Questions tagged [commutative-algebra]

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

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Observations on the associativity of the tensor product between two modules.

I have a little question about of the associativity of the tensor products of two modules. If $M, N, P$ are $A$-modules, we know that $(M \otimes N) \ P \cong M \otimes N \otimes P$ If i want to prove ...
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Tensor product of $n$ modules with minimal assumptions

Let $R_1,\ldots, R_n$ be (commutative) rings, and $M_1 , \ldots, M_n$ be modules. What are the minimal assumpsions about the rings and modules needed in order to define a tensor product $M_1 \otimes_{...
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Show that $x$ and $y$ are irreducible in $K[x,y]/(y^2 - x(x+1)(x-1))$ where $K$ is an algebraically closed field

I've already showed that for the affine smooth curve $X = V(y^2 - x(x+1)(x-1))$ the coordinate ring $K[X]$ is a domain and so what I've tried to show that the ideals $(x)$ and $(y)$ are prime ideals (...
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Example 4 in page 76 of Atiyah-Macdonald, Introduction to commutative algebra

In p.76 of this book, an example 4 says "Take a strictly decreasing sequence $F_1 \supset F_2 \supset \cdots $ of closed sets in $X$ ". I don't understand if there exists such the sequence. ...
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Hartshorne Thm5.1 [closed]

Let $A=k[x_1,x_2,...,k_n]$ be a polynomial ring over algebraic closed field k,$b$ be the prime ideal of A,and $b \subset a$ be the maximal ideal of $A$.Let $B:=(A/b)_a$ be a local ring by maximal ...
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Finding out the induced map via factoring through canonical map $A \to S^{-1}A$ of localization

I was trying to prove that $\varinjlim_{U \ni x} A(U) \cong A_{\mathfrak{p}}$ where $U$ are basic open sets containing $x = \mathfrak{p} \in \operatorname{Spec} A$, and $A(U) = A_f$ where $U = X_f$. $...
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Is this quotient of a polynomial ring local?

Let $k$ be a field, $R = k[X_1, \ldots, X_n]$ and $I = \langle X_1, \ldots, X_n\rangle$. I am trying to prove that the quotient ring $R/I^r$, $r\in\mathbb{N}$ is local, i.e., has only one maximal ...
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Name for algebra which is commutative up a group action

I am wondering if there is a name for an algebra which is commutative up to some group action. To be more concrete, assume $A= \bigoplus A_n$ is a graded algebra, so $A_n \cdot A_m \subset A_{n+m}$, ...
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In the proof of extension of a homomorphism, proposition 5.23 of Atiyah-Macdonald, Introduction to commutative algebra.

I don’t understand the part of the proof of prop5.23 of Atiyah-Macdonald, Introduction to commutative algebra. The author separates the proof into two cases, the case that x is transcendental or the ...
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Minimal elements under division in a ring

Let $p$ be an element of a commutative ring with unity. The following definition is natural: $p$ is minimal under division if its only divisors (up to equivalence) are $1$ and itself. That is, for ...
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Corollary (1.2.6) from EGA 1 (2nd edition)

Let $A$ and $A'$ be two commutative rings with unity, and let $\phi: A' \to A$ be a ring homomorphism. We can then define the map $^a\phi:$ Spec$(A) \to$ Spec$(A')$ by $^a \phi(\mathfrak{p}) = \phi^{-...
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Weakly finite and their relationship with f.g [closed]

Let N be submodule of M; How hom (R/I,M/N) is finitely generated? How hom (R/I,M ⊕ N) is finitely generated? Thank you for much, Javad Abdali, student PhD math in Iran
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Completion of $(k[x_1, \dots, x_n]/I)_{(x_1, \dots , x_n)/I}$ is $k[[x_1, \dots , x_n ]] / Ik[[x_1, \dots , x_n ]]$?

Seems simple question but don't know to prove rigorously. In Eisenbud's Commutative Algebra book, in chapter 7, p.179 he wrote that "More generally, if $k$ is a field and $R:=k[x_1, \dots , x_n]/...
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A theorem of Bertini and regular sequence.

In Exercise 10.4 (p.243) in Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud, it writes Let $a,b$ be a regular sequence in a domain $R$, and let $S=R[x]$ be the polynomial ...
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Integral domain containing non-prime irreducibles (hence not a UFD) where all factorizations into irreducibles are unique

An integral domain is called a UFD if (1) every non-zero non-unit element factors into irreducibles, and (2) every element that factors into irreducibles does so uniquely (up to units and order). It ...
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Counterexample of $f^* : \operatorname{Spec}B \to \operatorname{Spec}A$ injective implies every prime ideal of $B$ extended

This is exercise 3.20, (ii) of Atiyah & Macdonald. $f^*$ is the induced map on $\operatorname{Spec}$ of $f: A \to B$, a ring homomorphism. I have seen a counterexample on MathSE, stating that $k[t^...
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filtered inverse limit of finite presented modules

Let $A$ be a ring, for example, a group ring over a field, then I'd like to ask, when is the filtered inverse limit of finite presented modules also a finite presented module? Is there some criterion ...
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when is a representation flat $k[G]$-module

Let $k$ be a field of characteristic $0$ or positive, and $G$ be a group, finite or $p$-adic analytic, then I'd like to ask when a $k$-representation of $G$ is flat, in the sense that it is flat as $k[...
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Why are the underlying polynomials of a rational matrix function commutative?

