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

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Module of differentials of a tensor of algebras

Fix a base ring $A$. I am looking at Lemma 6.1.11 in Liu. Let $B_1, B_2$ be $A$-algebras, $R = B_1 \otimes_A B_2$. Then there is a canonical isomorphism $$ \varphi: (\Omega_{B_1 / A} ...
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1answer
44 views

Localization of $\mathbb{Z}_6$ with respect to the powers of $2$

I want to localize $\mathbb Z_6$ with respect to the powers of $2$. Now if $\frac{2}{2} = \frac{a}{b}$, by $c(2a-2b)= 0$ and letting $c=3$ we conclude if $a \neq 0$ then $\frac{a}{b} =1$. Thus ...
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1answer
20 views

determinant annihilates ring vector

Let $R$ be a ring, and let $A$ be an $n\times n$ matrix with coefficients from $R$. Suppose for $r\in R^n$ we have $Ar=r$. Prove that $\det (A-I)\cdot r=0$. It is actually part of a bigger problem ...
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5answers
177 views

$K[[X]]$ is not a finitely generated $K[X]$-module.

How can I prove that $K[[X]]$ is not finitely generated over $K[X]$ as a module, where $K$ is a field. What I tried: if above is not true then $K[[X]]$ is integral extension over $K[X]$. But I ...
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1answer
25 views

What is the Hilbert Series of $R/I$ for a regular sequence? [closed]

What is the Hilbert series of $R/I$ for $I = (F,G)$ where $F,G$ is a regular sequence on $R = k[x,y]$ with $\deg F \leq \deg G?$ Definition: A sequence $F,G$ is regular on $R$ if $F$ is a nonzero ...
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1answer
45 views

Hilbert function of $k[x_1, x_2, x_3]/ (x_1^2,x_2^2x_3,x_2^3)$

If $R = k[x_1, x_2, x_3]$ is a polynomial ring and $I = (x_1^2,x_2^2x_3,x_2^3)$ how do you see that the Hilbert function $H(R/I,i) = 4$ for $i \geq 4$? So the free resolution of $R/I$ is $$0 \to ...
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0answers
30 views

Any characterization for commutative rings over which “projective modules” equal “free modules”?

As far as I know, over any PID, an polynomial rings over a field, or an local ring, projective modules are always free. This kind of results make me curious about if there are any overall ...
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1answer
72 views

Hilbert function and series.

If $f$ is a homogeneous polynomial of degree $d$ in a polynomial ring in $t$ variables over a field, and it generates an ideal $I$. Then the Hilbert function of $R/I$ is $$H(R/I,n) = ...
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1answer
53 views

Do we have $\mathbb{C}[V^*] \cong S(V)$ or $\mathbb{C}[V] \cong S(V)$? [closed]

Let $V$ be a vector space over $\mathbb{C}$ and $V^*$ its dual vector space. Let $\mathbb{C}[V^*]$ (resp. $\mathbb{C}[V]$) be the coordinate ring of $V^*$ (resp. $V$) and $S(V)$ the symmetric algebra. ...
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0answers
33 views

Ideal generated by a set of polynomials $X^{a/b}$ where each monomial having $a$ and not having $b$

Let $$\mathcal R=\mathbb Z_2[x_1,\dots,x_n]/\langle x_1^2-x_1,\dots,x_n^2-x_n\rangle.$$ I want to learn ideal arithmetics to deal with polynomials of the forms such as Consider a set of ...
2
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0answers
35 views

Characterization of the zero set of an elementary symmetric polynomial over finite field

Consider an elementary symmetric polynomial of degree $d$ in $n$ variables over a finite field $F_q$. Is there a nice characterization of its zero set? We are particularly interested in the zero set ...
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0answers
34 views

Is there an accepted definition of coprimality in commutative ring theory?

