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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 7th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexPlease give me a sanity check: Dualizing the [[Quillen-Suslin theorem]] and passing to the graded context: would it be true that finitely generated injective graded-comodules over a cofree graded-co-commutative coalgebra are all co-free?

   Please give me a sanity check: Dualizing the Quillen-Suslin theorem and passing to the graded context: would it be true that finitely generated injective graded-comodules over a cofree graded-co-commutative coalgebra are all co-free?

2. • CommentRowNumber2.
   • CommentAuthorRichard Williamson
   • CommentTimeMar 7th 2017
   • PermaLink
   Author: Richard Williamson
   Format: MarkdownItexI am no expert, but was intrigued by the question, and looked up a few things. The graded case is actually much easier (already proven by Serre), and the proof that I glanced at (in section 4 of Lam's book _Serre's problem on projective modules_) seems like it could be written in an element free way, and hence would dualise. Writing up such a proof would seem a worthy topic for the nLab.

   I am no expert, but was intrigued by the question, and looked up a few things. The graded case is actually much easier (already proven by Serre), and the proof that I glanced at (in section 4 of Lam’s book Serre’s problem on projective modules) seems like it could be written in an element free way, and hence would dualise. Writing up such a proof would seem a worthy topic for the nLab.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 8th 2017
   • (edited Mar 8th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, Richard. By the way, the reason I ask is Positelski's [[model structure on dg-comodules]] "of the second kind". The fibrant objects are those dg-comodules whose underlying graded comodule is injective. Now if the dg-coalgebra in question is the Chevalley-Eilenberg chain algebra of an $L_\infty$-algebra $\mathfrak{g}$, then its underlying graded-cocommutative coalgebra is co-free, and so if the graded dual Qullen-Suslin theorem were true, then the Positelski fibrant comodules over it would be exactly the $L_\infty$-modules over $\mathfrak{g}$.

   Thanks, Richard.

   By the way, the reason I ask is Positelski’s model structure on dg-comodules “of the second kind”. The fibrant objects are those dg-comodules whose underlying graded comodule is injective.

   Now if the dg-coalgebra in question is the Chevalley-Eilenberg chain algebra of an $L_\infty$-algebra $\mathfrak{g}$, then its underlying graded-cocommutative coalgebra is co-free, and so if the graded dual Qullen-Suslin theorem were true, then the Positelski fibrant comodules over it would be exactly the $L_\infty$-modules over $\mathfrak{g}$.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 10th 2017
   • (edited Mar 10th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexJust to say that I found and wrote up a proof, for the version of the statement relevant for the application in #3: For $C$ a graded coalgebra of the form $\vee^\bullet \mathfrak{g}$ (i.e. one of the form underlying the Chevalley-Eilenberg coalgebras of an $L_\infty$-algebra $\mathfrak{g}$), then every injective comodule over $C$ is cofree. Thanks to Jon Pridham for hints, and thanks to Vincent Schlegel for discussion. I'll add this to the $n$Lab later.

   Just to say that I found and wrote up a proof, for the version of the statement relevant for the application in #3:

   For $C$ a graded coalgebra of the form $\vee^\bullet \mathfrak{g}$ (i.e. one of the form underlying the Chevalley-Eilenberg coalgebras of an $L_\infty$-algebra $\mathfrak{g}$), then every injective comodule over $C$ is cofree.

   Thanks to Jon Pridham for hints, and thanks to Vincent Schlegel for discussion. I’ll add this to the $n$Lab later.

5. • CommentRowNumber5.
   • CommentAuthorRichard Williamson
   • CommentTimeMar 11th 2017
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   Author: Richard Williamson
   Format: MarkdownItexGreat!

   Great!