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Questions tagged [ac.commutative-algebra]

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

3,743 questions
37 views

Why there are elementary equations that are not solvable in closed form

Elementary equations and closed-form solutions can be important in mathematics and in the natural sciences, e.g. for discovering and describing certain relationships. $\mathbb{L}$ denote the ...
82 views

Finite dimensional commutative algebras containing infinitely many nilpotents whose $d$-way products are nonzero

I'm interested in the following strange question: for some $d > 1$, what is the minimum dimension of a commutative $\mathbb{C}$-algebra containing infinitely many elements that square to zero, but ...
73 views

Catenarity and epimorphisms of rings

Let $R$ be a commutative ring. The following are well-known: If $R$ is catenary and $\mathfrak{a}\subseteq R$ is an ideal, then $R/\mathfrak{a}$ is catenary. If $R$ is catenary and $S\subseteq R$ is ...
64 views

What is the difference between total integral closure and integral closure?

I was advised here to make this a new question: What is the difference between total integral closure and integral closure (geometrically, in the context of rigid analytic geometry)? I have read in ...
109 views

Significance of integrally closed in an affinoid algebra

A Tate affinoid k-algebra is defined as a pair $(R,R^+)$ where $R$ is a Tate algebra and $R^+$ is an open and integrally closed subring of $R$ contained in the ring of powerbounded elements. See for ...
38 views

Non-separated Nygaard filtration

Let $S$ be a quasiregular semiperfectoid ring, then on its prism we may define the Nygaard filtration (Definition 12.1 in this preprint). What is an explicit example where it is not separated?
118 views

Properties of rings of global functions of open subschemes

It is known (although maybe not so well) that there are nice algebraic varieties whose ring of global functions is not finitely generated over the ground field. One can find examples on the web, but ...
162 views

Generalised CRT - How to compute the cokernel?

Let $R$ be a commutative ring of dimension one with minimal prime ideals $P_1,\ldots,P_n$. We have the canonical injective map $$\phi_n: R/(P_1 \cap \ldots \cap P_n) \to \prod_{i=1}^n R/P_i.$$ My ...
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Let $P$ be an irreducible polynomial of $\mathbb F_q[T]$ of degree $2$. Does there exist two polyomials $\alpha,\beta\in\mathbb F_q[T]$ (not both zeroes) such that the sequence $(\beta T^{q^{2n}}-\... 1answer 214 views Is completion of isolated singularity isolated? Let$K$be an algebraically closed field and let$A=K[x_1,\dots,x_n]/I$be a$K$-algebra of finite type which has only an isolated singularity at the origin. Let$\mathfrak{m}=(x_1,\dots,x_n)$and ... 0answers 76 views Intersection of an ideal and a subring Is there a Grobner basis method that can compute the intersection of an ideal$I$of a polynomial ring$R$and its subring$R^\prime$? For example, I have an ideal$I=(x+y+z^2,1+xyz+yz+xz)$of$\...
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Example of a ring where every module of finite projective dimension is free?

I'm interested in seeing an example of a ring which is not self-injective where every module admitting a finite projective resolution is free, or at least projective. Note that self-injectivity says ...
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This question is in some sense a continuation to this question: Derived Nakayama for complete modules For the setting: Let $A$ be a ring and let $I$ be some finitely generated ideal in $A$. Let $f\... 1answer 99 views Ideal in ring of power series Let$K$be a field of characteristic$p$and$A_n \colon= K[[X_1,\ldots,X_n]]$be a$n$-variable formal power series ring over$K$such that$n, p \geq 3$. Consider the ideal$I$defined by \begin{... 0answers 66 views Proper ideals are invertible I am reading through Cox's book Primes of the form$x^2+ny^2$and I am stuck with some proofs in Chapter 7 (I have the 2nd edition). There, the author presents the following Lemma: Lemma 7.5: Let$...

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