Comparing injective dimensions in a short exact sequence

Question

If $0→A→B→C→0$ is an exact sequence in the category of $R$-modules ($R$ commutative having unity) with injective dimensions of $A$ and $C$ both $≤n$, is that of $B$ also $≤n$? It seems to me that $Ext^{n+1} (N,-)$ may work.

Answer 1

Yes. The short exact sequence $0\rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ yields a long exact sequence $... \rightarrow Ext^n(N,A) \rightarrow Ext^n(N,B) \rightarrow Ext^n(N,C) \rightarrow Ext^{n+1}(N,A) \rightarrow Ext^{n+1}(N,B) \rightarrow Ext^{n+1}(N,C) \rightarrow ...$

Since $Ext^{m}(N,A)$ and $Ext^{m}(N,C)$ are $0$ for all $N$ and $m>n$, so is $Ext^{m}(N,B)$.