Locally free sheaves
Posted: Sun Dec 17, 2017 12:58 am
Hi I´m stuck with the following proposition, could you help me to solve it?
Suppose $0\rightarrow{\cal{E}}'\rightarrow{\cal{E}}\rightarrow{\cal{E}}''\rightarrow0$ is an exact sequence of locally free sheaves of ranks $r'\,, \, r$ and $r''$. Then \[\Lambda^{r}{\cal{E}}\cong\Lambda^{r'}{\cal{E}}'\otimes\Lambda^{r''}{\cal{E}}''\,.\]
Suppose $0\rightarrow{\cal{E}}'\rightarrow{\cal{E}}\rightarrow{\cal{E}}''\rightarrow0$ is an exact sequence of locally free sheaves of ranks $r'\,, \, r$ and $r''$. Then \[\Lambda^{r}{\cal{E}}\cong\Lambda^{r'}{\cal{E}}'\otimes\Lambda^{r''}{\cal{E}}''\,.\]