I can think of at least three possible definitions of coprimality in commutative ring theory: $a,b \in R$ are coprime iff if $c \mid a$ and $c \mid b$, then $c \mid 1$. if $a \mid c$ and $b \mid c$, ...
2
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1answer
43 views

Projective curve $x^3+y^3=2z^3$ in $\mathbb P^2$ singular?

Is the projective curve $x^3+y^3=2z^3$ in $\mathbb P^2$ (defined over $\mathbb{C}$) singular or nonsingular? If singular, what are the types of these singularities? For an affine curve, one would ...
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0answers
16 views

Associated primes of quotient rings [duplicate]

Let $R$ be a commutative, Noetherian ring with unit. Let $r\in R$ neither a unit nor a zero divisor. I want to show that $AP(R/(x))=AP(R/(x^n))$ for all $n\in\mathbb{N}$. ($AP$ denotes associated ...
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1answer
31 views

Is there an easy example that shows that the initial ideal of a radical ideal is not necessarily a radical ideal itself?

Is there an easy example that shows that the initial ideal of a radical ideal is not necessarily a radical ideal itself? This is the converse of if the initial ideal of an ideal is radical, then ...
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1answer
23 views

Proving the Nilradical of a Commutative Ring is closed under addition. [duplicate]

Let $R$ denote a commutative ring and $Nil(R)$ is the ideal consisting of all nilpotent elements in $R$. I am attempting to prove that $Nil(R)$ is closed under addition. My work so far is ...
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1answer
85 views

When is $\operatorname{Hom}(M, E)$ injective?

Let $R$ be a commutative Noetherian ring with non-zero identity, $M$ be an $R$-module, and $E$ be an injective $R$-module. When is $\operatorname{Hom}(M, E)$ an injective $R$-module?
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1answer
32 views

Connected affine scheme

I am trying to find a simple algebraic proof of the fact that if the affine scheme $X$ associated to a ring $A$ is connected, then $A$ is a nontrivial product of rings. I know that if $\mathrm{spec}\ ...
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0answers
69 views

infinitely $p$-divisible elements in $A\otimes \mathbb{Z}_p$

Let $A$ be a (possibly non-finitely generated) torsion-free abelian group. Suppose that $A$ contains no infinitely $p$-divisible elements, then does the same hold for $A\otimes \mathbb Z_p$, where ...
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1answer
39 views

Singular homogeneous polynomial [closed]

Let $p$ be a homogeneous polynomial in $\mathbb{K}[x_1, \dots, x_n]$, where $\mathbb{K}$ is an algebraically closed field. If $\frac{\partial p}{\partial x_1}, \dots, \frac{\partial p}{\partial x_1}$ ...
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0answers
23 views

Change of scalars and field of fractions

Does someone know a reference in a textbook for the following fact? Let $k \subseteq k'$ be an algebraic field extension and $A$ a $k$-algebra such that the extension of scalars $A \otimes_k k'$ ...
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1answer
30 views

Associated primes of a non-zero module always exist if corresponding ring is Noetherian

Let $R$ be a commutative Noetherian ring. Let $M$ be a non-zero $R$-module (not necessary finitely generated). Prove that set of associated primes of the module is not empty, ...
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1answer
97 views

Show that a radical ideal has no embedded prime ideals. [closed]

Let $A$ be a commutative ring and $I$ a decomposable ideal. Let $I=\bigcap_{k=1}^{n} I_k$ be a minimal primary decomposition. Show that if $I=\sqrt{I}$ then $I$ has no embedded prime ideals. (I ...
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1answer
34 views

Almost Noetherian module

An $R$-module $M$ is called an almost Noetherian $R$-module if every proper submodule of $M$ is finitely generated. My question is the "if" part of (ii): it says that in order to show that $K$ is ...
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0answers
47 views

When do the characters of an integral domain form an algebraically independent subset?

