I have a feeling that this may be a very easy question for some people on MO, but it isn't for me.


Take a finite pointed set $X$, with $\*$ the base-point. Build a cosimplicial set which in degree $n \ge 1$ is $X^n$ (the cartesian product of $n$ copies of $X$), and in degree $0$ is $\{ \* \}$; the cofaces are:


$d^0(x\_1, \ldots, x\_n) = (\*, x\_1, \ldots, x\_n)$


$d^i(x\_1, \ldots, x\_n) = (x\_1, \ldots, x\_i, x\_i, x\_{i+1}, \ldots, x\_n)$


$d^{n+1}(x\_1, \ldots, x\_n) = (x\_1, \ldots, x\_n, \*)$


Now apply the functor "free $k$-module on", where $k$ is your favorite ring. You get a cosimplicial $k$-module $A^\*$, so you can build the associated cochain complex where the differential is the alternating sum of the cofaces. Note that $A^n = (A^1)^{\otimes n}$. 



> 
> What is the cohomology of this complex?
> 
> 
> 


Ideally someone will say something like "this is the cobar construction, it computes the cohomology of the loop space on the discrete space $X$, so the cohomology is $0$ in positive degrees", or something close. And it would be awesome. (The buzzword "cobar construction" seems to show up a lot among the papers I've skimmed through online.)


Thank you so much for your help!


Pierre



What is the cohomology of this complex?
If I've understood well your construction, the complex $\hom\_k(A,k)$ is the Hochschild complex of the cochain $k$-algebra $C^\star(X,k)$ of the (discrete) space $X$ with coefficients in the $C^\star(X,k)$-module $k$. The $C^\star(X,k)$-module structure on $k$ is given via the augmentation $C^\star(X,k)\rightarrow k$ induced by the inclusion of the base point in $X$. Therefore


$$H\_\star\hom\_k(A,k)\cong HH\_\star(C^\star(X,k),k).$$


Since $X$ is discrete then $C^\star(X,k)=k\times\stackrel{n}\cdots\times k=k^n$ concentrated in degree $0$, where $n=|X|$ is the number of points. Now the Künneth formula shows that


$$H\_0\hom\_k(A,k)\cong HH\_0(C^\star(X,k),k)\cong k,$$


$$H\_d\hom\_k(A,k)\cong HH\_d(C^\star(X,k),k)=0,\quad d\neq 0,$$


Therefore


$$H^0A\cong k,$$


$$H^dA=0,\quad d\neq 0.$$


