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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 $A=\Lambda_\mathbb{Q}[x,y]$?

@FrédéricC Not even that, since some tuples end in a comma, not an integer. Anyway, how can I get maximal simplices which have integers for vertices?

The code

n=2; x=simplicial_complexes.RealProjectiveSpace(n); 
fcts=x.facets(); out=file('P'+str(n)+'.txt','w');
for a in fcts: out.write(str(a)+'\n')
out.close();

writes a nice result in the file P2.txt:

(0, 2, 3)
(0, 3, 4)
(0, 1, 5)
(0, 4, 5)
(2, 3, 5)
(1, 2, 4)
(0, 1, 2)
(1, 3, 4)
(1, 3, 5)
(2, 4, 5)

This works for $n=1,\ldots,4$. However, for $n=5$, the file contains

((1, 5, 6), (1, 6), (1, 3, 5, 6), (6,), (2,), (2, 4))
((1, 2, 4, 6), (5,), (1, 4), (3, 5), (4,), (1, 4, 6))
((3, 4, 6), (1, 3, 4, 5, 6), (4, 6), (1, 3, 4, 6), (6,), (2,))
((1, 5, 6), (1, 2, 3, 4, 5, 6), (1, 3, 5, 6), (1,), (1, 5), (1, 3, 4, 5, 6))
((1, 6), (1, 3, 5, 6), (1,), (1, 2, 3, 5, 6), (1, 3, 6), (4,))
((1, 2), (3,), (3, 5, 6), (1, 2, 4), (3, 5), (2,))
((1, 2), (1,), (3,), (3, 6), (1, 2, 4), (1, 2, 4, 5))
((1, 2), (1,), (1, 2, 3, 6), (4, 5), (1, 2, 6), (4,))
((2, 3, 4), (2, 3, 4, 5), (1, 2, 3, 4, 5), (2, 3), (6,), (2,))
((1, 3, 4, 5), (1, 3, 4, 5, 6), (5,), (3, 5), (2,), (3, 4, 5))
...

What am I doing wrong here? Why don't I get a list of 5-dimensional simplices? I would like to obtain only the facets of the first 10 projective spaces.

I installed it to /home/leon/sage-8.8. How do I create a command sage that runs the program? I can't find $HOME/.bashrc in the home folder. Btw, I have Linux Mint Mate 19.1.

Now it works, thank you for your help and patience! : ) Using the above code, I've managed to load 7GB of sparse matrices without crashing (but when loaded, they take up 23GB of RAM, a bit strange).

Hmm, I still haven't been able to import the file, but probably due to my incompetence. Could you give me specific instructions on how (in what form) to export the file from Mathematica and how to import it in Sage? Do I export it to file.json? Should its content be e.g. [ "bdrs={", "1: matrix(ZZ,1,7,{}),", "2: matrix(ZZ,7,21,{(3,3):-1, (3,9):-1, (3,14):-1}),", "};" ] How do I import this into sage? If I run load('file.json'), I get No such file or directory: '/home/file.json.sobj'. If I run json.loads('file.json'), I get ValueError: No JSON object could be decoded.

Oh, that's disappointing. I thought Sage was supposed to be much more efficient than Mathematica. Even when I try to import just the largest matrix (that takes up less that 800MB), Sage uses up 28GB of RAM and crashes.

In a 5GB file.sage, I stored a chain complex as a dictionary of sparse matrices (created it in Mathematica with no problems). When I run load(file.sage), the program uses up all 64GB RAM + 64GB swap and crashes. Why does Sage use that much of memory for a small file?

I tried splitting the file into three smaller ones and load one after another, but already with the first 1.3GB file, the system crashes after using all RAM, but without using swap. I get:

---------------------------------------------------------------------------
MemoryError                               Traceback (most recent call last)
<ipython-input-1-8cecfac681b8> in <module>()
----> 1 load('/home/leon/file.sage');

sage/structure/sage_object.pyx in sage.structure.sage_object.load (build/cythonized/sage/structure/sage_object.c:12879)()

/usr/lib/python2.7/dist-packages/sage/repl/load.pyc in load(filename, globals, attach)
    245             if attach:
    246                 add_attached_file(fpath)
--> 247             exec(preparse_file(open(fpath).read()) + "\n", globals)
    248     elif ext == '.spyx' or ext == '.pyx':
    249         if attach:

MemoryError:

Any ideas?

In Mathematica, the command MaxMemoryUsed[computation] tells you what the largest usage of RAM was when executing some computation. Is there a similar comand in SageMath?

Surely, ChainComplex() detects that these are sparse matrices, otherwise I wouldn't have enough RAM to store $10^6\times10^6=10^{12}$ entries... But still, thank you for your help!

Aaah, you're right, with check=False, it only takes 3sec. Sorry for not noticing this option in the documentation, and thank you for your help!

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 $\mathbb{Z}_p$)?

Yes, I'll be computing a lot of homology of chain complexes. Thank you for your answers!!

Great, it works! Could you also please answer the question about RAM, before I accept?

I have all my commands in a text file, and I would just like to run them in Sage. What is the easiest way to do this?

For example, in sage.txt, I have written:

d0=matrix(ZZ,4,4,{(0,0):1,(2,0):1,(1,1):1,(3,1):1,(1,2):1,(3,2):1});
d1=matrix(ZZ,4,4,{(1,0):-1,(2,0):-1,(3,1):-1,(1,2):1,(2,2):1,(3,3):1});

The command with open('/home/sage.txt','r') as ll: bdrs=[sage_eval(l.strip()) for l in ll]; returns

File "<string>", line 1
    d0=matrix(ZZ,Integer(4),Integer(4),{(Integer(0),Integer(0)):Integer(1),(Integer(2),Integer(0)):Integer(1),(Integer(1),Integer(1)):Integer(1),(Integer(3),Integer(1)):Integer(1),(Integer(1),Integer(2)):Integer(1),(Integer(3),Integer(2)):Integer(1)});
      ^
SyntaxError: invalid syntax

Also, does Sage have a limit on how much RAM it can use? If yes, how can I increase the available memory?