Work over a fixed commutative ring $R$. I'll just make sure we're all on the same page by quickly stating the relevant definitions: Definition 0. Given an $R$-algebra $X$ together with a subset ...
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1answer
35 views

Proving a sequence is split exact using Nakayama's lemma

Let $R=\bigoplus_{i=0}^{\infty}R_i$ be a graded ring with $R_0$ a field and let $M=\bigoplus_{i=0}^{\infty}M_i$ be a finitely generated graded module. Let $I \subseteq R$ be the homogeneous maximal ...
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2answers
62 views

Discrete valuation ring and finitely generated submodules

Let $R$ be a Discrete Valuation Ring with fraction field $K$. Will this imply any proper $R$-submodule of $K$ is finitely generated (hence a fractional ideal)? I know $K$ is not finitely ...
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1answer
26 views

Flatness in terms of the freeness of the pushforward of the structure sheaf…

Ravi gives a version of the claim below in exercise 24.4G in "Foundations...", but he adds the condition that $Y$ is reduced in order to conclude something additional (the equivalence of the condition ...
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1answer
38 views

Preservation of Krull dimension under inverse limit

Does inverse limit preserve Krull dimension of any inverse system of rings (at least in the case of zero-dimensional)? I mean, is the Krull dimension of the inverse limit zero when that of any ...
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1answer
37 views

Nakayama and locally free sheaves

Let $S$ be a noetherian scheme and $F,E$ be two coherent sheaves on $S$ with $E$ locally free. Suppose we have a morphisme $f : F \to E$ such that $f_s : F \otimes \kappa(s) \to E \otimes \kappa(s)$ ...
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1answer
50 views

Let $A$ a ring such that $0=m_1m_2…m_k$ where $m_i$ are maximal ideals of $A$. [closed]

Let $A$ a ring such that $m_1m_2...m_k=0,$ where $m_i$ are maximal ideals of $A$. Then $A$ is Noetherian if and only if $A$ is Artinian.
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36 views

If $R$ is $I$ and $J$-adically complete, then it is $(I+J)$-adically complete.

Let $R$ be a Noetherian ring with ideals $I$ and $J$. I already proved the following: Lemma: If $I \subseteq J$ and $R$ is $J$-adically complete, then $R$ is $I$-adically complete. And now I'm ...
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0answers
41 views

Is torsion of a topological module closed?

I was asking to myself the following question. Consider a $p$-adically complete and separated topological algebra $R$ over $\mathbb{Z}_{p}$. As $\mathbb{Z}_{p}$ is a domain, we know that the ...
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0answers
37 views

An irreducible variety is not composed of finitely many subvarieties

There is a lemma in commutative algebra: Let $\mathfrak{a}_1, \dotsc, \mathfrak{a}_n$ be ideals such that $\mathfrak{a}_n \cap \dotsb \cap \mathfrak{a}_n$ is contained in a prime ideal ...
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0answers
22 views

Algorithmic computing kernel of a graded homomorpism

For computing kernel of a module homomorphism we can use module-Grobner basis such as described in notes talking about computing SyZyGies. How can we compute kernel of a homomorphism between a graded ...
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1answer
38 views

The stalk of a specialization is a localization of the generization.

So I am trying to prove that in for a scheme locally of finite type over an algebraically closed field, smoothness and regularity are the same thing. I am working in Görtz and Wedhorn. In lemma 6.26, ...
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1answer
30 views

Minimal graded free resolution and matrix representations

In a graded $R$-module, let $$0 \to C_p \xrightarrow{\phi_p} C_{p-1} \xrightarrow{\phi_{p-1}}C_{p-2} \to \dots \to C_1 \xrightarrow{\phi_1} C_0 \xrightarrow{\psi} M \to 0$$ be a minimal graded free ...
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1answer
54 views

Representation of a linear map as a matrix.

Let $I = \left < f_1, \dots, f_n \right > \subset R$ be an ideal generated by homogeneous elements where $\deg(f_i) = d_i$ and $\phi$ be the graded $R$-mod homomorphism $$\phi: R(-d_1) \oplus ...
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2answers
46 views

Showing a Variety is Rational?

