# equivalence of categories between modules and vector space

Let $Map_k$ the category where the objects are triples $(V,W,f)$, where $V$ and $W$ are finite dimensional $K$-vector spaces and $f:V\rightarrow W$ is a $K$-linear map. A morphism from $(V,W,f)$ to $(V',W',f')$ in $Map_k$ is a pair $(h_1,h_2)$ of $K$-linear maps such that $h_2\circ f=f'\circ h_1$. If $(h_1',h_2')$ is a morphism from $(V',W',f')$ to $(V'',W'',f'')$ in $Map_k$, we set $(h_1',h_2')\circ(h_1,h_2)=(h_1'h_1,h_2'h_2)$.

Let $modA$ the category of finitely generated modules, the objects are finitely generated $A$-modules, the set of morphisms is the set of all $A$-module homomorphisms and the comosition of morphism is the compositions of maps. Where $A=\pmatrix{K & 0\\ K & K}$, where $K$ is a field

Let $\rho:modA\rightarrow Map_k$ a functor defined as follows

we note that $e_1=\pmatrix{1 & 0 \\ 0 & 0}$, $e_2=\pmatrix{0 & 0 \\ 0 & 1}$ and $e_{21}=\pmatrix{0 & 0 \\ 1 & 0}$ form a basis of $A$. It follows that every module $X$ in $modA$, viewed as a $K$-vector space, has a direct sum descomposition $X=Xe_1\oplus Xe_2$. Then $\rho(X)=(V_x,W_x,f_x)$ where $V_x=Xe_2$ and $W_x=Xe_1$ and $f_X(xe_2)=(xe_2)e_{21}=(xe_{21})e_1$. If $g:X\rightarrow Y$ is a homomorphism of $A$-modules, we define $\rho(g):\rho(X)\rightarrow\rho(Y)$ to be the pair $\rho(g)=(g_1,g_2)$, where $g_1:V_x\rightarrow V_y$ and $g_2:W_x\rightarrow W_y$ are the restrictions of $g$ to $V_x$ and to $W_x$.

I have to prove that $\rho$ is full and faithful, it means that i have to prove that $\rho(g)$ is inyective and suryective.

Let $h:X\rightarrow Y$ a homomorphism of $A$-modules and $h_1:V_x\rightarrow V_y$ and $h_2:W_x\rightarrow W_y$ are the restrictions of $h$ to $V_x$ and to $W_x$. I want to prove that $\rho$ is faithful so

\begin{align*} \rho(g)&=\rho(h)\\ (g_1,g_2)&=(h_1,h_2)\\ \end{align*}

I don't know why $g_1=h_1$ and $g_2=h_2$ and then $g=h$ and therefore $\rho$ is faithful. And how I can prove that $\rho$ is full.

• You want another notation for $A$. =) – Pedro Tamaroff Jul 27 '16 at 2:58
• what do you mean? I want to prove that $\rho$ is full and faithful – user292200 Jul 27 '16 at 3:06
• $A$ is a ring. What ring is it? – Pedro Tamaroff Jul 27 '16 at 3:23
• $A$ is the lower triangular matrix $K$-subalgebra of the matrix algebra $\mathbb{M}_2(K)$, where $K$ is a field. – user292200 Jul 27 '16 at 3:31
• Well, you have not written that in your post. – Pedro Tamaroff Jul 27 '16 at 3:46