ScalingStacks

Example 4.5.4.

Free modules are flat. In particular, if \(k\) is a field, all \(Z\)-graded \(k\)-vector spaces are flat, and if \(k\) is an ordinary commutative ring and \(n\geq 0\), the polynomial algebra \(k[x_1,\ldots, x_n]\) is flat, and hence it is also flat if considered as a \(\mathbb{Z}\)-graded \(k\)-algebra with generators \(x_i\) in some degree \(n_i \in \mathbb{Z}\).