I'm trying to show that the following varieties are rational: $V_1=V(y^2z-x^3)$ and $V_2=V(xyz-x^3-y^3)$. But I can't think of how to show they are birationally equivalent to $\mathbb{A}^n$ or ...
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1answer
29 views

Definition algebra over commutative ring (injectivity needed?)

I'm aware that many differently looking definitions exist for an algebra over a commutative ring $A$. For me, the most natural definition of an algebra over a commutative ring $A$ consists of a tuple ...
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0answers
25 views

Why are Newton polygons only ever treated over complete DVRs?

Let $R$ be a discrete valuation ring, and $f(x) := c_nx^n + \cdots + c_1x + c_0\in R[x]$ a polynomial with $c_nc_0\ne 0$. Then the newton polygon of $f(x)$ is the lower convex hull of the points ...
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1answer
27 views

Completion of a local ring, Vakil 29.3A

If $p$ is a point of $X$, which is a $\bar{k}$ variety of dimension $1$, $p$ is a node if the completion of $\mathcal{O}_{X,p}$ at $m_{X,p}$ is isomorphic to $\bar{k}[[x,y]]/(xy)$. If now ...
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1answer
53 views

Are rings of fractions of integral domains closed under finite intersection?

Let $D$ be an integral domain with fraction field $K$. Let $V$, $W$ be multiplicatively closed subsets of $D$. Consider the rings of fractions $V^{-1}D$ and $W^{-1}D$ as subrings of $K$. Is ...
3
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1answer
107 views

If $f$ is a polynomial and $g(n+1)-g(n)=f(n)$, then $g$ is a polynomial. [closed]

Assume that $f$ is a polynomial of degree $s$ which is not constant, and that for sufficiently large positive integers $n$, $g(n+1)-g(n)=f(n)$. Here $g$ is defined on the positive integers. Must ...
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0answers
33 views

Conditions for a ring to be a direct product of local rings

I recently came across a property of commutative rings which I could prove only for rings that are (isomorphic to) a direct product of (possibly infinitely many) local rings. It might be that my ...
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2answers
53 views

Minimal graded free resolution of the ideal $I = (x^r, y^s) \subset k[x,y]$

What is the minimal free-graded resolution of the ideal $I = (x^r, y^s) \subset k[x,y]=R$ for $r,s \in \mathbb{N}$? I tried reducing this down to $r = s = 1$ and I think it is $$0 \to R(-2) \to ...
1
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1answer
62 views

dimension of an affine variety over $ \mathbb C.$ [closed]

Consider the polynomials $f,g,h \in \mathbb C [x_1, \dots,x_n]$ defined by $ f(x)=x_1^2 + \cdots + x_{n-2}^2, \quad g(x)=x_{n-1},$ and $ h(x)=x_n.$ How can I compute the dimension of the variety ...
3
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1answer
87 views

Variant of Nakayama's lemma

I am trying to prove that if $M$ is an $R$-module, with $R$ complete w.r.t. an ideal $\mathfrak{m}$, and $M$ is separated ($\cap_k \mathfrak{m}^k M=0$) and the images of $m_1,\dots,m_n$ generate ...
2
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2answers
67 views

How is finiteness of solutions (zero-dimensionality) related to Krull's dimension?

I have encountered a lot the concept of zero-dimensional ideal: Let $k$ be a field. An ideal $I\subseteq k[x_1,...,x_m]$ is said to be zero-dimensional if its zero set $Z(I)$ has a finite number ...
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1answer
45 views

Induced maps between coordinate rings of $\mathcal Z(xy-z)$ and $\mathbb A^2$

I'm trying to understand the answer here to the question of finding the induced maps of coordinate rings corresponding to explicit isomorphisms between $\mathcal Z(xy-z)$ and $\mathbb A^2$. A ...