Denote by $HC(A,M)$ the Hochschild homological complex of an algebra $A$ with coefficients in an $A$-bimodule $M$, and let $B\rightarrow A$ be an $R$-flat extension of $R$-algebras, for some $CRing$ $R$.


On page 11 of this article:
<http://arxiv.org/pdf/1103.4377v2.pdf>
the author defines a filtration $G\_n^p$ of the Hochschild complex, and makes the claim that $\underset{p \in \mathbb{N}}{\varinjlim} G\_n^p$ is exactly the $n^{th}$ entry in the Hochschild complex $HC(A,M)$.


My question is short and simple.. why is this true?



Jacobi-Zariski exact sequence question
As observed in that paper, $G\_n^p=G\_n^{p+1}$ for $p\geq n$, so that the colimit, which is simply the union, is equal to $G\_n^n$. It is clear that $G\_n^n=CH\_n(\mathscr A,M)$, in the notation of the paper. This shows the claim you mention in each degree.


