vertb=0\partial^{\text{vert}} b = 0 so that ∂ vert∂ hora=0\partial^{\text{vert}} \partial^{\text{hor}} a = 0 rather than what’s written. In point 4, isn’t the representative we are looking for b−∂ horab - \partial^{\text{hor}} a (rather than +)?

    Another point is that in the proof of both Sharp 3x3 Lemmas we show stuff like the donor of B′ horB'^{\text{hor}} appearing in the zigzag of extramural maps as well as the donor of A corA^{\text{cor}}… Technically we just want the donor of B′B' and AA right? Or was this just to sort of informally point out that in our argument we will use the face that, for example, there’s an isomorphism between A vertA^{\text{vert}} and its donor?

    • CommentRowNumber39.
    • CommentAuthorUrs
    • CommentTimeSep 6th 2023
    • (edited Sep 6th 2023)

    Thanks again.

    Have now looked through the proof of the Salamander lemma and agree that there were these two glitches you point out (in item 3 and 4 of the proof).

    I went ahead and fixed them.

    Haven’t looked at the other issue that you highlight. But since you clearly know what you are talking about and if you have the energy, please feel invited to edit!

    Much appreciated.

    diff, v47, current

    • CommentRowNumber40.
    • CommentAuthorIsky Mathews
    • CommentTimeSep 6th 2023

    Added the edits discussed about the zigzags of extramural isomorphisms. Thanks for looking over what I wrote and thus helping with the first time I get my feet wet in the categorical community :) Definitely hope to be doing more categorical stuff in the future so who knows, might see more edits later down the line.

    Also might at some point add more homological algebra statements and their proofs using S-Lemma but this does seem pretty illustrative for now.

    diff, v48, current

    • CommentRowNumber41.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 7th 2023

    Thanks for joining in, Isky! Much appreciated.

    • CommentRowNumber42.
    • CommentAuthorAli Caglayan
    • CommentTimeJan 12th 2024

    Where it says “See Wise” there is a dead link that goes nowhere. Can this be fixed?

    • CommentRowNumber43.
    • CommentAuthorUrs
    • CommentTimeJan 12th 2024

    added pointer to:

    diff, v50, current

1 to 43 of 43

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