ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Wed, 06 May 2020 01:15:20 +0200Hochschild cohomology of a matrix subalgebrahttps://ask.sagemath.org/question/51290/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 $A=\Lambda_\mathbb{Q}[x,y]$?LeonWed, 06 May 2020 01:15:20 +0200https://ask.sagemath.org/question/51290/ChainComplex() runs 24 times slower than homology()https://ask.sagemath.org/question/44101/chaincomplex-runs-24-times-slower-than-homology/I load a list of matrices `bdrs` representing a chain complex, their dimensions are
{1, 21, 210, 1330, 5985, 20349, 54264, 116280, 203490, 293930, 352716, 352716, 293930, 203490, 116280, 54264, 20349, 5985, 1330, 210, 21, 1}, and the largest has density 3.91*10^-6. In total they take up 50MB of disk space. This finishes in 63sec.
When I run `chcx=ChainComplex(bdrs,base_ring=GF(2))`, it takes 7hrs20min, but `chcx.homology()` finishes in only 18min. **Why does it take so long to just store a few matrices?** At first I thought that `ChainComplex()` also does some simplifications/reductions, but `[chcx.free_module_rank(i) for i in range(0,21)]` shows the original dimensions of matrices :/.
**Is there a faster way to compute the homology** of a chain complex (over $\mathbb{Z}$ or $\mathbb{Z}_p$)?LeonSun, 28 Oct 2018 09:55:39 +0100https://ask.sagemath.org/question/44101/