Transition function of tensor product of two locally free sheaves

Question

It’s almost the same question here Transition functions of sheaf tensor product and just generalize the locally free sheaf $\mathcal{L}$ and $\mathcal{M}$ on $X$ to rank $l$ and $m$, respectively. Suppose the transition functions of the two locally sheaves between two trivialization open set $U,V$ are determined by $(s_{ij}) \in GL_n(\Gamma(U\cap V), \mathcal{O}_X)$ and $(t_{kl}) \in GL_m(\Gamma(U\cap V), \mathcal{O}_X) $, respectively. Then what is the transition function for $\mathcal{L} \otimes \mathcal{M}$ between these two trivialization open sets? It should be some element in $GL_{nm}(\Gamma(U\cap V), \mathcal{O}_X)$. And I calculated the element in position row “(i,k)”, column “(j,l)” to be $s_{ij}t_{kl}$, but I failed to simply it to some simple form. Is it the Kronecker product?

Although the backgroud is from locall free sheaves, it seems totally the same discussion the corresponding automorphism for tensor product of two vector spaces and two automorphism for them repsectively. That is, given vector spaces $L,M$ and automorphism $(s_{ij})$ and $(t_{jk})$ for $L,M$, what is the corresponding automorphsim matrix for $L \otimes M$?

Thank you in advance.

Answer 1

You are right, it is exactly the Kronecker product. Recall where the transition functions come from. If $U_i$ are open subsets of $X$ on which $\mathcal L$ and $\mathcal M$ are trivial, then there are isomorphisms $s_i\colon\mathcal O_{U_i}^l\to \mathcal L|_{U_i}$ and $t_i\colon\mathcal O_{U_i}^m\to \mathcal M|_{U_i}$. Then the transition functions on the intersections $U_{ij}:=U_i\cap U_j$ are the compositions$$s_{ij}\colon\mathcal O_{U_{ij}}^l\xrightarrow{s_i|_{U_{ij}}}\mathcal L|_{U_{ij}}\xrightarrow{s_j^{-1}|_{U_{ij}}}\mathcal O_{U_{ij}}^l,$$ which gives an element in $\mathrm{GL}_l(\Gamma(U_{ij},\mathcal O))$. The function $t_{ij}$ are defined similarly, as $t_j^{-1}|_{U_{ij}}\cdot t_i|_{U_{ij}}$. Then, $\mathcal L\otimes\mathcal M$ has trivializations $s_i\otimes t_i\colon\mathcal O_{U_i}^{lm}\to(\mathcal L\otimes\mathcal M)|_{U_{ij}}$, and the trivialization is $$(s\otimes t)_{ij}\colon \mathcal O_{U_{ij}}^{lm }\xrightarrow{s_i\otimes t_i|_{U_{ij}}}\mathcal L|_{U_{ij}}\xrightarrow{s_j^{-1}\otimes t_j^{-1}|_{U_{ij}}}\mathcal O_{U_{ij}}^{lm},$$ i.e. $$\begin{align*} (s\otimes t)_{ij}&=(s_j\otimes t_j)^{-1}|_{U_{ij}}\cdot(s_i\otimes t_i)|_{U_{ij}}\\ &=(s_j^{-1}|_{U_{ij}}\cdot s_i|_{U_{ij}})\otimes(t_j^{-1}|_{U_{ij}}\cdot t_i|_{U_{ij}})\\ &=s_{ij}\otimes t_{ij}. \end{align*}$$ This is just the Kronecker product.