EDIT: 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? (Poincare duality means that $A_j=0$ for $j>N$, $A_0=\mathbb{C}, A_N\simeq \mathbb{C}$, and the product $A_j\times A_{N-j}\to A_N\simeq \mathbb{C}$ is a perfect pairing for any $j$.)

When is an algebra defined by generators and relations finite-dimensional and satisfies Poincaré duality?

EDIT: 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: $$A=\mathbb{C}[x_1,\dots,x_k]/\langle f_1,\dots,f_l\rangle,$$ 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 Poincaré duality? (Poincaré duality means that $A_j=0$ for $j>N$, $A_0=\mathbb{C}, A_N\simeq \mathbb{C}$, and the product $A_j\times A_{N-j}\to A_N\simeq \mathbb{C}$ is a perfect pairing for any $j$.)

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?

EDIT: 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? (Poincare duality means that $A_j=0$ for $j>N$, $A_0=\mathbb{C}, A_N\simeq \mathbb{C}$, and the product $A_j\times A_{N-j}\to A_N\simeq \mathbb{C}$ is a perfect pairing for any $j$.)