# Questions tagged [ac.commutative-algebra]

Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

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### Enriched Categories: Ideals/Submodules and algebraic geometry

While working through Atiyah/MacDonald for my final exams I realized the following: The category(poset) of ideals $I(A)$ of a commutative ring A is a closed symmetric monoidal category if endowed ...
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### Is there a notion of polynomial ring in “one half variable”?

Let $C$ be the category of commutative rings. Is there a functor $F :C \to C$ such that $F(F(R)) \cong R[X]$ for every commutative ring $R$ ? (Here, we may assume those isomorphisms to be natural ...
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### The field of fractions of the rational group algebra of a torsion free abelian group

Let $G$ be a torsion free abelian group (infinitely generated to get anything interesting). The group algebra $\mathbb{Q}[G]$ is an integral domain. Let $\mathbb{Q}(G)$ be its field of fractions. ...
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### On the definition of regular (non-noetherian, commutative) rings

All rings are commutative with unit. A ring $R$ is called regular if it satisfies (Reg) Every finitely generated ideal of $R$ has finite projective dimension. Clearly this gives the usual ...
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### CH and automorphisms of ultrapowers of $\mathbb{Z}$ and $\mathbb{R}$

Notation and motivation. Given an algebraic structure $\mathbb{M}$ of cardinality at most the continuum and with countably many operations, and a nonprincipal ultrafilter $\cal{U}$ on a countably ...
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### If $A, B$ are abelian groups such that $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ for all abelian groups $G$, must $A$ and $B$ be isomorphic?

The question is in the title. If the isomorphism $\mathrm{Hom}(A, G) \cong \mathrm{Hom}(B, G)$ is natural in $G$ then this is just the Yoneda Lemma. If $A$ and $B$ are finitely generated this is also ...
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### Rigid Non-archimedean Real Closed Fields

Question. Is there a countable rigid non-Archimedean real closed field? Background: As usual, a structure is said to be rigid if the only automorphism of the structure is the identity map. It is ...
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### Scholze's infinite to finite type ring theory reductions?

In following essay "The Perfectoid Concept: Test Case for an Absent Theory" by Michael Harris, there is the following sentence I found to be quite striking. The most virtuosic pages in Scholze's ...
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### Higher Adeles of a scheme and related topics

Let $X$ be a noetherian scheme. I will describe a construction of a simplicial ring which I think is called the Bellinson higher Adeles complex (or something similar). Consider the augmented ...
Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...