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How to compute k-span iniside a k-algebra

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Juan Diego Rojas

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Sep 7, 2024, 6:44:06 PM9/7/24
to Macaulay2
Say I have B as quotient of a polynomial ring A over QQ. How do I compute the dimension QQ-span of some elements in B? Namely how do I tell Macaulay2 to compute using the underlying field?

Trevor Karn

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Sep 8, 2024, 10:11:25 PM9/8/24
to Macaulay2
If they are of different degrees, then they are automatically linearly independent over the field, so you may assume that your elements are all of the same degree.

>> restart
>> K = ZZ/32003
>> 
>> R = K[x,y,z,w]
>> I = ideal(x-y,w-z)
>> B = R/I
>> 
>> someEltsInB = {x,y,z,w}
>> degOfEltsInB = 1
>> 
>> J = ideal(someEltsInB)
>> 
>> M = module(J)
>> 
>> leadCoefficient hilbertSeries(M, Order=>degOfEltsInB + 1)

I'm sure there are better ways to do this but this works.

Trevor Karn

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Sep 9, 2024, 10:27:06 AM9/9/24
to Macaulay2
Sorry - "If they are of different degrees, then they are automatically linearly independent over the field, so you may assume that your elements are all of the same degree." is only true when you are modding out by a homogenous ideal. Hopefully that is the case you are in.
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