Some results on the structure of certain types of modules over regular local rings. These types of results and much more can be found in [Huneke-Sharp], [Lyubeznik], [Lyubeznik2].
Proof.
Choose a set $J$ and an isomorphism $M[\mathfrak m] \to \bigoplus _{j \in J} k$. Since $\bigoplus _{j \in J} E$ is injective (Dualizing Complexes, Lemma 47.3.7) we can extend this isomorphism to an $A$-module homomorphism $\varphi : M \to \bigoplus _{j \in J} E$. We claim that $\varphi $ is an isomorphism, i.e., bijective.
Injective. Let $z \in M$ be nonzero. Since $M$ is $\mathfrak m$-power torsion we can choose an element $f \in A$ such that $fz \in M[\mathfrak m]$ and $fz \not= 0$. Then $\varphi (fz) = f\varphi (z)$ is nonzero, hence $\varphi (z)$ is nonzero.
Surjective. Let $z \in M$. Then $x_1^ n z = 0$ for some $n \geq 0$. We will prove that $z \in x_1M$ by induction on $n$. If $n = 0$, then $z = 0$ and the result is true. If $n > 0$, then applying $D_1$ we find $0 = n x_1^{n - 1} z + x_1^ nD_1(z)$. Hence $x_1^{n - 1}(nz + x_1D_1(z)) = 0$. By induction we get $nz + x_1D_1(z) \in x_1M$. Since $n$ is invertible, we conclude $z \in x_1M$. Thus we see that $M$ is $x_1$-divisible. If $\varphi $ is not surjective, then we can choose $e \in \bigoplus _{j \in J} E$ not in $M$. Arguing as above we may assume $\mathfrak m e \subset M$, in particular $x_1 e \in M$. There exists an element $z_1 \in M$ with $x_1 z_1 = x_1 e$. Hence $x_1(z_1 - e) = 0$. Replacing $e$ by $e - z_1$ we may assume $e$ is annihilated by $x_1$. Thus it suffices to prove that
\[ \varphi [x_1] : M[x_1] \longrightarrow \left(\bigoplus \nolimits _{j \in J} E\right)[x_1] = \bigoplus \nolimits _{j \in J} E[x_1] \]
is surjective. If $d = 1$, this is true by construction of $\varphi $. If $d > 1$, then we observe that $E[x_1]$ is the injective hull of the residue field of $k[[x_2, \ldots , x_ d]]$, see Dualizing Complexes, Lemma 47.7.1. Observe that $M[x_1]$ as a module over $k[[x_2, \ldots , x_ d]]$ is $\mathfrak m/(x_1)$-power torsion and comes equipped with operators $D_2, \ldots , D_ d$ satisfying the displayed Leibniz rule. Thus by induction on $d$ we conclude that $\varphi [x_1]$ is surjective as desired.
$\square$
Proof.
Choose a set $J$ and an $A$-module homorphism $\varphi : M \to \bigoplus _{j \in J} E$ which maps $M[\mathfrak m]$ isomorphically onto $(\bigoplus _{j \in J} E)[\mathfrak m] = \bigoplus _{j \in J} k$. We claim that $\varphi $ is an isomorphism, i.e., bijective.
Injective. Let $z \in M$ be nonzero. Since $M$ is $\mathfrak m$-power torsion we can choose an element $f \in A$ such that $fz \in M[\mathfrak m]$ and $fz \not= 0$. Then $\varphi (fz) = f\varphi (z)$ is nonzero, hence $\varphi (z)$ is nonzero.
Surjective. Recall that $F$ is flat, see Lemma 51.17.6. Let $x_1, \ldots , x_ d$ be a minimal system of generators of $\mathfrak m$. Denote
\[ M_ n = M[x_1^{p^ n}, \ldots , x_ d^{p^ n}] \]
the submodule of $M$ consisting of elements killed by $x_1^{p^ n}, \ldots , x_ d^{p^ n}$. So $M_0 = M[\mathfrak m]$ is a vector space over $k$. Also $M = \bigcup M_ n$ by our assumption that $M$ is $\mathfrak m$-power torsion. Since $F^ n$ is flat and $F^ n(x_ i) = x_ i^{p^ n}$ we have
\[ M_ n \cong (M \otimes _{A, F^ n} A)[x_1^{p^ n}, \ldots , x_ d^{p^ n}] = M[x_1, \ldots , x_ d] \otimes _{A, F^ n} A = M_0 \otimes _ k A/(x_1^{p^ n}, \ldots , x_ d^{p^ n}) \]
Thus $M_ n$ is free over $A/(x_1^{p^ n}, \ldots , x_ d^{p^ n})$. A computation shows that every element of $A/(x_1^{p^ n}, \ldots , x_ d^{p^ n})$ annihilated by $x_1^{p^ n - 1}$ is divisible by $x_1$; for example you can use that $A/(x_1^{p^ n}, \ldots , x_ d^{p^ n}) \cong k[x_1, \ldots , x_ d]/(x_1^{p^ n}, \ldots , x_ d^{p^ n})$ by Algebra, Lemma 10.160.10. Thus the same is true for every element of $M_ n$. Since every element of $M$ is in $M_ n$ for all $n \gg 0$ and since every element of $M$ is killed by some power of $x_1$, we conclude that $M$ is $x_1$-divisible.
Let $x = x_1$. Above we have seen that $M$ is $x$-divisible. If $\varphi $ is not surjective, then we can choose $e \in \bigoplus _{j \in J} E$ not in $M$. Arguing as above we may assume $\mathfrak m e \subset M$, in particular $x e \in M$. There exists an element $z_1 \in M$ with $x z_1 = x e$. Hence $x(z_1 - e) = 0$. Replacing $e$ by $e - z_1$ we may assume $e$ is annihilated by $x$. Thus it suffices to prove that
\[ \varphi [x] : M[x] \longrightarrow \left(\bigoplus \nolimits _{j \in J} E\right)[x] = \bigoplus \nolimits _{j \in J} E[x] \]
is surjective. If $d = 1$, this is true by construction of $\varphi $. If $d > 1$, then we observe that $E[x]$ is the injective hull of the residue field of the regular ring $A/xA$, see Dualizing Complexes, Lemma 47.7.1. Observe that $M[x]$ as a module over $A/xA$ is $\mathfrak m/(x)$-power torsion and we have
\begin{align*} M[x] \otimes _{A/xA, F} A/xA & = M[x] \otimes _{A, F} A \otimes _ A A/xA \\ & = (M \otimes _{A, F} A)[x^ p] \otimes _ A A/xA \\ & \cong M[x^ p] \otimes _ A A/xA \end{align*}
Argue using flatness of $F$ as before. We claim that $M[x^ p] \otimes _ A A/xA \to M[x]$, $z \otimes 1 \mapsto x^{p - 1}z$ is an isomorphism. This can be seen by proving it for each of the modules $M_ n$, $n > 0$ defined above where it follows by the same result for $A/(x_1^{p^ n}, \ldots , x_ d^{p^ n})$ and $x = x_1$. Thus by induction on $\dim (A)$ we conclude that $\varphi [x]$ is surjective as desired.
$\square$
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