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Questions tagged [coherent-rings]

In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented.

In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented. Many theorems about finitely generated modules over Noetherian rings can be extended to finitely presented modules over coherent rings. At the same time, the class of coherent rings is wider than that of Noetherian rings since it contains, for example, all regular rings (in the sense of von Neumann) and the rings of polynomials over Noetherian rings in an arbitrary number of variables.

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Geometric intuition for coherent rings, modules, and sheaves

Throughout, all rings are commutative. Definition 1. A ring $R$ is coherent if the solutions $\mathbf x=(x_1,\dots,x_n)$ to a linear equation $\mathbf{rx}=0$ are a finitely generated $R$-submodule of $R^{n\times 1}$. So in categorical terms, $R$ is…
Arrow
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Coherent ring whose nilradical is not finitely generated.

Let $A$ be a commutative ring with $1$. We say that $A$ is coherent if and only if every finitely generated ideal of $A$ is finitely presented. Does there exist a coherent ring such that nil-radical of $A$ is NOT finitely generated? In other…
Anoop singh
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Infinite intersection of finitely generated ideals in a coherent ring.

It has been claimed without proof in several answers that an intersection of two finitely generated ideals in a coherent ring is finitely generated. Thus, the finitely generated ideals in a coherent ring form a lattice. However, can an infinite…
Morgan Rogers
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Do two similar boolean matrices have same number of non-zero entries?

I was just wondering: is it necessarily the case that if $A$ is a $(0, 1)$ matrix, and $SAS^{-1}=B$, where $B$ is also a $(0,1)$ matrix, then do $A$ and $B$ have the same number of $1$s? I have the gut feeling that this may not be true. However, let…
mirewine
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Is every ring/module the filtered colimit of its finitely presented/coherent/quasicoherent subrings/submodules?

Is every ring/module the filtered colimit of its coherent/quasicoherent subrings/submodules? What about finitely presented subobjects? What's the intuition behind each case? Notation. Let $I,J$ be two ideals of commutative ring. If $I=aR,J=bR$,…
Arrow
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Is a Noetherian sheaf of rings stalkwise Noetherian?

Let $X$ be a scheme such that $O_X$ is a coherent $O_X$-module. Assume that for every open subset $U\subset X$ and every family of coherent ideal sheaves $\{I_i\}_i$ of $O_U$, $\sum_iI_i$ is a coherent $O_U$-module. Do we have that $O_{X,x}$ is a…
Doug
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$I$ finitely presented nilpotent ideal of a commutative ring $R$ and $R/I$ coherent implies $R$ coherent

Let $R$ be a commutative ring, and let $I$ be an ideal. We assume that $I$ is nilpotent, so $I^n=0$ for some $n$. Moreover, we assume that it is finitely presented, namely it is the cokernel of some morphism of $R$-modules $R^n \to R^m$, in other…
Francesco Genovese
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Polynomial ring in infinitely many variables over a noetherian ring is coherent

If $R$ is noetherian, show that the polynomial ring of infinite variables $R[x_1,x_2,...]$ is coherent, i.e. every finitely generated ideal is finitely presented. I don't really know how to get started. I tried to use that over a noetherian ring…
MathStarter
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Rings where finitely generated ideals are closed under countable intersection

Does there exist a characterization of those rings $R$ such that finitely generated left ideals are closed under countable intersection? For example, any noetherian ring has this property, since all ideals are finitely generated. On the other hand,…
Piotr Pstrągowski
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Are coherent modules hopfian?

As it is well-known, a noetherian module over an arbitrary ring is hopfian. Are coherent modules also hopfian?
Mariano Suárez-Álvarez
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Can we control the number of homogeneous generators of a f.g. homogeneous ideal?

Let $G$ be an abelian group and $R$ be a $G$-graded ring. Is there a map $\phi:\mathbb{N}\rightarrow\mathbb{N}$ such that for every $n\in \mathbb{N}$ and any homogeneous ideal $I$ of $R$ generated by $n$ elements, $I$ can be generated by…
A. R.
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infinite indeterminates over a left noetherian ring is left coherent(want a basic proof)

A ring is left coherent if every finitely generated left ideal is finitely presented. Statement: If $R$ is a left noetherian ring, then $R[X]$ is left coherent where $X$ represents infinite many indeterminates. I do not see any basic proof without…
user45765
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(quasi)coherent rings for which $\dim R[T]\neq \dim R+1$

What are some some examples of (quasi)coherent rings for which $\dim R[T]\neq \dim R+1$? Why (hopefully geometrically) should we not always have equality? Notation. Let $I,J$ be two ideals of commutative ring. If $I=aR,J=bR$, denote the conductor…
Arrow
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Is a commutative perfect and coherent ring an Artin ring?

In stackexchange the questioner says that when R is a commutative ring with unity, the Chase's theorem states that any direct product of projective R-modules is projective iff R is Artinian. I doubt the above statement because from SC60,Theorem 3.3.…
Liang Chen
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$M$ is a coherent left $R$-module implies $M/IM$ is a coherent $R/I$-module.

Do you know how to prove that for a ring $R$, and a bilateral ideal $I$ of $R$, if $M$ is a coherent left $R$-module, then $M/IM$ is a coherent $R/I$-module? I know how to prove the fact if $I$ annihilates $M$: In that case, $_RM$ is finitely…
Carlos Alberto Arriaga Solrzan
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