2. Standard bases

Definition 2..1   We define ${\bf Loc_< K[\underline{x}]}:= S^{-1}_< K[\underline{x}]$ to be the localization of $K[\underline{x}]$ with respect to the multiplicative closed set $S_< = \{1 + g \mid g = 0$ or $g \in K[\underline{x}] \backslash \{ 0\}$ and $1 > L(g)\}$.

Remark 2..2  

1)

$K[\underline{x}] \subseteq \mbox{Loc}_< K[\underline{x}] \subseteq K[\underline{x}]_{(\underline{x})}$, where $K[\underline{x}]_{(\underline{x})}$ denotes the localization of $K[\underline{x}]$ with respect to the maximal ideal $(x_1, \ldots, x_n)$. In particular, $\mbox{Loc}_< K[\underline{x}]$ is noetherian, Loc $_< K[\underline{x}]$ is $K[\underline{x}]$-flat and $K[\underline{x}]_{(\underline{x})}$ is Loc $_< K[\underline{x}]$-flat.

2)

If < is a wellordering then x⁰ = 1 is the smallest monomial and Loc $_< K[\underline{x}] = K[\underline{x}]$ . If 1 > x_i for all i, then Loc $_< K[\underline{x}] = K[\underline{x}]_{(\underline{x})}$.

3)

If, in general, $x_1, \ldots, x_r < 1$ and $x_{r+1}, \ldots, x_n > 1$ then

$$1 + (x_1, \ldots, x_r) K[x_1, \ldots, x_r]\; \subseteq\; S_< \;\subseteq\; 1 + (x_1, \ldots, x_r) K[\underline{x}] =: S,$$

hence

$$K[x_1, \ldots, x_r]_{(x_1, \ldots, x_r)} [x_{r+1}, \ldots, x_... ...Loc}_< K[\underline{x}] \;\subseteq\; S^{-1} K[\underline{x}].$$