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?

  • $\begingroup$ Properties 1,2,3,5 are the ones that I've encountered in my experience. Could you give an example of a property of schemes of type 4? Or was it just an intermediate step you found natural to squeeze in? $\endgroup$
    – babubba
    May 12, 2010 at 7:00
  • $\begingroup$ the usual local-condition is the second one. your claims are correct. you can find some general theorems of this type in the stacks project math.columbia.edu/algebraic_geometry/stacks-git/properties.pdf § 3 $\endgroup$ May 12, 2010 at 7:20
  • 1
    $\begingroup$ you could also consider the following 6: Regard $P$ as a full subcategory of $Rng$. Then the full faithful embedding from schemes to $Set^{Rng}$ should restrict to a full faithful embedding from schemes with $P$ to $Set^P$. This fits well with the other conditions for some nice $P$. $\endgroup$ May 12, 2010 at 7:27

1 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 $\Gamma_x$?


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}.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.