# Hochschild cohomology of a matrix subalgebra

Let $M_n(R)$ be the $n\times n$ matrix algebra over the polynomial ring $R=K[t]$. It is spanned by ${e_{ij}; i,j\in[n]}$.

Let $A$ be a subalgebra of $M_n(R)$, spanned by ${p_{ij}e_{ij}}$, where $(i,j)$ ranges over some subset of $[n]\times[n]$ and $p_{ij}\in R$ are some polynomials. How can I compute with SageMath the hochschild (co)homology $HH^\ast(A;A)$?

So far, I know how to do some basics:

R.<t>=GF(3)[];
A.<x,y> = ExteriorAlgebra(QQ);
C = A.hochschild_complex(A); print(type(A),'\n',type(C))
show(C.homology(0),', ',C.homology(1),', ',C.homology(2))
show(C.cohomology(0),', ',C.cohomology(1),', ',C.cohomology(2))


However, I don't know how to create my matrix subalgebra $A$ over $R$. Also, I don't think this really computes hochschild cohomology. Is it dualised over $\mathbb{Q}$ or over $A=\Lambda_\mathbb{Q}[x,y]$?

edit retag close merge delete