Zariski site is subcanonical?

Question

I want to show that the zariski site is subcanonical as an exersice of the book "Sheaves in geometry and logic" and I need help with it... To be honest I didnot really understand the definition of what a zariski site exactly is... I tried to take the structure sheaf $O=Hom(A,-)$(where $A$ is an $k-$algebra) which is equivalent to $Hom(-,V(A))$ as an arbitrary pre-sheaf and and claim it is actually a sheaf... I took $f_1,...f_n\in A$ such that $A=<f_1,...,f_n>$ And then I defined a matching family $\alpha_i : V(A_{f_i} )\rightarrow (V_A)$ and I draw the pull back diagram consisting of $$V(A_{f_i})\rightarrow V(A)$$ , $$V(A_{f_j})\rightarrow V(A)$$, $$V(A_{f_i,f_j})\rightarrow V(A_{f_i})$$ and $$V(A_{f_i,f_j})\rightarrow V(A_{f_j})$$ ......

Please help....

Answer 1

First let me tell you what is the zariski site. consider the category of affine schemes(which is the opposite category of rings). for giving a site I have to tell which morphisms in this category are the cover: we declare a map $A\to A_{f_1}\times... A_{f_n}$ as a cover if $A=(f_1,f_2,...,f_n)$. this is equivalent to saying that $spec A=\cup spec A_{f_i}$ in the ordinary zariski topology. the only difference is that for zariski site you consider a different category: instead of open subsets of a fixed scheme you consider the category of all rings.

Now back to the question you want to prove that $Hom(B,-)$ give a sheaf on zariski site. for this assume that you have the zariski cover $A\to A_{f_1}\times...\times A_{f_n}$,and the maps $\phi_i:B\to A_{f_i}$ such that the induced map $\phi_{ij}:B\to A_{f_if_j}, \phi_{ji}:B\to A_{f_{i}f_{j}}$ are equal(this maps $\phi_{ij}$ are defined by combining $\phi_i$ by the canonical map $A_{f_i}\to A_{f_if_j}$). by definition you have to prove that you can "glue" this $\phi_i$ together to get a map $\phi:B\to A$.

Let $b\in B$ we want to define $\phi(b)$. we know that $\phi_{i}(b)|_{A_{f_if_j}}=\phi_j(b)|_{A_{f_if_j}}$. now lets accept this lemme for the moment:

Lemma: if $x_i\in A_{f_i}$ such that $x_i|_{A_{f_if_j}}=x_j|_{A_{f_i}A_{f_j}}$ then there is an unique element $x$ of $A$ such that $x|_{A_{f_i}}=x_i$.

Now take $x_i=\phi_i(b)$ and define $\phi(b)=x$ in the notation of the above lemma. so we defined the unique map $\phi:B\to A$. and hence $Hom(B,-)$ is a sheaf.

for the proof of lemma see the proof of proposition 3.1 in page 42 of liu book.