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

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1answer
51 views

The height of a prime ideal in the $\kappa[[X]][Y]$

Let $\kappa$ be a field and $S=\kappa[[X]]$ be the ring of power series which depends on the indeterminate $X$. Now consider the ring $S[Y]$, the ring of polynomials with coefficients in $S$ and ...
6
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1answer
141 views

Ext of an $\mathfrak{m}$-primary ideal

Let $(A,\mathfrak m,k)$ be a Noetherian local ring, $M$ a finitely generated $A$-module, and $I$ an $\mathfrak{m}$-primary ideal. If $\operatorname{Ext}^{i}_{A}(A/\mathfrak{m},M)=0$ then ...
0
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1answer
74 views

Localization of two rings which is an integral extension, then integral extension still holds?

Question seems simple, but I just can't find the solution. Let $A/B$ be an integral ring extension and let $P$ be a prime ideal of $B$. By lying-over theorem, there is $Q$, a prime ideal of $A$, ...
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1answer
58 views

$\mathbb Z_p[T]/(T^a,p^b)\cong\mathbb Z_p[[T]]/(T^a,p^b)$

Let $a,b\in \mathbb N,$ then $$\mathbb Z_p[T]/(T^a,p^b)\cong\mathbb Z_p[[T]]/(T^a,p^b)$$ 1.What is this isomorphism ? 2.How to prove that $|\mathbb Z_p[[T]]/(T,p)^t|=p^{t(t+1)/2}$ Now ...
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2answers
79 views

Calculation of codimension

I've just learnt the idea of codimension, then I try to find some exercises to calculate the codimension of some maximal ideal $J$ in the ring $R=k[x,y,z]/I$, I find it somewhat tricky. For example, ...
3
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0answers
70 views

Depth for intersection of prime ideals

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over field $K$. How can one compute $\operatorname{depth}(R/\bigcap_{i=1}^r p_j)$, where each $p_j$ is generated by some variables $x_i$ and have a ...
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0answers
58 views

what classes of modules admit finite free resolutions?

As i understand, finitely generated graded modules over Noetherian graded rings admit a finite free resolution (FFR). What are other classes of modules that admit a FFR? How about finitely generated ...
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1answer
92 views

Lemma 1.3.4(b) in Bruns and Herzog

My question refers to the proof of the second of the following lemma given in Cohen-Macaulay rings by Bruns and Herzog. Lemma 1.3.4 (Bruns and Herzog): Let $(R,m,k)$ be a local ring, and $\phi:F ...
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1answer
92 views

Algebraic independence in $ k[x,y]$

Let $k$ be a field, then $x$ and $y$ are algebraically independent in polynomial ring $k[x,y]$, so I would guess that 2 is the maximal number of algebraically independent elements in $k[x,y]$ But I ...
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1answer
44 views

Explanation about notation

Can anyone explain to me what $C[x^2,x^3]_{(x^2,x^3)}$ means? It is connected with localizations but it is unclear to me what it means exactly.
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1answer
34 views

$\dim_{\mathbb F_p}(\mathbb F_p[[T]]/(T^l))=l$

Let $\mathbb F_p[[T]]$ be the ring of formal series over $\mathbb F_p$, and $l\in\mathbb N.$ How to prove that: $\dim_{\mathbb F_p}(\mathbb F_p[[T]]/(T^l))=l$ ?
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1answer
36 views

proving $(M\otimes N)/J (M \otimes N) \cong M \otimes \left(N/JN \right)$

Let $f: R \rightarrow S$ be a ring homomorphism of Noetherian rings, $M$ finite $R$-module and $N$ finite $S$-module which is flat over $R$. Let $J$ be an arbitrary ideal of $S$. I want to prove that ...
4
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2answers
148 views

$\mathbb{C}[x,y]/(x^2+y^2+1)$ is an integral domain.

I stuck in the following question. Prove that $ \mathbb{C}[x,y]/\langle x^2+y^2+1 \rangle $ is an integral domain, using the following: Let $\mathbb{F}$ be a field, $c \in \mathbb{F} $. ...
0
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1answer
69 views

Reduceness and faithfully flatness

I have a "well known question" for which I do not find a reference. Let $A$ and $B$ be a commutative rings and $A\rightarrow B$ be a faithfully flat morphism. Let $C$ be a ring over $A$. Is it ...
2
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0answers
60 views

Regular sequence in local rings

Assume that $(R,m)$ is a local ring and $J\subset I$ are proper ideals of $R$. If $I/J$ is generated by regular sequence in $R/J$, I want to show that $$J\cap I^t=JI^{t-1}\ \forall t\geq1.$$
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2answers
139 views

$||x||=1$ in $K/\mathbb{Q}$ implies $x$ is a root of unity.

