Hochschild cohomology of a matrix subalgebra
Let Mn(R) be the n×n matrix algebra over the polynomial ring R=K[t]. It is spanned by eij;i,j∈[n].
Let A be a subalgebra of Mn(R), spanned by pijeij, where (i,j) ranges over some subset of [n]×[n] and pij∈R are some polynomials. How can I compute with SageMath the hochschild (co)homology HH∗(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 Q or over A=ΛQ[x,y]?