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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 29th 2017
   • (edited Sep 29th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI gave _[[Koszul complex]]_ and Idea-section and stated two key [Properties](https://ncatlab.org/nlab/show/Koszul+complex#Properties) in citable form (but without proof), one of them the statement that a sufficient condition for the Koszul complex to be a resolution of $R/(x_1, \cdots, x_n)$ is that $R$ is Noetherian, the $x_i$ are in the Jacobson radical, and the cohomology in degree -1 vanishes. Finally I stated the special case of this ([here](https://ncatlab.org/nlab/show/Koszul+complex#KoszulComplexForFormalPowerSeriesAlgebras)) where $R$ is a formal power series algebra over a field and the elements $x_i$ are formal power series with vanishing constant term. (I have added the relevant facts as citable numbered examples at _[[Noetherian ring]]_ and at _[[Jacobson radical]]_.) This happens to be the case that one need in [[BV-formalism]] in [[field theory]]. I am writing this out now at _[[A first idea of quantum field theory]]_ ([here](https://ncatlab.org/nlab/show/geometry+of+physics+--+A+first+idea+of+quantum+field+theory#GaugeSymmetries)).

   I gave Koszul complex and Idea-section and stated two key Properties in citable form (but without proof), one of them the statement that a sufficient condition for the Koszul complex to be a resolution of $R/(x_1, \cdots, x_n)$ is that $R$ is Noetherian, the $x_i$ are in the Jacobson radical, and the cohomology in degree -1 vanishes.

   Finally I stated the special case of this (here) where $R$ is a formal power series algebra over a field and the elements $x_i$ are formal power series with vanishing constant term.

   (I have added the relevant facts as citable numbered examples at Noetherian ring and at Jacobson radical.)

   This happens to be the case that one need in BV-formalism in field theory. I am writing this out now at A first idea of quantum field theory (here).

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeOct 12th 2017
   • (edited Oct 12th 2017)
   • PermaLink
   Author: Urs
   Format: MarkdownItexDoes [this fact](https://ncatlab.org/nlab/show/Koszul+complex#KoszulResolutionForNoetherianRngAndElementsInJacobson) remain true for _graded-commutative_ rings: *** Let $R$ be a [[commutative ring]] and $(x_1, \cdots, x_d)$ a [[sequence]] of elements in $R$, such that 1. $R$ is [[Noetherian ring|Noetherian]]; 1. each $x_i$ is contained in the [[Jacobson radical]] of $R$ then the following are equivalent: 1. the [[cochain cohomology]] of the Koszul complex $K(x_1, \cdots, x_d)$ vanishes in degree $-1$; 1. the Koszul complex $K(x_1, \cdots, x_d)$ is a [[free resolution]] of the [[quotient ring]] $R/(x_1, \cdots, x_d)$, hence its [[cochain cohomology]] vanishes in all degrees $\leq -1$; *** Does this remain true for $R$ a $\mathbb{Z}$-graded-commutative ring?

   Does this fact remain true for graded-commutative rings:

   ---

   Let $R$ be a commutative ring and $(x_1, \cdots, x_d)$ a sequence of elements in $R$, such that

   1. $R$ is Noetherian;

   2. each $x_i$ is contained in the Jacobson radical of $R$

   then the following are equivalent:

   1. the cochain cohomology of the Koszul complex $K(x_1, \cdots, x_d)$ vanishes in degree $-1$;

   2. the Koszul complex $K(x_1, \cdots, x_d)$ is a free resolution of the quotient ring $R/(x_1, \cdots, x_d)$, hence its cochain cohomology vanishes in all degrees $\leq -1$;

   ---

   Does this remain true for $R$ a $\mathbb{Z}$-graded-commutative ring?