Let $K/\mathbb{Q}$ a finite (i.e. algebraic and finitely generated) extension. Let $x \in K$, such that $||x||=1$ for all normalized absolute values of $K$ but at most one. Then $x$ is a root of ...
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1answer
58 views

an equivalence of functors involving tensor and hom

Let $\phi:(R,m,k) \rightarrow (S,n,l)$ be a morphism of local Noetherian rings. Let $M$ be a finite $R$-module and $N$ a finite $S$-module that is flat over $R$. Question: Why is it true that ...
4
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1answer
102 views

Example of Artinian ring which is not a finitely generated k-algebra

In wikipedia it says: Let $A$ be a commutative Noetherian ring with unity. Let $k$ be a field and $A$ finitely generated $k$-algebra. Then $A$ is Artinian if and only if $A$ is finitely generated ...
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1answer
54 views

Direct sums and Dedekind domains.

Suppose that $D$ is a Dedekind domain and $I,J,K$ are nonzero ideals of $D$. Is there any necessary and sufficient condition for $I \oplus J \cong D \oplus K$?
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1answer
230 views

Torsion-free and projective modules over a Dedekind domain

Suppose that $A$ is a Dedekind domain (and integral domain). I am trying to prove that if $M$ is a torsion-free $A$-module, then it is projective and vice versa. Suppose that $M$ is projective. Then ...
3
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2answers
137 views

Proper ideal containing power of maximal ideal is primary?

In Eisenbud's Commutative Algebra, Chapter 3.9, it says, "If $P$ is a maximal ideal of $R$ and $I$ is any proper ideal containing a power of $P$, then $I$ is $P$-primary: For in this case $P$ is the ...
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1answer
86 views

When a function field is a regular extension of the field of coefficients?

Let $A$ be an integral affine $k$-algebra with field of fractions $K$. I am wondering when the extension $K/k$ is regular. In particular, is the following statement correct? $K/k$ is regular ...
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0answers
73 views

For this morphism of integral schemes $X\to Y$, if $X$ is geometrically reduced (irreducible) then is $Y$ also geometrically reduced (irreducible)?

Let $k$ be an arbitrary field. Suppose that $C/k$ is an integral curve which is birationally equivalent to a projective line. Is it true that $C$ is geometrically reduced and irreducible? This is of ...
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1answer
48 views

An easy question (I think) about the subrings of $S^{-1}R$

Let $S$ be a multiplicative subset of a commutative ring $R$. Now consider the homomorphism $\phi_S :S^{-1}R \mapsto R$ where $\frac{r}{s} \mapsto r$ for any $s\in S$. Now my question is: Does this ...
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1answer
73 views

Equivalent characterizations of Dedekind domains

I have two characterizations of Dedekind domains, namely 1) Every nonzero proper ideal of $A$ is invertible, and $Q \neq A$, where $Q$ is a field of fractions 2) $A$ is a Noetherian ring and it has ...
4
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1answer
151 views

Hartshorne's definition of structure sheaf

Hartshorne at page $70$ defines the structure sheaf on Spec $A$. The elements of $\mathcal O_{\textrm{Spec}A}(U)$ are particular functions $s:U\longrightarrow\coprod_{p\in U}A_p$. With the symbol ...
4
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2answers
123 views

Eisenbud Unmixedness Example

I am struggling with the following example in Chapter 18 of Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry , in which the author uses the unmixedness theorem to show that a genus ...
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1answer
99 views

Power series ring over a ring of integers

Let $K/\mathbb {Q}_p$ be a finite extension, $\mathcal{O} := \mathcal{O}_K$ the ring of integers of $K,$ $\frak p$ the maximal ideal of $\mathcal{O}$, and $\pi$ a uniformizer, i.e., $\frak{p} = ...
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2answers
58 views

Irreducibility in $\operatorname{Proj}S$

$\newcommand{\proj}{\operatorname{Proj}}\newcommand{\spec}{\operatorname{Spec}}$ It is well know that in $\spec A$, $V(I)$ is irreducible if and only if $\sqrt{I}$ is a prime ideal. Is it also true ...
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2answers
117 views

Properties of ideals preserved under extension of scalars

The motivation for this question comes from a question in a book by a certain R.H dealing with geometrically reduced and irreducible schemes. Let $k \subset K$ be algebraically closed fields and let ...
2
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1answer
161 views

In a Noetherian ring every non-zero prime ideal is invertible implies every non-zero proper ideal is invertible.

Suppose that $R$ is an integral domain and $R$ is Noetherian. How to show that if every non-zero prime ideal of $R$ is invertible, then every non-zero ideal of $R$ is invertible? Actually, I am ...
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1answer
92 views

Forms in the ideal generated by linear forms.

I'm trying to show the following: Let $F_1,\dots,F_m$ be forms of degree one in $K[x_1,\dots,x_{n+1}]$ with $K$ an algebraic closed field and $m\leq n$. Then all the forms of the same degree can ...
4
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2answers
513 views

Nilpotent/invertible polynomial over commutative ring. [duplicate]

Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial over a commutative ring $R$. Prove that (a) $p$ is unit in $R[x]$ iff $a_0$ is unit and $a_1,a_2,\ldots,a_n$ are nilpotent in ...
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2answers
313 views

Existence of prime ideals in rings without identity

Let $R$ be a commutative ring (not necessarily containing $1$). Say that $R$ is the trivial ring if it has trivial (zero) multiplication. If $R$ is the trivial ring, then $R$ has no prime ideals (as ...
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2answers
122 views

Noetherian ring and prime ideal contained in an invertible maximal ideal.

