Finite generated algebra cohomology

asked 2018-03-20 21:16:16 +0200

SahDoum gravatar image

I have an algebra and a differential over this algebra. I want to construct cohomology ring and find the annihilators of some elements. How could I do that with sage?

My algebra looks like:

variables =  ', '.join(['u{}'.format(i+1) for i in range(n)] + ['v{}'.format(j+1) for j in range(n)])
F = FreeAlgebra(ZZ, n+n, variables)
gens = F.gens()
u = gens[:n]
v = gens[n:]
print u
print v

I = []

# Koszul
for i in range(5):
    for j in range(i+1, 5):
        I.append(u[i]*u[j] - u[j]*u[i])
# Stanley-Raysner
for i in range(5):
    for j in range(i+1, 5):
        if j - i != 1 and j-i != 4:
            I.append(v[i]*v[j])

A = F.quotient(F * I * F)
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Comments

So in the first line above, n=5, to avoid the immediate crash. One can also define F over 

B.<u1,u2,u3,u4,u5> = PolynomialRing(ZZ)

then add freely the v-variables. Now we only need to know the graded pieces (the weighting of the variables) and the differential to isolate the $\mathbb Z$-modules in each degree. (Sage may do the job only for some first degrees.)

dan_fulea gravatar imagedan_fulea ( 2018-03-21 19:14:57 +0200 )edit
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