# Questions tagged [ac.commutative-algebra]

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

3,688 questions
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### $P$ is projective if and only if $P\otimes N\cong Hom(Hom(P,R),N)$

Let $R$ be a commutative Noetherian ring and $P$ be finitely generated $R$-module. How to prove the following. $P$ is projective if and only if $P\otimes N\cong Hom(Hom(P,R),N)$ for all finitely ...
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### tangent cone and local picture

Let $X\subset \mathbb P^n_{\mathbb C}$ be a closed algebraic subset and $x\in X$. Suppose tha tangent cone of $X$ at $x$ is the union of (say) a plane and a line (meeting at the origin). Can we ...
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### $\omega$-categorical algebra

Let us consider a 1-category $C$. For any commutative and unital ring $k$ the the free $k$-module generated by the morphisms of $C$ can be equipped with an algebra structure by setting $fg$ to be ...
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### Strongly isovariant (aka fixed point reflecting / stabiliser preserving) morphisms

I have some questions about what feels like basic topics in quotients of schemes by group actions. Consequently I suspect there are well-known references; I couldn't find them, though. First ...
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### Tensor square of duals over a domain

The title is motivated by my needs ($M=N$ in the sequel). Linked to the question here and there (in the case of products) is the following. Let $M,N$ $k$-modules ($k$ a commutative ring), then we ...
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### Intersection of a reduced projective variety with a general hyperplane is reduced

Let $X\subset \mathbb{P}^n$ be a reduced closed subscheme. For a general hyperplane $H$, $X\cap H$ is again reduced (and of dimension one less). Is there an easy proof of this result? Algebraically, ...
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### Valuation Rings and Ultrafilters

I notice there is a certain similarity between the definition of a valuation ring and the definition of an ultrafilter. To begin, take a field $K$ and let $\mathcal{A}$ be the set of subrings of $K$. ...
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### Coordinate free expression for the determinant of a $2 \times 2$ matrix

Let $E\rightarrow X$ be a rank two vector bundle on a variety $X$. How can one write an abstract map $Sym^2(E)\rightarrow (\wedge^2 E)^{\otimes 2}$ that in local coordinates, when we fix a frame of $E$...
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### Extension of alg closed fields = limit of smooth algebras

In the proof of https://stacks.math.columbia.edu/tag/0F0B, it is a claim that if $K/k$ is an extension of algebraically closed fields, then $K$ is a limit of smooth $k$-algebras. This is justified ...
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### Graded analogues of theorems in commutative algebra

Many theorems in commutative algebra hold true in a ($\mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word commutative ...
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### Is the formal power series ring integrally closed?

Let $k$ be a field and $s$ and $t$ be variables. Is the ring $k[s][[t]]$ integrally closed in $k[s,s^{-1}][[t]]$?
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### Are there enough curves (to connect 'points' of f.g. algebras)?

(Intuition: any two points in a connected space may be connected by a path. I would like to know if something like this holds in certain category of `connected algebraic spaces'. I formulate the ...
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### A question on dominant morphism of affine schemes

Let $A \subseteq B$ be a ring extension where $A,B$ are both finitely generated $\mathbb C$-domain of the same Krull dimension. Also assume $A$ is regular (i.e. $A_{ \mathfrak p}$ is regular local ...
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### Graded Betti numbers $\beta_{n,j}$ for points in $\mathbb{P}^n$

Let $S = \mathbb{C}[z_0, \dots, z_n]$, and let $X$ be a set of points in $\mathbb{P}^n$. I'm looking for references concerning results for the graded Betti numbers $\beta_{n,j}(S/I(X))$, i.e., the ...
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### Algebra for algebraic topology

My research is in analysis, but it moved to the area that requires algebraic topology. I have some working knowledge in that area, but I always feel that I am on a shaky ground and I need to go back ...
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### Explicit description of injective hulls

Let $k$ be a field, let $R:=k[x_1,\ldots,x_n]$, and consider the $R$-module $M:=R/{(x_1,\ldots,x_n)}\cong k$. Then the injective hull $I_M$ of $M$ admits the following explicit description:  I_M = k[...
Vaguely, an arithmetic surface is an algebraic curve with coefficients in some "ring of integers". More formally it is an integral normal scheme $X$ of dimension 2 of finite type over a Dedekind ...