I read about Cohomology ring of finite-dimensional Grassmanian, and author used algebraic lemma as "every algebraist know about it, it is very easy". I've tried to prove it myself, search through the Internet, ask a few algebraist from my faculty, but they don't know about this statement. Lemma is the following: Let $\mathbb{k}[x_1,\dots,x_m]$ be a polynomial ring over a field, $\deg x_i = d_i$. Let $\theta_1, \dots, \theta_m$ be homogeneous elements of degree $\deg\theta_i = s_i$ such that $\mathbb{k}[x_1,\dots,x_m]/(\theta_1,\dots,\theta_m)$ is finite dimensional vector space. Then $\mathbb{k}[x_1,\dots,x_m]/(\theta_1,\dots,\theta_m)$ is Poincaré algebra of formal dimension $\sum\limits_{i=1}^{m}(s_i-d_i)$. (author uses definition of Poincaré algebra of formal dimension $d$ for graded algebra, such that it has exactly $d$ nonzero homogeneous components, $d$-th component is one-dimensional and there is non-degenerate pairing similar to Poincaré duality in topology).
Actually, I think I can prove fact about dimension using Poincaré series and Koszul resolution, but how can I prove the last facts?