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

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$A{ss}_R(R/I)=\left\{\sqrt{I}\right\}$ [duplicate]

I want to prove the above statement for any (necessarily noetherian?) ring $R$ and $I$ a primary ideal of $R$. I know that since $R$ is primary, $\forall x,y\in R:xy\in I \implies x\in I \text{ or } ...
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1answer
20 views

Why the ideal norm is multiplicative

Let $I\subseteq B$ be an ideal, we define the ideal norm of $I$ as the ideal in $A$ generated by the elements $N_{E/K}(\alpha)$ where $\alpha \in I.$ We denote it by $N_{E/K}(I).$ If $\mathfrak{p}$ ...
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3answers
76 views

What does Hom(M,N) mean? Atiyah Macdonald proposition 2.9

In Atiyah Macdonald, "Introduction to commutative Algebra" it says: Proposition 2.9:i) Let $M' \xrightarrow[]{u}M \xrightarrow[]{v} M'' \rightarrow 0$ be a sequence of A-modules and homomorphisms. ...
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1answer
51 views

Do two rational parametric curves intersect only finitely many times?

Suppose there are two rational parametric curves $f = (f_1, \ldots, f_n)$ and $g = (g_1, \ldots, g_n)$ in $\mathbb{R}^n$. I read somewhere that such a parametric expression can always be transformed ...
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0answers
24 views

Finite module over Noetherian ring faithfully flat?

If I have Noetherian rings $B=A^G\subset A$ (for some action of finite group $G$, maybe not relevant) and $A$ is finite as $B$-module. Is it always true that $A$ is faithfully flat over $B$? EDIT: ...
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0answers
26 views

Krull's intersection theorem and closedness of submodules

In the book "Formal and rigid geometry" by Bosch, page 70, there is a claim that if $\varphi :\operatorname{Sp}A\to \operatorname{Sp}A^{\prime}$ is a map of affinoid spaces, $x\in ...
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2answers
34 views

Auslander-Buchsbaum formula without minimal/finite resolutions

Does anybody know a proof of Auslander-Buchsbaum's formula that uses only projective/injective/flat resolutions and homological functors Ext and Hom without using minimal/finite resolutions?
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0answers
35 views

Certain map of modules is iso [Mumford Abelian Varieties]

I have trouble showing the following in the proof of Prop. 2 in Abelian Varieties (pg.70 my edition, Chapter about quotients by finite groups): Suppose you have a Noetherian ring $B=A^G$ as ...
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1answer
27 views

There always exists a finite, increasing chain of R-submodules of M isomorphic to R/P. Can we describe P?

So I've been studying some commutative algebra and I came across the following theorem Theorem : Let R be a Noetherian ring. Let $M$ be a non trivial $R$-module, finite over $R$. There exists a ...
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0answers
63 views

How do ring theorists think about square roots?

Let $R$ denote a commutative ring. Then it seems to me that we can adjoin to $R$ a square-root of $4$ as follows: $$R[\sqrt{4}] = R[x]/(x^2-4)$$ This defines a functor $\mathbf{CRing} \rightarrow ...
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34 views

Relationship of maximal ideals and associated primes in the total ring of fractions

Let $R$ be a ring, $S\subset R$ the set of non-zero-divisors. The total ring of fractions is then defined as $Q(R):=S^{-1}R$. Now I want to show that $A{ss}_R(R)$ being finite implies that there are ...
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2answers
64 views

Give an example of a commutative von Neumann regular ring which is not a product of fields

One knows that every commutative von Neumann regular ring with a finite Boolean algebra of idempotents is a product of fields. Give an example of a commutative von Neumann regular ring which is ...
4
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1answer
123 views

Algebraically Closed Quotient Fields

It is well-known that if the quotient field of a commutative noetherian integrally closed domain $R$ is algebraically closed, then $R$ is a field. The proof is easy: let $r_0 \in R$ and choose ...
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1answer
32 views

Saturation of homogeneous ideal

Let $I \subset S=k[x_0,...,x_n]$ be a homogenous ideal. The saturation of $I$, $\bar{I}$ is defined to be $\{s \in S: \exists m \; s.t. \; \forall i \; x_i^m s \in I\}$ Is it true that $\bar{I}=(s ...
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0answers
41 views

What is extension of scalars used for in algebraic geometry?

