The quotient of an algebra by an ideal is an algebra again. If we deform the ideal, the quotient might or might not be a flat algebra deformation of the original quotient, depending on whether conditions are met. What is the most common way to formulate such conditions?
Starting point Let $ A $ be an algebra over $ k $ and $ I ⊂ A $ an ideal. Assume $ (B, \mathfrak{m}) $ is a local noetherian unital $ k $-algebra with residue field $ B/\mathfrak{m} = k $. Let $ I_q $ be deformation of $ I $ in the sense that $ I_q + \mathfrak{m} A = I + \mathfrak{m} A $. That is, $ I_q $ is the same as $ I $ up to infinitesimal terms.
Definition Let us say $ (B \hat{\otimes}_k A) / I_q $ is an algebra deformation of $ A/I $ if there exists a deformation $ *_q $ of the natural product on $ A/I $ such that there is a $ B $-linear algebra isomorphism $$ (B \hat{\otimes}_k A) / I_q \xrightarrow{\sim} (B \hat{\otimes}_k (A/I), *_q). $$
Criterion I can show that $ (B \hat{\otimes}_k A) / I_q $ is an algebra deformation of $ A/I $ if and only if
- $ I_q $ is pseudoclosed: The image of the multiplication map $ B \hat{\otimes}_k I_q → B \hat{\otimes}_k A $ lies in $ I_q $.
- $ I_q $ is flat: We have $ I_q ∩ \mathfrak{m}A ⊂ \mathfrak{m} I_q $.
Question To investigate deformed ideals in detail, I need “flatness terminology” for ideals, instead of only for algebras. The formulation of the above criterion and the terminology pseudoclosed and flat is my own, but I consider it better to follow literature if available. I'm looking for:
- correct names for pseudoclosed and flat,
- more common ways to state the criterion,
- suggestions which literature I should look at.
Thanks a lot!
Where I have searched I've googled and searched math.sx on queries like "[deformation-theory] flat ideal". I've consulted two fellow students and my advisor. I'll try to find out whether Martin Markl's book is of help. I'm also reaching out to further network.