# Tagged Questions

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### Does projective duality preserve arithmetic-Cohen-Macaulay-ness?

Let $V$ be a vector space over $\mathbb{C}$. Suppose $X\subset \mathbb{P} V$ is an algebraic variety, and consider its projective dual variety $X^\vee \subset \mathbb{P} V^*$. If the coordinate ring ...
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### Bound for the height of equations defining the singular locus of a variety

Fix positive integers $m, n, d$. In what follows, the height of an algebraic number will mean the absolute multiplicative height. Let $V \subset \bar{\mathbb{Q}}^n$ be an affine algebraic variety ...
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### $\Gamma_Z(\widetilde M)\cong\widetilde{ \Gamma_Z(M)}$

Let $R$ be a Noetherian ring and let $M$ is an $R$-module. Consider the associated affine scheme $(\text{Spec R},\mathcal{O}_{\text{Spec R}})$ and Suppose $Z\subset X$ is a closed subset of ...
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### commutative algebra, diagonal morphism

can anyone help me with the following statement (it is part of a bigger proof where it is not explained). Let $B$ be a finite type $A$-algebra and consider the kernel $I$ of the diagonal ...
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### generalization Abhyankar's lemma

This question is related to a question I already asked on MO (smooth quotient out of a singular variety?), but I realized later that the hypotheses where not precise enough in my former question. Let ...
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### Is every Noetherian *connected* ring a quotient of a Noetherian domain?

This question is a strengthening of this question (answered negatively), and arose due to David Speyer's answer here. Geometrically, this asks if every Noetherian connected affine scheme can be ...
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### Existence of a ring with specified residue fields

Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$? To prevent things from being too easy, I ...
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### Classification (and automorphisms) of torsion-free modules/sheaves

I would like to know what can be said about the classification of torsion-free modules. For my purposes, we can assume that $R$ is the function ring of a smooth affine variety over a field. How does ...
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### Minimal fields of isomorphism for varieties

Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...
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### Reference: A nowhere vanishing section of a vector bundle is locally split

Well-known fact: If $(A, \mathfrak{m})$ is a local Noetherian ring, $E$ is a finitely generated free $A$-module, and $e\in E$ is an element not contained in $\mathfrak{m}E$, then $E/eA$ is also a ...
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### Is the completion of an arbitrary ring w.r.p. a maximal ideal a local ring? [closed]

Give an arbitrary commutative ring (not necessarily noetherian) $A$ with $\mathfrak{m}$ a maximal ideal. Is the completion $\hat{A}$ w.r.t. the ideal $\mathfrak{m}$ a local ring?　If so, is the maximal ...
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### Arithmetic Cohen-Macaulayness of curves/surfaces defined by weighted power sums in 3 variables

Pick $p,q,r$ complex numbers (I am most interested in the case when they are positive integers). Define the function $P_i = px^i + qy^i + rz^i$ where $x,y,z$ are coordinates. I have a few related ...
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### stability of discriminant curve of nonsingular nets of quadrics

The following are from this question: A pencil of quadrics in $\mathbb{P}^n$ is a line in $\mathbb{P}^N$, where $N=\frac{n(n+3)}{2}$. So the space of pencil of quadrics is the Grassmannian ...
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### Descent of functions along finite birational morphisms

Let $A\to B$ be a morphism of (unitary commutative) rings such that $B$ is module-finite over $A$ and there exists $f\in A$ which is a nonzerodivisor in $A$ and in $B$, with $A[1/f]\to B[1/f]$ an ...
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### Reference for comparison of heart cohomology with standard cohomology

I'm looking for a reference for the following fact (which I believe to be true and should be easy for people who understand how spectral sequences arise from filtrations). Let A,B be two hearts of ...
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### A weak version of Bass' conjecture

Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the Bass conjecture states that $K_0(A)$ is a finitely generated abelian group. As far as I am aware, this ...
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### Are Abelian varieties (sometimes) globally $F$-split?

As defined by Karen Smith here, beginning of section 3? If $E$ is an elliptic curve, then it is when $E$ is ordinary. I wonder about higher dimension cases. Any references would be greatly ...
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### Let $f \colon X \to Y$ be an étale morphism of schemes. If $Y$ is integral, then is $X$ integral? [closed]

Let $f \colon X \to Y$ be an étale morphism of schemes. We know: (1) if $Y$ is normal, then $X$ is normal. (2) if $Y$ is regular, then $X$ is regular. (3) if $Y$ is reduced, then $X$ is ...
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### Are quotients of affine schemes by finite groups faithfully flat?

Let $R$ be a (Noetherian) ring, and $G$ a finite group acting on $R$. Consider the subring $R^G$. Is the map $R^G\rightarrow R$ faithfully flat? If not, does this become true if we restrict to ...
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### Degree bounds for Grobner Basis

Let $I= \langle f_1, \ldots f_n \rangle \subset K[x_1,\ldots, x_n]$ be a homogeneous ideal and $\operatorname{deg}(f_i) \leq d$ then it has Grobner Basis where degree of each generator is less than or ...
Let $R$ be an local Artinian ring, with maximal ideal $\mathfrak{m}$. Let $e$ be the smallest positive integer for which $\mathfrak{m}^e=(0)$. Let $t$ be the smallest positive integer for which ...