Extending properties of commutative rings to schemes

Question

I'm trying to pin down the various ways we can extend a property of commutative rings to a corresponding property for schemes. Let $P$ be a property of commutative rings. We could define a scheme $(X,\mathcal{O}_X)$ to have property $P$ in one of the following ways:

1. $\mathcal{O}_X(U)$ has $P$ for every open subset $U\subset X$.
2. $\mathcal{O}_X(U)$ has $P$ for every affine open subset $U\subset X$.
3. there exists an affine open cover ${U_i}$ of $X$ such that $\mathcal{O}_X(U_i)$ has $P$ for all $i$.
4. for each $x\in U\subset X$ with $U$ an open subset, there exists an affine open $V\subset X$ with $x\in V\subset U$ such that $\mathcal{O}_X(V)$ has $P$.
5. $\mathcal{O}_{X,x}$ has $P$ for all $x\in X$.

Evidently, $(1)\Rightarrow (2)\Rightarrow (3)$. If the property $P$ is stable under inversion of single elements (that is, $A$ has $P$ $\Rightarrow$ $A[1/s]$ has $P$ for any element $s\in A$) then $(3)\Rightarrow (4)$. Furthermore, if the property $P$ is stable under arbitrary localizations (that is, $A$ has $P$ $\Rightarrow$ $S^{-1} A$ has $P$ for any multiplicative subset $S$ of $A$) then $(4)\Rightarrow (5)$.

Thus, for many properties of commutative rings $(1)\Rightarrow (2)\Rightarrow (3)\Rightarrow (4)\Rightarrow (5)$.

Now we need to consider going in the other direction. I believe it can be shown that if $P$ is a local property in the sense that $A$ has $P$ iff $A_{\mathfrak{p}}$ has $P$ for each $\mathfrak{p}\in$ Spec$(A)$ then $(5)\Rightarrow (2)$. Now, it seems to me that if $P$ is a local property in this sense then the property is stable under localization by arbitrary multiplicative subsets. Thus if $P$ is a local property in this sense then $(2) \Leftrightarrow (3) \Leftrightarrow (4) \Leftrightarrow (5)$.

Finally, here is another notion of a property being local. Suppose that $P$ is such that $A$ has $P$ implies that $A[1/s]$ has $P$ for each $s\in A$ and that on the other hand, if $s_1,\ldots,s_n \in A$ are such that Spec$(A)=D(s_1)\cup D(s_2) \cup \cdots \cup D(s_n)$ then $A[1/s_i]$ has $P$ for all $i=1,..,n$ implies that $A$ has $P$. Then I believe it can be shown that $(4)\Rightarrow (2)$ and thus for such a property we have that $(2) \Leftrightarrow (3) \Leftrightarrow (4)$.

Does this seem right to you? I haven't seen any books on algebraic geometry discuss this question to my satisfaction and I am nervous that there may be some holes in my proofs, so if anyone knows off the top of their head that what I have described seems right then I would be happy to hear from you. Do you have any further comments to make about this process of extending a property of commutative rings to schemes?

Answer 1

The $U$-sections funtctors $\Gamma_U: Sheaves(X, Ring)\to Ring: \mathscr{O} \mapsto \mathscr{O}(U) $ preserve (and reflect) only limits ($Ring$ mean commutative rings by unit).

The Stalk functors $\Gamma_x: Sheaves(X, Ring)\to Rings: \mathscr{O} \mapsto \mathscr{O}_x $ $x\in X$ preserve (and reflect) also colimits.

Then the question is what kind of propriety is preserved (and reflexed) by the funtors $\Gamma_U$ or $\Gamma_x$?

Formally:

1) Is: $\mathscr{O}$ is $P$ $\Leftrightarrow$ $\Gamma_U(\mathscr{O} )$ is $P$ $\forall U\in \tau_X$ ?

2) Is: $\mathscr{O}$ is $P$ $\Leftrightarrow$ $\Gamma_x(\mathscr{O} )$ is $P$ $\forall x\in \tau_X$ ?

These questions involving the “logic inside a category” concept (for formalizing what you mean about property $P$).

The (1) has the answere: for all $P$ definible in terms of “Cartesians formulas”

The (2) has the answere: for all $P$ definible in terms of “geometrical formulas”

The fist step in this direction was the cap.III of Monique Hakim's book "Schemas relatifs et Topos anelles”, for a more easy lectures can read

Saunders MacLane, Ieke Moerdijk Sheaves in Geometry and Logic: A First Introduction to Topos Theory

But for example if $P$ is the $local-ring$ proprierty the this means that $\mathscr{O}_{X,x} $ a unique maximal ideal but dont means that $\mathscr{O}_X(U) $ has a unique maximal ideal, but a more general condiction on the sheaf $\mathscr{O}$ : for any $ U\tau_X$ and $s\in \mathscr{O}(U) $ we have $U=U_s \cup U_{1-s}$ where $U_t \subset U$ ($t\in \mathscr{O}(U) $) is the maximal open of $U$ where $t$ is invertible (this definitions coincides by the usal for sheaves on thee trivial one-point space, i.e. and the funtor $\Gamma_x$ is just the restriction the the space {x}.