Let $K$ be a field and consider the polynomial ring $K[x_1,x_2,\cdots, x_n]$ in $n$ variables. Consider the ideal $(f_1,f_2, \cdots, f_n)$, then the set $$V(f_1,f_2,\cdots, d_n)=\{x \in K^n| f_1(x)=\cdots=f_n(x)=0 \} \subset K^n$$ defines a variety of the polynomials $f_i, \ i=1(1)n$, where $x=(x_1,x_2,\cdots, x_n)$.
My question-
What happened to the case $K[[x_1,x_2,\cdots, x_n]]$ of formal power series ie.,e when each $f_1, f_2,\cdots , f_n$ are formal power series ?
Do we still call the new set a variety ?