I am reading the Eisenbud, Commutative Algebra, Proposition 21.4 and stuck at certain statement. In this question, we shall assume that all the rings considered are Noetherian.
First I arrange assoicated definitions and theorems.
Definition 1. Let $(A,P)$ be a local ring. Let $M$ be an (finitely generated) $A$-module. Then the socle of $M$, $\operatorname{soc}(M)$ is defined as $$ \operatorname{soc}(M):=\{m\in M : am=0, \forall a \in P\}$$
, or equivalently, as the sum of all the simple submodules of $M$.
Definition 2. Let $R$ be a ring and let $M \subseteq E$ be $R$-modules. We sat that $M$ is an essential submodule of $E$, or that $E$ is an essential extension of $M$ if every nonzero submodule of $E$ intersects $M$ nontrivially.
Proposition-Definition A3.10. Let $R$ be a ring.
c. There is, up to isomorphism, a unique essential extension $E$ of $M$ that is an injective $R$-module ; this is callled the injective envelope ( or injective hull ) of $M$, written $E(M)$.
Proposition-Definition 21.1. Let $(A,P)$ be a local zero-dimensional ring. If $E$ is any dualizing functor from the category of finitely generated $A$-modules to itself, then there is an isomrophism of functors $E(-) \cong \operatorname{Hom}_A(-, E(A))$. Further, $E(A)$ is isomorhpic to the injective hull of $A/P$. Thus there is up to isomorphism at most one dualizing functor.
We define the canonical module $\omega_A$ of a local zero-dimensional ring $A$ ( not necessarily containing a field ) to be the injective hull of the residue class field $A$.
Second, now I propose statement of Proposition 21.4 and write its proof.
Proposition 21.4. Let $A$ be a zero-dimensional ring. Suppose that for some local ring $B$, $A$ is a $B$-algebra that is finitely generated as a $B$-module and the maximal ideal of $B$ maps into that of $A$. If $E$ is the injective hull of the residue class field of $B$, then $$ \omega_A = \operatorname{Hom}_B(A,E)$$ In particular, if $B$ is zero dimensional, then $\omega_A = \operatorname{Hom}_B(A,\omega_B)$.
Proof. By Appendix A3.8, $\operatorname{Hom}_B(A,E)$ is an injective $A$-module. To show that it is the injective hull of the residue class field $k$ of $A$, it suffices to show that it is an essential extension of the residue class field $k$ of $A$. Let $P$ be the maximal ideal of $A$, and let $P_B$ be the maximal ideal of $B$. By hypothesis, the preimage of $P$ is $P_B$, so there is an induced homomorphism of the residue class field $k_B$ of $B$ to $k$. As $k$ is a finite dimensional vector space over $k_B$, we have $k=\omega_k \cong \operatorname{Hom}_{k_B}(k,k_B)$ as $k$-modules. ($\because$ By the introduction of section 21.1 ( refer ), $\operatorname{Hom}_{k_B}(-,k_B)$ is a dualizing functor of the category of finitely generated $k$-module. And apply the Proposition-Definition 21.1 above . Anyway we accept this statement. ) This is in fact the special case of the proposition where $A$ and $B$ ar fields.
Let $S\subseteq \operatorname{Hom}_B(A,E)$ be the $A$-submodule of homomorphisms $\varphi$ whose kernel contains $P$, or equivalenty, such that $P \varphi = 0$. ( $\divideontimes$ $\operatorname{Hom}_B(A,E)$ has $A$-module structure by $a\varphi ( a') := \varphi ( a a')$. ). The module $S$ is the socle of $\operatorname{Hom}_B(A,E)$ as an $A$-module. If $\varphi \in S$, then since $P_B A \subseteq P$, the image of $\varphi$ is annihilated by $P_B$ ; that is, the image of $\varphi$ is in the socle of $E$ as a $B$-module, and since $E$ is the injective hull of $k_B$, this is $k_B$ ( ; i.e., $\operatorname{soc}(E) := \operatorname{soc}(E(k_B)) =\operatorname{soc}(k_B) = k_B$. C.f. Are Socle of a module and it's injective hull same? ). Since the homomorphism in $S$ all factor through the projection $A\to A/P =k$, we have $S \cong \operatorname{Hom}_B(k,k_B)\cong k$.
If $\varphi : A\to E$ is any nonzero $B$-module homomorphism, then since $P$ is nilpotent ( $\because$ $A$ is zero-dimensional ; C.f. Theroem 2.12 in the Eisenbud's book ), $\varphi$ is annihilated by a power of $P$, and thus there is a multiple $a\varphi \neq 0$ that is annihilated by $P$ ( $\because$ Let $k_0 := \operatorname{min}\{k\in \mathbb{N} : P^{k}\varphi =0 \}$. Then $0 \neq a \in P^{k_0 -1}$ such that $a \varphi \neq 0$ may play such a role.) Thus $S$ is an essential $A$-submodule of $\operatorname{Hom}_B(A,E)$, as required. QED.
Q.1. I am stuck at understanding the bold statement.
Q.1-1). We proved in the first paragraph that $k\cong \operatorname{Hom}_{k_B}(k, k_B)$ as $k$-modules. $\operatorname{Hom}_B(k, k_B) \cong \operatorname{Hom}_{k_B}(k, k_B)$ is true? Or $\operatorname{Hom}_{k_B}(k, k_B)$, instead of $\operatorname{Hom}_B(k, k_B)$, is more correct notation? ( Did Eisenbed made mistake? )
Q.1-2). For the isomorhpism $S\cong \operatorname{Hom}_{B}(k, k_B) $ ( or $S\cong \operatorname{Hom}_{k_B}(k, k_B) $, if it is more correct ), does it refer to isomorphism between modules on which ring? As an $A$-module? Some confusion : We have $k \cong \operatorname{Hom}_{k_B}(k,k_B)$ as '$k$-modules'. Then it also induces an $A$-module isomorphism? If so, then $S \cong k$ as an $A$-modules and by the argument in the final paragraph of the proof, $ k \cong S$ is an essential $A$-submodule of $\operatorname{Hom}_{B}(A,E)$, as required.
Can anyone help?
