# 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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### Testing whether a finite dimensional algebra is a complete intersection with QPA

Let $A$ be a local commutative finite dimensional $K$-algebra for a field $K$. We can thus assume that $A$ is given by quiver and admissible relations $KQ/I$. It is easy to check when $A$ is ...
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### Are algebras of smooth functions formally smooth?

Let $M$ be a manifold. Then is the ring of smooth functions $C^\infty(M,\mathbb{R})$ formally smooth over $\mathbb{R}$? If it helps, feel free to assume that $M$ is compact. (This is not a joke ...
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### On $\text{depth}_S\left(\dfrac {S}{JS+xS}\right)$, when $\text{depth}_R(R/J)=0$, and $R\to S$ is a certain flat map of local rings

Let $(R, \mathfrak m) \xrightarrow{\phi} (S,\mathfrak n)$ be a flat homomorphism of local rings such that $\mathfrak n=\mathfrak m S +xS$ for some $x\in \mathfrak n \setminus \mathfrak n^2$. Let $J$ ...
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### Commutative local rings which satisfy Krull-Remak-Schmidt

Question 1: Can the class of local (always noetherian and commutative) rings be classified for which the Krull-Remak-Schmidt theorem (KRS) holds for finitely generated modules? They contain for ...
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### Projectivity of the fundamental ideal of Witt groups

Suppose $k$ is a field. I wonder when the Witt ring of the quadratic forms $\textbf{W}(k)$ has a projective fundamental ideal, which is the kernel of the rank modulo 2 morphism. Here I want a ...
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### For which fields $k$ with $k^\times$ not $p$-divisible, does there exist finite $l/k$ such that $l^\times$ is $p$-divisible?

Is there a prime $p$ and a field $k$, not real closed, with $k^\times$ not $p$-divisible, such that there exists a finite extension $l/k$ such that $l^\times$ is $p$-divisible? This question came up ...
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### Example of a secondary representation of a module that is not a direct sum

Let $A$ be a commutative ring. An $A$-module $M$ is said to be secondary if $M\neq 0$ and for each $a\in A$, the endomorphism $\phi_a:M\to M$ defined by $\phi_a(m)=am$ for $m\in M$ is either ...
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### Smooth surjective morphism to integral scheme

Suppose that $f : X \rightarrow Y$ is a smooth (or even étale) surjective morphism over a field $k$ to a scheme $Y$ of finite type over $k$. I want to show that $X$ is locally integral, i.e. (in the ...
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### Situations where extension of contraction is well-behaved?

It is known that one cannot expect the extension of the contraction of an ideal under a ring homomorphism $f:A\to B$ to be the original ideal, except in very special scenarios like surjections or ...
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### Uniqueness of the "algebraic closure" of a commutative ring

There are several ways to generalize the notion of "algebraic closure" from fields to arbitrary commutative rings. A good overview is On algebraic closures by R. Raphael. I am more ...
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### Linear resolution in quotient ring

Say $I$ is a homogeneous monomial ideal in the polynomial ring $S = k[x_1,...x_n]$ with a linear resolution (over $S$). Consider the quotient ring $R = k[x_1,...,x_n]/(x_1^2,...,x_n^2)$. This is my ...
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### Tensor product of finite type UFD algebras over an algebraically closed field is again UFD?

Let $K$ be an algebraically closed field, $A$ and $B$ two finite type $K$-algebras which are assumed to be UFD. Is $A \otimes_K B$ again a UFD? This question has been already asked here and here, but ...
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### Is there any analogue of $\pi$ for non-Archimedean Euclidean fields?

Let us recall that a Euclidean field is an ordered field in which every positive element has a square root. Given a Euclidean field $F$, we can consider the Euclidean'' plane $F^2$ endowed with the ...
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### What is a PBW algebra? (I.e., an algebra generalising properties of $U(\frak{g})$)

I am reading a paper where they refer to a certain algebra as a PBW algebra. What does this mean exactly? I would infer from the $U(\frak{g})$ setting that this means the existence of an ordered ...
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### Can a Dedekind domain have a power basis over a ring that isn't a Dedekind domain?

This aim of this question is to determine whether there exists a proof (or some counterexample) to the following statement : "If $R$ is a subring of Dedekind domain $S$, such that $S$ has a power ...
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### Classification of 2-periodic triangulated categories

Let $T$ be an algebraic triangulated (k-linear over a field, Hom-finite, idempotent-complete) category. Call $T$ 2-periodic if $\Omega^2(X) \cong X$ for all $X \in T$. Question 1: Is there a ...
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### When is $R$ a direct summand of Frobenius pushforwards?

Let $(R,\mathfrak m)$ be a reduced Noetherian local ring of prime characteristic $p$. For integer $e>0$, let $F^e_* R$ denote the $R$-module which is $R$ as an abelian group, but the $R$-module ...
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### Top local cohomology - recommendations

I need some background in local cohomology to read a certain paper, which exploits the structure of the top local cohomology. Even after acquainting myself to local cohomology, I fail to understand ... 109 views

### Is a closed subsecheme contained in a Cartier divisor?

Let $X$ be a variety over a field $k$. For a closed subscheme $Z\hookrightarrow X$ and a closed point $x\in X$ such that $\text{codim}_XZ \geq 1$ and $x\notin |Z|$, is there an effective Cartier ...
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### A question on curves on effective divisors

Let $X$ be a smooth projective variety over $\mathbb{C}$ with $\dim X=3$ and $\mathrm{Pic}(X)=\mathbb{Z}\cdot D$, where $D$ is a very ample effective Cartier divisor on $X$. Let $Z$ and $C$ be two ...
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### On the noetherianess of some subalgebras of an affinoid algebra

$\DeclareMathOperator\Sp{Sp}$Let $X=\Sp(A)$ be a connected smooth affinoid rigid space over a discretely valued non-archimedean field $K$. Let $\mathcal{R}$ be a valuation ring of $K$, and fix a ...
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### Non-tree models of Lagrange inversion polynomials

The specific Lagrange inversion / series reversion polynomials (LIPs) I'm addressing are illustrated in OEIS A134685 with a general linear term and in Lang's pdf for A176740 with the coefficient of ...
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### Nullstellensatz with nilpotents and $I=J(V(I))$

Let $R$ be the ring $$\mathbb{R}[t_1,t_2\ldots]/(t_1^2,t_2^2,\ldots)$$ Let $p_1,\ldots p_\ell$ be polynomials in $R[x_1,\ldots,x_n]$ whose constant terms are 0. Let $f$ be a polynomial which is zero ...
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### When can we choose non-zero-divisor $x\in \mathfrak m$ in a reduced local ring $(R,\mathfrak m)$ such that $R/xR$ is also reduced?

Let $(R,\mathfrak m)$ be a local Cohen-Macaulay reduced ring of dimension at least $2$. Then, can we find a non-zero-divisor $x\in \mathfrak m$ such that $R/xR$ is again a reduced ring? If needed, I ...
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### Lifting module homomorphisms imposing conditions on characteristic polynomials

Suppose that we are in the setting described in the first two paragraphs of this MSE post. My question wants to deal with an instance of the study of the amount of freedom that the choice of the ...
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Let $X$ be an algebraic surface with normal crossing singularities. Suppose the singular locus of $X$ is a smooth curve. Let us denote it by $C$. Suppose $D$ is another smooth curve in $X$ which ...
$\DeclareMathOperator\CM{CM}\DeclareMathOperator\Spec{Spec}$Let $k$ be an algebraically closed field and let $G$ be an algebraic group over $k$. My question: is $G$ Cohen–Macaulay? If not, are there ...