There is an equivalence between the category of commutative finite dimensional Frobenius algebras and 2 dimensional topological quantum field theories, see for example the book by Joachim Kock, which I read a bit (but I have no idea about the topological part).
Do module theoretic notions have a topological interpretation using this equivalence? For example an open problem for commutative finite dimensional Frobenius algebras A is wheter $Ext_A^1(M,M) \neq 0$ for any non-projective module $M$. Does this have a topological interpretation?
Given a commutative f.d. Frobenius algebra $A$, then its trivial extension algebra $T(A)$ (see for example https://math.stackexchange.com/questions/229412/trivial-extension-of-an-algebra for the definition) is again a commutative Frobenius algebra with twice the dimension of $A$. What is the topological interpretation of the trivial extension construction using the equivalence?
For two f.d. commutative Frobenius algebras $A,B$ one has $rad(A \otimes B)=rad(A) \otimes B + A \otimes rad(B)$ (here rad dentotes taking the Jacobson radical of an algebra). This reminds me of the formula for the boundary of the product of two topological spaces. Is this just random, or is there a connection between those two formulas?