# Criteria for coherence of rings

I'm trying to collect pointers into the literature about coherent rings. Recall that a ring is left coherent if its finitely generated left ideals are finitely presented. This condition was introduced by Stephen Chase in [Stephen U. Chase, Direct products of modules, Trans. Amer. Math. Soc. 97 (1960), 457–473. MR0120260] as a characterization of rings whose class of flat right modules is closed under arbitrary direct products.

There are many results out there giving criteria for coherence. In particular, providing conditions which, when satisfied, ensure that a ring constructed in some way (polynomials, formal series, &c) from another ring is coherent. There are also interesting negative results: the nicest one I can find is [Jean-Pierre Soublin, Anneaux et modules cohérents, J. Algebra 15 (1970), 455–472 MR0260799] which shows that the Hilbert basis theorem is false for coherence.

Coherence is a more or less technical thing, so it tends to appear a bit 'hidden' in various contexts, so I am pretty sure I am missing a big part of the literature in spite of having spent a non-negligible time following links on MathSciNet. For example, it appears that model theorist have (had?) an interest in coherence.

May the amazing collective MO erudition help me find relevant results?

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1+. I don't think that coherence is just a technical condition. It's the best condition for that the category of finitely generated modules is abelian. Due to laziness, we often restrict to noetherian rings for this. –  Martin Brandenburg Jan 15 '11 at 8:45
+1 Martin! Coherence is introduced very early on in EGA in this context (so presumably Grothendieck knew about the notion in the early 60s). Somehow one reason it's less widely-known is, I guess, that when Hartshorne wrote his book he stuck to the Noetherian case for simplicity, so just defined a coherent sheaf of modules as one which locally looked like a f.g. module, so one doesn't see the subtlety of the definition in the non-Noeth case. Course this is all in the commutative case, but EGA is a place to look [look in index for "coherent"...] –  Kevin Buzzard Jan 15 '11 at 9:14
Also, see Serre "Faisceaux Algebriques Coherent", Annals 1955, page 210 for a definition of coherence for a sheaf of commutative rings. –  Donu Arapura Jan 15 '11 at 13:50
My guess is that the idea must have preexisted FAC in the analytic case. Cartan, maybe? (Although that's pretty much like saying «in Euler»...!) –  Mariano Suárez-Alvarez Jan 15 '11 at 17:03
Yes, right. I think the idea of coherence of sheaves goes back to the work of Cartan and Oka in several complex variables. –  Donu Arapura Jan 15 '11 at 20:58

Dear Mariano,

One class of coherent rings that I know is the following:

If $\{A_n\}$ is an inductive sequence of left Noetherian rings, with right flat transition maps, then the inductive limit $A$ of the $A_n$ is left coherent. (This is an exercise; one place it comes up is in Berthelot's theory of overconvergent $\mathcal D$-modules, when he shows that the ring $\mathcal D^{\dagger}_{\mathbb Q}$ of overconvergent differential operators on a $p$-adic formal scheme is coherent.)

[Added: As Mariano notes in his comment below, one can replace left Noetherian by left coherent in the hypotheses on the $A_n$, as is proved in the paper of Soublin referenced in the original post.]

Here is another:

Let $A$ be a Noetherian commutative ring, and let $F: A \to A$ be a flat endomorphism. If $A[F]$ denotes the twisted polynomial ring, with commutativity relation $F \cdot a = F(a) \cdot F$, then $A[F]$ is left coherent. This is proved in my preprint On a class of coherent rings etc.. One application is when $A$ is a smooth finite type $k$-algebra for a field $k$ of char. $p$, and $F$ is Frobenius (either absolute or relative).

Added: Mark Hovey notes in this answer that any von Neumann regular ring is coherent.

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Soublin, in the paper I mentioned in the question, and Bourbaki in an exercise show that a flat direct limit of coherent commutative rings is coherent (and Soublin says commutativity is not necessary---I have not checked) –  Mariano Suárez-Alvarez Jan 15 '11 at 21:24
Thanks, by the way :) –  Mariano Suárez-Alvarez Jan 15 '11 at 21:25
Dear Mariano, Thanks for the remark re. coherence vs. Noetherian. This makes perfect sense; I simply never had cause to deduce coherence by taking a direct limit of coherent algebras that weren't actually Noetherian! Best wishes, Matt –  Emerton Jan 15 '11 at 21:28
Also, you're welcome! –  Emerton Jan 15 '11 at 21:28

Exercises 11 and 12 for section 2 of Ch. 1 of Commutative Algebra by Bourbaki may be useful. 12 (g) states a ring is left coherent iff 1. the left annihilator of every element is finitely generated 2. the intersection of any two finitely generated left ideals is finitely generated.

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What are the criteria given there? –  J.C. Ottem Jan 15 '11 at 20:56

The book Commutative Coherent Rings by Sarah Glaz is nice to read and a good source for references.

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Here are some further interesting references:

• C. Faith, Coherent rings and annihilator conditions in matrix and polynomial rings, Handbook of algebra, Vol. 3 (2003).

• M. E. Harris, Some results on coherent rings, Proc. Amer. Math. Soc. 17 (1966), 474-479.

• M. E. Harris, Some results on coherent rings II, Glasgow Math. J. 8 (1967), 123-126.

• E. Matlis, Commutative semi-coherent and semi-regular rings, J. Algebra 95 (1985), 343-372.

• G. Sabbagh, Coherence of polynomial rings and bounds in polynomial ideals, J. Algebra 31 (1974), 499-507.

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