My question is not very concrete. Rather I am wondering if there is anything known in the following direction.

Assume we are given a commutative algebra $A$ (say over complex numbers) which is graded and is given by its homogeneous generators and homogeneous relations: $$A=\mathbb{C}[x_1,\dots,x_k]/<f_1,\dots,f_l>,$$ where $f_i$ are homogenous polynomials in $x_1,\dots,x_k$.

1. Are there sufficient conditions on $f_i$'s to guarantee $A$ to be finite dimensional?

2. Are there sufficient conditions on $f_i$'s to guarantee $A$ satisfies Poincare duality?