Question

let $A$ be a ring and endow the closed subsets of $Spec A$ with the grothendieck topology of finite covers. it would be nice if $V \mapsto A/I(V)$ defines a sheaf. this is not true in general and is related (equivalent?) to the following pure algebraic question:

in which rings $A$ are the radical ideals closed under sum?

if $A$ is a noetherian integral domain of dimension $0$ or $1$, it's true. you cannot omit integral here (consider $k[x,y]/(x^2 y + y^2)$ localized at $(x,y)$), nor the dimension (in $k[x,y]$, consider $(x^2 + y)+(y) = (x^2,y)$). are there other interesting examples/counterexamples or approaches for general classification?

Answer 1

With regards to the original question - I don't have any good ideas, probably valuation rings satisfy this (but they are not generally Noetherian except in the cases already outlined).

With regards to other classes of ideals that satisfy this (now in characteristic zero), the ideals of ``unions of log canonical centers'' satisfy this property in characteristic zero (this is mostly due to a result of Florin Ambro I think). See for example this preprint.

Answer 2

Allen Knutson has a nice recent preprint which, among other things, discusses a class of rings (of prime characteristic) for which a certain supersubclass of the radical ideals is closed under sum. They're called "Frobenius split rings." I guess they're originally defined by Brion and Kumar. They are defined to be those rings $R$ of characteristic $p$ admitting an additive map $\phi\colon R \rightarrow R$ such that $\phi(a^pb) = a\phi(b)$ and $\phi(1)=1$. An ideal $I$ is "compatibly split" if $\phi(I) \subseteq I$. Compatibly split ideals are radical, prime components of a compatibly split ideal are compatibly split, and if $I$ and $J$ are compatibly split, then so is $I+J$.

Answer 3

Recently, there is a very interesting preprint by Schwede and Tucker where they address Frobenius splitting from a algebraic point of view. They proved a general statement (apply to rings of char 0 as well) which seems to be related to what you want. That is Theorem 4.2, which says:

Let $C$ be a collection of prime ideals in an excellent local ring (this covers almost all local rings of interest) and embedding dimension $n$. Suppose that the set $I = \cap_{P\in C'} P | C' \text{a finite subset of}\ C$ is closed under sum. Then the number of primes $P\in C$ of dimension $d$ is less than $n \choose d$.

The paper contains many related results as well.