Let $R$ be a Noetherian ring which is an integral domain, and $M$ be an invertible maximal ideal. Suppose that $P$ is a prime ideal and $P<M$. How to show then that $P=PM$? I was trying to ...
5
votes
2answers
186 views

surjective map of rings with same dimension

Let $A \to B$ be a surjective homomorphism between (unital) noetherian commutative rings with the same Krull dimension. Is the kernel of this map nilpotent ? Thanks to Makoto Kato and Martin ...
1
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1answer
85 views

Dedekind ring characterization via projective modules

I am looking for a book or course notes proving the following result: Let $R$ be an integral domain. Then $R$ is a Dedekind ring if and only if every submodule of a projective $R$-module is ...
0
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1answer
53 views

graded homomorphism of finitely-generated, free modules with minimality condition

The present question is a follow up of this question: finite-free-graded modules and the grading of their duals. Let $k$ be a field, $S=k[x_1,\dots,x_n]$ and $\phi: F \rightarrow G$ a graded ...
5
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2answers
181 views

Tensor product of algebraic closure with an algebraic extension of a field

Suppose $K$ be a field, $\bar{K}$ its algebraic closure, and $L$ some algebraic extension of $K$. I need to compute $\hbox{Spec}(L \otimes_K \bar{K})$. Is there some result from algebra which ...
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1answer
72 views

Castelnuovo-Mumford regularity and Betti numbers: an existence question

Let $k$ be a field and $M$ a finitely generated, graded module over the graded ring $S=k[x_1,\dots,x_n]$. Let $\cdots \rightarrow F_j \rightarrow F_{j-1} \rightarrow \cdots F_1 \rightarrow F_0 ...
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1answer
139 views

Tensor product of a module and a localized ring

Let $A$ be a commutative ring with unity. Let $S$ be a multiplicative subset of $A$. Let $M$ be an $A$-module. Let $x \in M$. Suppose $x\otimes 1 = 0$ in $M\otimes_A S^{-1}A$. Then there exists $s \in ...
2
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1answer
144 views

Finite free graded modules and the grading of their duals

Let $S$ be a $\mathbb{Z}$-graded ring and $F$ a $\mathbb{Z}$-graded module that is free of finite rank $n$. Then we can write $F = \oplus_{i=1}^n S(\nu_i)$, where $S(\nu_i)$ is a graded ring ...
3
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1answer
44 views

An invariant of homogeneous ideals in polynomial rings

Let $k$ be a field, $S=k[x_1,\dots,x_n]$ and $I$ a homogeneous ideal of $S$. Let $f_1,\dots,f_l$ be a minimal generating set of $I$ and let $d$ be the maximal degree among the degrees of the $f_i$. ...
2
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1answer
57 views

isomorphic quotient rings of polynomial ring and Hilbert functions

Let $k$ be a field, $R=k[x_1,\cdots,x_n]$ and $I,J$ homogeneous ideals of $R$. Denote by $H_I(s), H_J(s)$ the Hilbert functions of $I,J$ respectively. If $R/I, R/J$ are isomorphic as graded rings, ...
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0answers
153 views

Every finite map is surjective

I'm trying to understand the proof of the theorem which states that every finite map is surjective in Shafarevich's book: I didn't understand why the part underlined in red is true. I need a ...
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1answer
56 views

How to show that $\operatorname{Spec}(S^{-1}A)=\operatorname{Spec}(O_{X, p})=\cap_{s\not\in p}\operatorname{Spec}(A_s)$?

Let $X=\operatorname{Spec}(A)$ be an affine scheme, $p \in X$ a prime ideal, $S=\{s \in A \mid s \not\in p\}$. $O_{X}$ is the structure sheaf of $X$ and $O_{X, p}$ is the stalk of $O_X$ at $p$. ...
7
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1answer
87 views

Where is $k$ algebraically closed used?

Suppose $k$ is algebraically closed, $A$, $B$ are $k$-algebras and $A$ is an affine $k$-algebra. It is known that then $A\otimes_k B$ is a domain if $A$ and $B$ are domains. This can be found in ...
5
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1answer
155 views

What ideal is this?

Let $k$ be a field and $R = k[X]$ all polys over $k$ in $X$. Choose $p \in R$ and define $I_p = \{ f \in R : f\circ p(X) \in I \}$, where $I$ is some ideal in $R$. Then $I_p$ is an additive ...
1
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1answer
104 views

Hilbert's Nullstellensatz implies $I(Z(I)) = \sqrt{I}$

Let $k$ be an algebraically closed field. Let $I$ be the map that takes algebraic sets in $k^n$ to the ideal generated by them: $I : \{$ algebraic sets $\} \to \{$ ideals of $k[x_1,\dots, x_n] \}$, ...
2
votes
1answer
47 views

Compactness of $\operatorname{Proj}S$

It is well-known that for any ring $A$, $\operatorname{Spec}A$ is quasi-compact. Is it true in general that $\operatorname{Proj}S$ is quasi-compact, where $S$ is an $\mathbb{N}$-graded ring? ...