I am currently reading the book "The Theory of Matrices" where one chapter deals with the extension of scalar-valued functions to quadratic matrices. In the book, a matrix function is ...
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A characterization of a maximal ideal in a polynomial ring

Let $I$ be a proper ideal in $R=\mathbb{C}[x,y]$, generated by two elements $f_1,f_2$, $I=\langle f_1,f_2 \rangle$. I wonder if the following claim is true or false: Claim: If for every $c_1,c_2,d,e \...
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How to use the hint of Atiyah-Macdonald, Exercise 3.12 (iv)?

Here is the question: If $M$ is any $A$-module, then $T(M)$ is the kernel of the mapping $x \mapsto 1 \otimes x$ of $M$ into $K \otimes_A M$ where $K$ is the field of fractions of $A$. Here, $A$ is ...
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Proof verification of Atiyah-Macdonald, Exercise 2.26

Exercise 2.26: Let $N$ be an $A$-module. Then $N$ is flat if and only if $\operatorname{Tor}_1(A/\mathfrak{a},N) = 0$ for all finitely generated ideals $\mathfrak{a}$ in $A$. I think the proof is ...
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Does there exist a prime ideal $\mathfrak q\notin X$ with $\mathfrak q\subseteq\bigcup_{\mathfrak p\in X}\mathfrak p$?

I'm reading "Algebraic Number Theory" by Neukirch. The following paragraph is from P.70. To end this section, we now want to compare a Dedekind domain $\mathcal O$ to the ring $$\mathcal O(...
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Recover direct summands in derived category?

Let $E,F\in D^b(X)$, where $D^b(X)$ denotes the derived category of coherent sheaves on some smooth variety $X$. I am thinking about the following question: If $E \oplus E[1] \simeq F \oplus F[1]$, ...
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Short exact sequence of finitely generated R-modules

Assume that M',M'' are finitely generated R-modules (here R is a commutative unitary ring) and M is another R-module. The initial problem is that if there exists a s.e.s. $0\rightarrow M'\xrightarrow{...
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If a longest chain of prime ideals can be chosen to go through any given prime, does the same hold for inclusions of primes?

Let $R$ be a commutative ring (not necessarily Noetherian) of finite Krull dimension. Suppose that for any prime ideal $p$, $\dim(R) = \dim(R/p) + \mathrm{height}(p)$ (call this condition $DIM$) . ...
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Examples of PM rings

Def. A ring R is called a pm ring if each prime ideal is contained in exactly one maximal ideal. I asked AI to give some examples of pm rings. The answer is: Examples of PM rings in algebra include: ...
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Error in Jet Nestruev, Smooth Manifolds and Observables, Proposition 9.30?

Let $R$ be a commutative ring and $A$ a commutative $R$-algebra (with units). For an $R$-algebra homomorphism $\varphi: A \to R$ let $\mu_\varphi = \ker \varphi$. Let $T_{\varphi}^* A = \mu_{\varphi}/\...
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A ring is reduced if and only if it can be embedded into a product of fields

One of my homework questions asks me to prove that a commutative ring $A$ is reduced if and only if there exists fields $\{k_s\}_{s\in S}$ and an injective ring homomorphism $A\rightarrow \prod_{s\in ...
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Confusion on the proof of Oscar Zariski's Commutative Algebra, Theorem 24 in Chapter 4

I got stuck on the proof of Oscar Zariski's Commutative Algebra, Theorem 24 in Chapter 4. The theorem says: (I modified the expression, expanding some terms) Let $R$ be a noetherian ring. $\mathfrak{...
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There is a bijection between $\operatorname{Hom}_{Ring}(A,\mathbb{F}_2)$ and $\operatorname{Spec} A$

Let $A$ be a Boolean ring. One of my homework problems asks me to prove that the map of sets $\operatorname{Hom}_{Ring}(A,\mathbb{F}_2)\to \operatorname{Spec} A$ defined by $$ \phi\mapsto \ker(\phi) $$...
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Module over infinite product of rings

I ask this question yesterday, i wish it is more clear now. Let $X=Spec(R)$ the spectrum of a ring $R$ In the article "Modules projectifs et espaces fibrés à fibre vectorielle". Jean pierre ...
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Question about If $B$ is a $A$-algebra and $N$ is a $B$-module, then can $N$ naturally become a $A$-module?

