Induced topology on site + Reconstructing global sections of a scheme (Orlov)

Question

Let $(C,T,O)$ be a ringed site. Let $X$ be a presheaf on C. We get an induced ringed site $(C/X,T_X,O_X)$. C/X is the over category wrt the presheaf X. The topology $T_X$ is the biggest topology making the functor $j_X:C/X\rightarrow C$ continues and $O_X=j_X^*O$. See paragraph 1.1,1.5 and 1.6.

Given any sieve $J$ on $R\rightarrow X$ in $C/X$, how can one determine if it is covering or not?

Orlov points to SGA1, but I am a mere commoner. I want a procedure/criteria using only my original topology $T$ and not all sheafs as in the definition.

$J$ is covering if the canonical map $$Hom(J,G_X)\rightarrow Hom(R\rightarrow X,G_X)$$ is bijective for any pullback $G_X=j_X^*G$ of any sheaf $G$ on $(C,T)$.

This is a prereq for my 'real' question. In the Orlov paper, $X$ is a scheme and $C$ is the category of affines. In this set-up, how would you calculate the 'global' sections of $O_X$? I would like to work out in detail the equivalence between his set-up and the usual one.

Answer 1

First things first: we need a more tractable definition of "continuous".

Let $\mathcal{C}$ and $\mathcal{D}$ be categories, let $J$ be a Grothendieck topology on $\mathcal{C}$, and let $K$ be a Grothendieck topology on $\mathcal{D}$. The following are equivalent for a functor $u : \mathcal{C} \to \mathcal{D}$:

• For every $K$-sheaf $Y$ on $\mathcal{D}$, $u^* Y$ is a $J$-sheaf on $\mathcal{C}$.
• For every object $C$ in $\mathcal{C}$ and every $J$-covering sieve $\mathfrak{U}$ on $C$, the morphism $u_! \mathfrak{U} \to u_! h_C$ becomes a split epimorphism after $K$-sheafification.

Assuming $u_! : \hat{\mathcal{C}} \to \hat{\mathcal{D}}$ preserves monomorphisms (which happens if e.g. $\mathcal{C}$ has finite limits and $u : \mathcal{C} \to \mathcal{D}$ preserves finite limits), then the previous conditions are also equivalent to the following conditions:

• For every object $C$ in $\mathcal{C}$ and every $J$-covering sieve $\mathfrak{U}$ on $C$, the morphism $u_! \mathfrak{U} \to u_! h_C$ becomes an isomorphism after $K$-sheafification.
• For every object $C$ in $\mathcal{C}$ and every $J$-covering sieve $\mathfrak{U}$ on $C$, the sieve on $u(C)$ generated by $\{ u(f) : f \in \mathfrak{U} \}$ is a $K$-covering sieve.

Now, let us consider the case of the projection $u : \mathcal{C}_{/ X} \to \mathcal{C}$. It is well known that $(\mathcal{C}_{/ X})ˆ$ is equivalent to $\hat{\mathcal{C}}_{/ X}$, and under this identification, $u_! : \hat{\mathcal{C}}_{/ X} \to \hat{\mathcal{C}}$ is the projection. In particular, $u_!$ preserves monomorphisms, so the Grothendieck topology on $\mathcal{C}_{/ X}$ is the "obvious" one: a sieve in $\mathcal{C}_{/ X}$ is a covering sieve if and only if its image in $\mathcal{C}$ is a covering sieve. (Note that the image of a sieve is automatically a sieve in this case!)

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Your second question is actually independent of the first one – Grothendieck topologies and sheaf conditions are irrelevant here. Again, we start with a general fact:

Let $\mathcal{C}$ and $\mathcal{D}$ be categories, let $u : \mathcal{C} \to \mathcal{D}$ be a functor, and let $Y$ be a presheaf on $\mathcal{D}$. We have the following natural bijection, $$\hat{\mathcal{D}} (u_! 1, Y) \cong \hat{\mathcal{C}} (1, u^* Y) \cong \Gamma (Y)$$ where $1$ is the terminal object in $\hat{\mathcal{C}}$.

In particular, for the projection $u : \mathcal{C}_{/ X} \to \mathcal{C}$, we have $u_! 1 \cong X$, so $\Gamma (Y) \cong \hat{\mathcal{C}} (X, Y)$ in this case.

Incidentally, if $\mathcal{C}$ is the category of affine schemes and $X$ and $Y$ are the functor of points of schemes, then $\hat{\mathcal{C}} (X, Y)$ can be identified with the set of scheme morphisms $X \to Y$.