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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 17th 2017
   • (edited Dec 17th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexFor $(V^\bullet,d)$ a cochain complex in characteristic zero, the cohomology of its graded symmetric dg-algebra should be the graded symmetric algebra on its cohomology $$ H^\bullet(Sym(V^\bullet,d)) \;\simeq\; Sym( H^\bullet(V,d) ) \,. $$ (now recorded at _[[symmetric algebra]]_: [here](https://ncatlab.org/nlab/show/symmetric+algebra#CochainCohomologyOfSymmetricAlgebraOnCochainComplex)) I just want to cite this from the literature. This must be a textbook fact (in char = 0 at least); but the only reference I find is this MO discussion [here](https://mathoverflow.net/questions/90684/how-to-compute-homology-of-symmetric-products-of-complexes/91243#91243). Anyone has a more canonical citation (for char = 0) at your fingertips?

   For $(V^\bullet,d)$ a cochain complex in characteristic zero, the cohomology of its graded symmetric dg-algebra should be the graded symmetric algebra on its cohomology

   $$H^\bullet(Sym(V^\bullet,d)) \;\simeq\; Sym( H^\bullet(V,d) ) \,.$$

   (now recorded at symmetric algebra: here)

   I just want to cite this from the literature. This must be a textbook fact (in char = 0 at least); but the only reference I find is this MO discussion here.

   Anyone has a more canonical citation (for char = 0) at your fingertips?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 17th 2017
   • (edited Dec 17th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have spelled out the proof that for finite groups in char = 0 (at least) taking invariants commutes with taking cochain cohomology: [here](https://ncatlab.org/nlab/show/invariant#TakingInvariantsForFiniteGroupCommutesWithTaingHomologyInCharZero). And made this a proof of the above fact: [there](https://ncatlab.org/nlab/show/symmetric+algebra#CochainCohomologyOfSymmetricAlgebraOnCochainComplex)

   I have spelled out the proof that for finite groups in char = 0 (at least) taking invariants commutes with taking cochain cohomology: here. And made this a proof of the above fact: there