An interesting example arises in the consideration of the $n$th symmetric power of a flat scheme morphism (such as for "directly" constructing the Hilbert scheme of $n$ points on a curve and relating it to the Picard scheme, building on one of Weil's original approaches to constructing the Jacobian of a smooth curve). More specifically, if $A$ is a flat $R$-algebra and $S_n$ denotes the $n$th symmetric group then the subalgebra $(A^{\otimes n})^{S_n}$ is $R$-flat and its formation commutes with any base change on $R$. (Thus, more globally, if $X$ is a flat projective scheme over a ring $R$ then the projective $R$-scheme ${\rm{Sym}}^n(X) := (X^n)/S_n$ is $R$-flat and its formation commutes with any base change on $R$.) A key point is that we do not impose any "lazy" hypothesis concerning the size of $S_n$ being a unit in $R$.

To see such properties for the subalgebra of symmetric tensors, one forgets about the algebra structure and aims to show more generally that if $M$ is any flat $R$-module then $(M^{\otimes n})^{S_n}$ is $R$-flat and its formation commutes (via the evident map) with any base change on $R$. This module problem is compatible with direct limits in $M$, so by the Lazard theorem we are reduced to the case when $M$ is finite free, which in turn is clear by inspection! See pp. 252-254 in "Neron Models" for a discussion (with references) for the application to Hilbert schemes of curves.

Another place where the symmetric power of algebras arises is in the construction of the relative Verscheibung morphisms ${\rm{Ver}}_{G/S}: G^{(p)} \rightarrow G$ for any flat commutative group scheme $G \rightarrow S$ over an ${\mathbf{F}}_p$-scheme $S$, compatible with any base change on $S$. For this, one uses the $p$th symmetric power of suitable affine opens in the underlying $S$-scheme. See 4.2-4.3 in Exp. VII$_{\rm{A}}$ in SGA3 for further details (Lazard's theorem arises at the end of 4.2).

An interesting example arises in the consideration of the $n$th symmetric power of a flat scheme morphism (such as for "directly" constructing the Hilbert scheme of $n$ points on a curve and relating it to the Picard scheme, building on one of Weil's original approaches to constructing the Jacobian of a smooth curve). More specifically, if $A$ is a flat $R$-algebra and $S_n$ denotes the $n$th symmetric group then the subalgebra $(A^{\otimes n})^{S_n}$ is $R$-flat and its formation commutes with any base change on $R$. (Thus, more globally, if $X$ is a flat projective scheme over a ring $R$ then the projective $R$-scheme ${\rm{Sym}}^n(X) := (X^n)/S_n$ is $R$-flat and its formation commutes with any base change on $R$.) A key point is that we do not impose any "lazy" hypothesis concerning the size of $S_n$ being a unit in $R$.

To see such properties for the subalgebra of symmetric tensors, one forgets about the algebra structure and aims to show more generally that if $M$ is any flat $R$-module then $(M^{\otimes n})^{S_n}$ is $R$-flat and its formation commutes (via the evident map) with any base change on $R$. This module problem is compatible with direct limits in $M$, so by the Lazard theorem we are reduced to the case when $M$ is finite free, which in turn is clear by inspection! See pp. 252-254 in "Neron Models" for a discussion (with references) for the application to Hilbert schemes of curves. In a similar spirit, if $M$ is a flat $A$-module then its symmetric and exterior powers are $A$-flat (as we see by using Lazard's theorem to pass to the direct limit on the elementary case of finite free modules).

Another place where the symmetric power of algebras arises is in the construction of the relative Verscheibung morphisms ${\rm{Ver}}_{G/S}: G^{(p)} \rightarrow G$ for any flat commutative group scheme $G \rightarrow S$ over an ${\mathbf{F}}_p$-scheme $S$, compatible with any base change on $S$. For this, one uses the $p$th symmetric power of suitable affine opens in the underlying $S$-scheme. See 4.2-4.3 in Exp. VII$_{\rm{A}}$ in SGA3 for further details (Lazard's theorem arises at the end of 4.2).

For yet another application, if $A \rightarrow B$ is a flat map of rings and $I$ is an ideal of $A$ equipped with a divided power structure $\gamma$ then the divided power structure extends (visibly uniquely) to $IB$. The proof involves reducing the existence (or rather, well-definedness) problem to one in which $B$ intervenes only through its underlying $A$-module structure, so one can use Lazard's theorem to reduce the newly formulated problem to the case of finite free modules, where the necessary computations are easy.