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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.

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        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 ...
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        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 ...
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        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 ...
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        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 ...
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        votes
        1answer
        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 ...
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        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?
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        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 ...
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        1answer
        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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        1answer
        70 views

        Converging sequence of polynomials

        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}}-\...
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        votes
        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 ...
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        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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        1answer
        235 views

        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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        1answer
        100 views

        Mapping cone and derived tensor product

        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\...
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        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{...
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        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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