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Let $A$ be a commutative ring and endow the closed subsets of $\mathrm{Spec}(A)$ with the Grothendieck topology of finite covers. One may ask if the presheaf $V \mapsto A/I(V)$ is a sheaf. This is not true in general and is related (but not equivalent) to the following pure algebraic question:

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

The property can be checked locally. It holds in dimension $0$, and also for integral domains of dimension 1. It doesn't hold for the $2$-dimensional ring $k[x,y]$ (consider $(x^2 + y)+(y) = (x^2,y)$), nor for the 1-dimensional ring $\bigl(k[x,y]/(x^2 y + y^2)\bigr)_{(x,y)}$.

Are there other interesting examples/counterexamples or approaches for a general classification? I think the property has some algebro-geometric interpretation: All intersections of closed subschemes are transversal. See also SE/322872.

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For finite-type schemes, I don't think the classification can be too hard. If the dimension is at least $2$, we can set up a non-transverse intersection somewhere in the smooth part of $\operatorname {Spec} (A/\sqrt{(0)})$ without any difficulty. For a curve, this holds if and only if the intersection of every pair of unions of irreducible components is transverse. –  Will Sawin Sep 2 '13 at 23:21

3 Answers 3

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.

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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 $I+J$.

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<i>I guess they're originally defined by Brion and Kumar.</i> No, they just wrote the textbook. Frobenius splitting was introduced by Mehta and Ramanathan, and refined by Ramanan and Ramanathan. –  Allen Knutson Jan 5 '10 at 21:40

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.

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