$\def\O{\mathcal O}\def\N{\operatorname N}\def\tr{\operatorname{Tr}}\def\Q{\mathbb Q}\def\Z{\mathbb Z}$Let $K$ be a number field and $\O_K$ be its ring of integers. The norm $\N(I)$ of an ideal $I\neq 0$ of $\O_K$ is defined by $N(I)=|\O_K/I|=[\O_K\colon I]$. In Tian's note of algebraic number theory, I find two propositions (Proposition 3.12 and 3.14 in page 27 and 28) of norms difficult to understand, especially their proofs, which state as follows.
Proposition 3.12 If $I=(x)$ for some $x\in\O_K$, then $\N(I)=|\N_{K/\Q}(x)|$. Proof. Let $(α_1,\cdots,α_n)$ be a $\Z-basis$ of $\O_K$. There exists a matrix $C\in M_{n\times n}(\Z)$ such that $$(xα_1,\cdots,xα_n)=(α_1,\cdots,α_n)C.$$ It follows that $$\N(I)=[\O_K\colon I]=[\sum_i\Zα_i\colon\sum_i\Z xα_i]\color{red}{=}|\det(C)|.$$
And
Proposition 3.14 We have $N(\delta_K)=|\Delta_K|$. Proof. Let $(α_1,\cdots,α_n)$ be a $\Z$-basis of $\O_K$, and $(α_1^\vee,\cdots,α_n^\vee)$ be its dual nasis w.r.t $\tr_{K/\Q}$. Then $\delta_K^{-1}=\oplus_i\Zα_i^\vee$ and $α_i=\sum_j\tr_{K/\Q}(α_iα_j)a_j\vee$. Therefore $$|\Delta_K|=|\det(\tr_{K/\Q}(α_iα_j))|\color{red}{=}[\bigoplus_i\Zα_i^\vee\colon\bigoplus_i\Zα_i]=[\delta_K^{-1}\colon\O_K]\color{red}{=}[\O_K\colon\delta_K]=\N(\delta_K).$$
Here the absolute different $\delta_K$ of $K$ is defined by $(\delta_K^{-1})^{-1}$, where $\delta_K^{-1}=\{x\in K\mid\tr_{K/\Q}(xy)\in Z,\forall y\in\O_K\}$ is a fractional ideal containing $\O_K$.
My question is about these red equal signs. For the first two, it seems that $|\det(C)|=[\bigoplus_i \Z\beta_i\colon \bigoplus_i\Z\gamma_i]$ if $\beta_i, \gamma_i\in K$ are $\Z$-independent and $\bigoplus_i \Z\beta_i\supset \bigoplus_i\Z\gamma_i$ with $C$ being the transition matrix? For the last equal sign, how can I see $[\delta_K^{-1}\colon\O_K]=[\O_K\colon\delta_K]$? Thanks!