# Whether Morita equivalence holds the following properties?

Let $A,B$ be two K-algebras over a field K.

1. $A$ and $B$ are said to be $Morita$ $equivalent$ if the category $Mod A$ and $Mod B$ are equivalent.
2. $A$ and $B$ are said to be $derived$ $equivalent$ if $\mathcal{D}^b(Mod A)$ and $\mathcal{D}^b(Mod B)$ are equivalent as triangulated categories.
3. Given a minimal injective resolution of $A$ as an $A$-module$$0 \rightarrow A \rightarrow I_0 \rightarrow I_1 \rightarrow \dots$$ If n is maximal with the property that all modules $I_j$ are projective for $j<n$, then n is called the $dominant$ $dimension$ of $A$.

The following are my questions：

1) Is there any relation between $Morita$ $equivalent$ and $derived$ $equivalent$? (I think $A$ and $B$ are $Morita$ $equivalent$ can induces $A$ and $B$ are $derived$ $equivalent$. Is it right? Conversely, if $A$ and $B$ are $derived$ $equivalent$, when $A$ and $B$ are $Morita$ $equivalent$?)

2) If $A$ and $B$ are $Morita$ $equivalent$, someone tells me that the dominant dimensions of $A$ and $B$ are also equal. But I don't know the reason. Who can tell me? Thank you for any help.

• @Mare For question 2), I know that Morita equivalence holds these properties. But given a minimal injective resolution of $A$, can you make sure the equivalence maps $A$ to $B$? – Xiaosong Peng Oct 22 '16 at 13:16