# Questions tagged [commutative-algebra]

Questions about commutative rings, their ideals, and their modules.

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### For a projective cover $(\sigma, P)$ of a module $M$, $P$ is indecomposable implies $M$ is indecomposable

We say that a module $M$ is indecomposable if for $M=M_{1} +M_{2}$ (not direct sum) we have that $M_{1}=M$ or $M_{2}=M$. Let $\sigma:P \to M$ a projective cover of $M$, this means that $\sigma$ is an ...
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### direct product decompositions of semi-rings

(With Hilbert's Basis Theorem in mind.) Is it true that every finitely presented commutative semi-ring with unit is a finite direct product of directly indecomposable factors?
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### A puzzling computation by Mumford of a module of Kähler differentials.

Mumford in his Red Book gives on page 144 an example of computation of a module of Kähler differential forms. Namely, he lets $k$ be a field and considers the quotient algebra $B=k[X,Y]/(XY)$. He ...
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### Integral closure of $k[x^3,x^2y,y^3]$ in field of fractions

Let $A = k[x^3,x^2y,y^3] \subset k [x,y]$. I want to find the integral closure of $A$ in its field of fractions. To do so, I first want to find the field of fractions $\mathrm{Frac}(A)$. I think ...
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### A bundle pull-back along itself

Let $X$ be a scheme, $\mathcal{E}$ a locally free $\mathcal{O}_X$-module of finite rank and $p: E\to X$ the corresponding geometric vector bundle (with global sections $\mathcal{E}$). Do we have an ...
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### Localizations of $k[x,y]/(f)$ UFD

Let $k[x,y]$ be a polynomial ring in two indeterminants and $f \in k[x,y] \backslash k$ a non constant poynomial. I want to know if there exist any nice criteria to answer the question when the ...
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Let $B \to A, B \to B'$ be injective, finite ring homomorphisms (finite means that $A$ and $B'$ are finite $B$-modules). Suppose that $A$ and $B$ are integral domains. Denote by $N$ the nilradical of ...
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### finiteness of Koszul groups

A basic question about Koszul homology from Matsumura's Commutative Ring Theory In Theorem 16.5(ii) it is assumed that $(A,m)$ is a local ring and $x_1,\ldots,x_n \in m$, and $M$ is a finite $A$-...
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### Blow-up and regular sequence

I'd like how to deduce, if it's possible that: the blowup of an affine variety $X$ along $V(g_1,\ldots,g_k)$ is $V(t_i g_j-t_j g_i)_{i,j}\hookrightarrow X\times\text{Proj}(k[t_1,\ldots,t_k])$ if the ...
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### A subspace of a set of bounded and continuous functions is closed in this set and the ideal of this set

Question: Let $(X, d)$ be a compact metric space. For a given $x_0 \in X$, define $C_{x_0}(X,\mathbb{R})$ by $$C_{x_0}(X,\mathbb{R}) = \{f \in C(X, d):f(x_0) = 0\}$$ Note that $C(X,\mathbb{R})$ ...
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### List of equations in MAGMA

I have some integer $n$, some ambient affine space $\mathbb{A}^n$, and a list $L$ of equations $f_{ij}$ cutting out a variety $X$ in the ambient space. I have problems defining the list $L$ correctly. ...
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### Tensor product restriction of scalars and isomorphism

Let $R$ be a commutative ring and $S\subset R$ a subring. Let $M$ and $N$ be two $R$-modules. 1) Via restriction of scalars, we can also view $M$ and $N$ as $S$-modules. Show that we have a ...
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### Checking irreducibility of polynomials in two variables

There are a few exercises in Hartshorne about checking singularity of an affine curve. For example, $Y$ defined by $x^2 = x^4 + y^4$ over a field $k$ (with ${\mathrm{char}}k \neq 2$). This is easy. ...
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### Coprime polynomials of consecutive degrees

Let $f,g \in k[t]$ be two polynomials, $k$ a field of characteristic zero. Assume that $f$ and $g$ satisfy the following two conditions: (i) $f$ and $g$ are coprime, namely, they have no common root (...
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### Stalks of Plane Conic $C \cong \mathbb{P}^1$ are UFD

Assume $k$ is a alg closed field. Then it is easy to check that the morphism $\phi: \mathbb{P}^1 \to \mathbb{P}^2, (x_0:x_1) \mapsto (x_0^2: x_0x_1:x_1^2)$ induces an isomorphism between $\mathbb{P}^1$...
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### going between property

Let $R'\in R$ be an integral extension of rings with $R'$ a $K$-algebra finitely generated . Consider a chain of different prime ideals in $R'$ , $P_{1}\subsetneq P_{2}\subsetneq P_{3}$ Such that ...
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### Can commutative local rings have any non-zero zero divisors? [closed]

Can commutative local rings have any non-zero zero divisors? Is this possible?
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### Computing projective dimension over hereditary rings (modules).

I want to prove this example from Rotman's which I have not found proved in literature yet and Im curious about. Let $_{R}M$ be a left module over a left herditary ring $R$, then $p.d(_{R}M) \leq 1$,i....
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### $\mathbb{R}[X]$ is an integral extension of $\mathbb{R}[X^2-1]$

I am trying to prove that every polynomial of $\mathbb{R}[X]$ satisfies a monic polynomial equation with coeffients in $\mathbb{R}[X^2-1]$ that is every polynomial $b(x)= x^m+b_{m-1}x^m-1+...+b_{0}$ ...
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### Is the canonical map $U^* \otimes V^* \to (U \otimes V)^*$ always injective? [duplicate]

Let $U$ and $V$ be modules over a commutative ring $K$. Is the canonical map $U^* \otimes V^* \to (U \otimes V)^*$ always injective? I'm a differential geometer so I'm usually dealing with finite-...
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### Ideals whose union is an ideal [duplicate]

Does someone have an example of three ideals such that no one is contained in another, yet their union is an ideal?
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