So I think the only point is to show how to build the $A$-module structure on $N.$ We know that $B$ is an $A$-algebra which is also an $A$-module where the $A$-module structure is given by the scalar ...
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An easy property for extension and contraction of ideal

So given ring homomorphism $f:A\rightarrow B,$ for commuatative ring with identity. My problem is about the property $\mathfrak{a}\subseteq\mathfrak{a}^{ec}.$ It's easy to make myself convinced about ...
Beginner's user avatar
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Prove that a field/commutative ring is IBN (invariant basis number).

I am searching all over and it keeps coming up how trivial it is, but I don't actually see the proof of a field being IBN. I would also like to know where a proof of a nonzero commutative ring is IBN ...
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Generating a prime ideal of a polynomial ring using polynomials that involve few variables after a change of variables

Let $A=\mathbb{C}[x_1,\dotsc,x_n]$. For $f\in A$, let $v(f)$ be the smallest $t$ such that there are $y_1,\dotsc,y_t$, where each $y_i$ is a linear combination (over $\mathbb{C}$) of $x_1,\dotsc,x_n$, ...
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Degree zero part of localized graded ring in computer algebra

Does anyone know how to do the following computation in a standard computer algebra package (MAGMA, SAGE, Macualy2) Let $S$ be a graded ring given by explicit generators (not necessarily all in degree ...
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Difficulty understanding the claim: base change according to a purely inseparable field extension is a homeomorphism

$\newcommand{\Frac}{\operatorname{Frac}}\newcommand{\spec}{\operatorname{Spec}}\newcommand{\trdeg}{\operatorname{trdeg}_k}\newcommand{\p}{\mathfrak{p}}\newcommand{\q}{\mathfrak{q}}$In Liu's algebraic ...
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Explicit proof of the fact that a non integrally closed domain is not a UFD

Context. Let $R$ be a domain. It is well-known that $R$ is a PID $\Rightarrow$ $R$ is a UFD $\Rightarrow$ $R$ is integrally closed (in its field of fractions). In other words, we have $R$ is not ...
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Algorithms to decompose a graded module over R[x], where R is a PID

I have a certain class of objects, which can be thought of either as modules over a ring $R[x]$ or as functors, and I am looking for algorithms to decompose these objects into indecomposable summands. ...
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Small generating set for the unique minimal prime ideal of a finitely generated $\mathbb{C}$-algebra

Let $A$ be a $\mathbb{C}$-algebra, generated by $n$ elements ($n$ finite). Assume that $A$ has a unique minimal prime ideal $\mathfrak{p}$. Write $t$ for the minimal number of generators for the ideal ...
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Tensor product of two free resolutions yields a free resolution (closed binomial edge ideals)

I am trying to read the paper Closed binomial edge ideals by Irena Peeva: https://www.degruyter.com/document/doi/10.1515/crelle-2023-0048/html My question is related to Lemma 4.6, where it is claimed ...
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Proving that $Rad ( \bigoplus_{i \in I} M_{i})= \bigoplus_{i \in I} Rad (M_{i})$

I want to prove that $ Rad ( \bigoplus_{i \in I} M_{i})= \bigoplus_{i \in I} Rad (M_{i})$, according to how Kasch sketched the proof in Modules and Rings in Corollary 9.1.5. I got the gist of the ...
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Are one object involutive categories commutative monoids?

The nlab has this paragraph on dagger categories (from: https://ncatlab.org/nlab/show/dagger+category) : In enriched category theory, involutive categories have also been called symmetric categories. ...
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Projective and free modules

In the graded context, if $R = K[x_1,...,x_n]$ where $K$ is a field, is a projective R-module a free R-module?
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Example of two group filtrations that induce different topologies but the same completion

This is a question about completion in the sense of commutative algebra, i.e. inverse limit of quotients. Let $G$ be a topological abelian group, $G_0 \supseteq G_1 \supseteq \cdots$ and $H_0 \...
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R-module whose endomorphism ring equals R

For a commutative ring $R$, there is a natural map $m:R\to\text{End}_R(M)$ of any $R$-module $M$. Assume $M$ is finitely generated. If the map $m$ is an isomorphism, one speculates more or less that $...
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Computing Riemann Roch space for $\mathbb{P}^n$

Suppose $X=\mathbb{P}^2$ and we take $V=V(F)$, where $F$ is a homogeneous polynomial of degree $d$, and I'd like to compute a basis for the space $H^0(X,V)$. By definition, we have $f\in H^0(X,V)$ ...
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Counterexamples to $\hat{A} \otimes_A M \cong \hat{M}$

I'm looking for counterexamples to Proposition 10.13 in Atiyah-Macdonald when the hypotheses are not satisfied (throughout, $A$ is a ring, and $\hat{\phantom{M}}$ denotes the $I$-adic completion for ...
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Free summand of a module vs. finite direct sum of copies of it

Let $M$ be a finitely generated module over a commutative Noetherian ring $R$. If $R$ is a direct summand of $M^{\oplus n}$ for some $n\ge 1$ , then is $R$ a direct summand of $M$? I can easily prove ...
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