computing an intersection of submodules over different rings

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Avi Steiner

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Apr 1, 2022, 7:43:36 PMApr 1
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Let f : R -> S be a map of rings. Let M be an S-module, N an R-module, and g : N-> M a map of R-modules (with respect to f). Let L be an S-submodule of M. How do I compute a generating set for g^-1(L)?

For concreteness, here's an explicit setup in M2 code:

R = QQ[m_0..m_2]
S = QQ[x,y,z,a,b,c]
f = map(S, R, {y*c-z*b, -x*c+z*a, x*b-y*a})
M = S^1
N = R^{3:-1}
g = map(M, N, f, {{x,y,z}})
L = module ideal (x,y,z)


Best,
Avi

jche...@gmail.com

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Apr 1, 2022, 8:12:04 PMApr 1
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Hi Avi,

What about computing the kernel of g, after replacing M with M/L? (E.g. "gens ker map(M/L, N, f, {{x,y,z}})" in your example.)

Justin

Avi Steiner

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Apr 1, 2022, 8:19:16 PMApr 1
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Thanks! That works! :)
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