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nLab > Latest Changes: derived category

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeNov 11th 2009
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Author: Urs
Format: Markdownadded reference to [[derived category]]

added reference to derived category
- CommentRowNumber2.
- CommentAuthoradeelkh
- CommentTimeMar 23rd 2013
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Author: adeelkh
Format: MarkdownI added the original reference to Verdier's thesis (also created a page for him), and some references for applications of derived categories in algebraic geometry. At some point I plan to make a separate page for this and I will then move these references there.

I added the original reference to Verdier's thesis (also created a page for him), and some references for applications of derived categories in algebraic geometry. At some point I plan to make a separate page for this and I will then move these references there.
- CommentRowNumber3.
- CommentAuthoradeelkh
- CommentTimeJan 17th 2015
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Author: adeelkh
Format: MarkdownItexSome light editing to the introductions of [[derived category]] and [[homological algebra]].

Some light editing to the introductions of derived category and homological algebra.
- CommentRowNumber4.
- CommentAuthorDmitri Pavlov
- CommentTimeApr 27th 2020
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Author: Dmitri Pavlov
Format: MarkdownItexRedirects. <a href="https://ncatlab.org/nlab/revision/diff/derived+category/28">diff</a>, <a href="https://ncatlab.org/nlab/revision/derived+category/28">v28</a>, <a href="https://ncatlab.org/nlab/show/derived+category">current</a>

Redirects.

diff, v28, current
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeApr 7th 2023
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Author: Urs
Format: MarkdownItexThe first paragraph of the Idea-section used to claim that > By the [[stable Dold-Kan correspondence]], [the derived category of an abelian category] may be viewed as a linearization of the [[stable (infinity,1)-category of spectra]]. Since this didn't quite make sense as stated(?), I have adjusted this as follows: > In the case that $\mathcal{A} \simeq$ [[Mod|$R Mod$]] (cf. the [[Freyd-Mitchell embedding theorem]]), the [[stable Dold-Kan correspondence]] says that the derived $\infty$-category of $\mathcal{A}$ is equivalently the [[stable (infinity,1)-category of spectra|stable $\infty$-category of]] [[Eilenberg-MacLane spectrum|$H R$]]-[[module spectra]]. <a href="https://ncatlab.org/nlab/revision/diff/derived+category/31">diff</a>, <a href="https://ncatlab.org/nlab/revision/derived+category/31">v31</a>, <a href="https://ncatlab.org/nlab/show/derived+category">current</a>

The first paragraph of the Idea-section used to claim that

By the stable Dold-Kan correspondence, [the derived category of an abelian category] may be viewed as a linearization of the stable (infinity,1)-category of spectra.

Since this didn’t quite make sense as stated(?), I have adjusted this as follows:

In the case that $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>𝒜</mi><mo>≃</mo></mrow><annotation encoding="application/x-tex">\mathcal{A} \simeq</annotation></semantics></math>$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math>$ (cf. the Freyd-Mitchell embedding theorem), the stable Dold-Kan correspondence says that the derived $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>$ -category of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>$ is equivalently the stable $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>$ -category of $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">H R</annotation></semantics></math>$ -module spectra.

diff, v31, current
- CommentRowNumber6.
- CommentAuthorUrs
- CommentTimeApr 7th 2023
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Author: Urs
Format: MarkdownItexadded pointer to: * [[Sergei Gelfand]], [[Yuri Manin]], Section III of: *[[Methods of homological algebra]]*, transl. from the 1988 Russian (Nauka Publ.) original, Springer (1996, 2002) [[doi:10.1007/978-3-662-12492-5](https://doi.org/10.1007/978-3-662-12492-5)] <a href="https://ncatlab.org/nlab/revision/diff/derived+category/31">diff</a>, <a href="https://ncatlab.org/nlab/revision/derived+category/31">v31</a>, <a href="https://ncatlab.org/nlab/show/derived+category">current</a>
added pointer to:
- Sergei Gelfand, Yuri Manin, Section III of: Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original, Springer (1996, 2002) [doi:10.1007/978-3-662-12492-5]
diff, v31, current
- CommentRowNumber7.
- CommentAuthorperezl.alonso
- CommentTimeJan 31st 2024
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Author: perezl.alonso
Format: MarkdownItexfixed reference: * Theo Bühler, _An introduction to the derived category_ ([pdf](https://ncatlab.org/nlab/files/intro-derived.PDF)). (I uploaded the file given that the previous link was broken, I'm however unsure whether this is technically allowed, or whether I should only redirect to [here](https://web.archive.org/web/20150701090745/http://www.uni-math.gwdg.de/theo/intro-derived.pdf) instead) <a href="https://ncatlab.org/nlab/revision/diff/derived+category/33">diff</a>, <a href="https://ncatlab.org/nlab/revision/derived+category/33">v33</a>, <a href="https://ncatlab.org/nlab/show/derived+category">current</a>
fixed reference:
- Theo Bühler, An introduction to the derived category (pdf).
(I uploaded the file given that the previous link was broken, I’m however unsure whether this is technically allowed, or whether I should only redirect to here instead)

diff, v33, current
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeJan 31st 2024
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Author: Urs
Format: MarkdownItexGood, thanks. Uploading files worth retaining is better than pointing to external repositories for which we have no guarantee that they will last.

Good, thanks.

Uploading files worth retaining is better than pointing to external repositories for which we have no guarantee that they will last.
- CommentRowNumber9.
- CommentAuthorlinuxmetel
- CommentTimeMar 1st 2025
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Author: linuxmetel
Format: MarkdownItexfix link <a href="https://ncatlab.org/nlab/revision/diff/derived+category/37">diff</a>, <a href="https://ncatlab.org/nlab/revision/derived+category/37">v37</a>, <a href="https://ncatlab.org/nlab/show/derived+category">current</a>

fix link

diff, v37, current

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nLab > Latest Changes: derived category