Given $n \in \mathbb{N}$, there exists an algebra $A_n$ such that $\text{Findim}(A_n) = n - 1$ but $r.\text{gl.dim}(A_n) = \infty$.

Consider the following quiver algebra $A_n$:

with the relation that the composition of any two arrows is zero, i.e., $r^2 = 0$.

Clearly, $A_n$ is a monomial algebra. According to the paper Finitistic dimensions of finite dimensional monomial algebras, it follows that $\text{Findim}(A_n) = n-1$. Moreover, it is straightforward to compute that $r.\text{gl.dim}(A_n) = \infty$.

Therefore, we have $\text{Findim}(A_n) = n-1 < \infty = r.\text{gl.dim}(A_n)$ for $n \geq 1$.

Specifically, when $n=1$, $A_1 \cong k[t]/(t^2)$, so $\text{Findim}(k[t]/(t^2)) = \text{Findim}(A_1) = 1-1 = 0$.

Given $n \in \mathbb{N}$, there exists an algebra $A_n$ such that $\text{Findim}(A_n) = n - 1$ but $r.\text{gl.dim}(A_n) = \infty$.

Consider the following quiver algebra $A_n$: with the relation that the composition of any two arrows is zero, i.e., $r^2 = 0$.

Clearly, $A_n$ is a monomial algebra. According to the paper Finitistic dimensions of finite dimensional monomial algebras, it follows that $\text{Findim}(A_n) = n-1$. Moreover, it is straightforward to compute that $r.\text{gl.dim}(A_n) = \infty$.

Therefore, we have $\text{Findim}(A_n) = n-1 < \infty = r.\text{gl.dim}(A_n)$ for $n \geq 1$.

Specifically, when $n=1$, $A_1 \cong k[t]/(t^2)$, so $\text{Findim}(k[t]/(t^2)) = \text{Findim}(A_1) = 1-1 = 0$.

Given $n \in \mathbb{N}$, there exists an algebra $A_n$ such that $\text{Findim}(A_n) = n - 1$ but $r.\text{gl.dim}(A_n) = \infty$.

Consider the following quiver algebra $A_n$:

with the relation that the composition of any two arrows is zero, i.e., $r^2 = 0$.

Clearly, $A_n$ is a monomial algebra. According to the paper Finitistic dimensions of finite dimensional monomial algebras, it follows that $\text{Findim}(A_n) = n-1$. Moreover, it is straightforward to compute that $r.\text{gl.dim}(A_n) = \infty$.

Therefore, we have $\text{Findim}(A_n) = n-1 < \infty = r.\text{gl.dim}(A_n)$ for $n \geq 1$.

Given $n \in \mathbb{N}$, there exists an algebra $A_n$ such that $\text{Findim}(A_n) = n - 1$ but $r.\text{gl.dim}(A_n) = \infty$.

Consider the following quiver algebra $A_n$:

with the relation that the composition of any two arrows is zero, i.e., $r^2 = 0$.

Clearly, $A_n$ is a monomial algebra. According to the paper Finitistic dimensions of finite dimensional monomial algebras, it follows that $\text{Findim}(A_n) = n-1$. Moreover, it is straightforward to compute that $r.\text{gl.dim}(A_n) = \infty$.

Therefore, we have $\text{Findim}(A_n) = n-1 < \infty = r.\text{gl.dim}(A_n)$ for $n \geq 1$.

Specifically, when $n=1$, $A_1 \cong k[t]/(t^2)$, so $\text{Findim}(k[t]/(t^2)) = \text{Findim}(A_1) = 1-1 = 0$.