Given a ring homomorphism $f:A \rightarrow B$ and an $A$-module $M$, one can construct and $A$-module with the tensor product: $M_B=B \otimes_A M$ which has a $B$-module structure. This is said to be ...
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1answer
23 views

Commutative ring which is essential extension of each of its non-zero ideals

Let $R$ be a commutative ring with unit. Assume $R$ is an essential extension of each of its non-zero ideals. I feel that there should be something in the literature about this, but I could not find ...
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2answers
57 views

Showing the ideal $\left \langle yz,xz,yx+ay,x^2+ax \right \rangle$ is radical for all $a\neq 0$

Let $I_a = \left \langle yz,xz,yx+ay,x^2+ax \right \rangle$ be an ideal of $k[x,y,z]$, where $a \neq 0$. Show that $I_a$ is radical. What is the geometric meaning of the elements in ...
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1answer
41 views

Zeros of specialization of a family of polynomials [on hold]

Let $k$ be an algebraically closed field, and $K\supset k$ be an algebraically closed extension. Let $a\in K^n$ be a tuple, we call $a^\prime\in k^n$ a specialization of $a$ if for any $f(X)\in k[X]$ ...
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1answer
34 views

$B/I$ and $B/J$ flat $A$-algebras; does $I=J$ hold?

Let $A\to B$ be a ring homomorphism. Consider $I$ and $J$ ideals of $B$ such that $B/I$ and $B/J$ are flat $A$-algebras. We know furthermore that there exists a non zero-divisor $t\in A$ such that ...
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2answers
36 views

Flat module and finite intersection of submodules

Let $R$ be an integral domain, $F$ be a flat $R$-module, and $A$ and $B$ are two $R$-submodules of $Q$, where $Q$ is the quotient field of $R$. How can we show that $F\otimes (A \cap B) = (F\otimes A) ...
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0answers
14 views

Integral basis of an extension of complete fields

Let $\mathcal{O}_K$ be a complete discrete valuation ring with quotient field $K = \text{Quot}(A)$. Let $L | K$ be an arbitrary finite field extension. Because $K$ is henselian, the integral closure ...
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1answer
24 views

Singular ideals and rings

In Lam's book, Corollary (7.4)(2) says that for a nonzero ring $R$ we have $Z(R_R)≠ R$, where $Z(R_R) $ stands for the singular ideal of $R$.. But, some nonzero commutative rings are "singular" in the ...
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3answers
78 views

Recommended books on commutative algebra stressing links with algebraic geometry

Can someone recommend some books on commutative algebra stressing links with algebraic geometry? My concern is this. It seems to me that most of commutative algebra was formulated at least initially ...
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2answers
44 views

Help in showing that the cusp $(y^2-x^3)\subset \mathbb{C}^2$ is not isomorphic to $\mathbb{C}$

Let $X:=(y^2-x^3)\subset \mathbb{C}^2$ be the vanishing of the polynomial $f(x,y)=y^2-x^3.$ I have proved an exercise in Hartshorne: If $\varphi:\mathbb{C} \to X, \ t \mapsto (t^2,t^3)$ is the ...
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1answer
97 views

normalization of a curve, simplest example

I am learning about normalization of nodal curves and I am trying to understand the simplest example: $xy=0$ As far as I understand its coordinate ring is $k[x]\oplus k[y]$ (let $k$ be an ...
2
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0answers
49 views

Non-zero ideal in algebraic integers generated by two elements

I've been doing past questions for my exams next week and would like to check an answer: Let $I$ be a non-zero ideal of the algebraic integers and let $0\neq a \in I$. Show that $\exists b \in I$ ...
3
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1answer
54 views

What does $(0:x)$ mean?

The following excerpt is from Eisenbud's "Commutative Algebra with a view toward Algebraic Geometry" on pg. 424 We can decide whether an element $x\in R$ is a nonzerodivisor from the homology of ...
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0answers
57 views

Traces of powers of a matrix $A$ over an algebra are zero implies $A$ nilpotent.

I would like to have a result similar to "Traces of all positive powers of a matrix are zero implies it is nilpotent". Namely: Let $R$ be a commutative $\mathbb{C}$-algebra, $A \in ...
4
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2answers
51 views

Integrally Closed domain and Principal Ideal

Let $R$ be an integrally closed local domain. Suppose there is a $y\in I^n$ such that $yI^n=I^{2n}$ for some $n$. I would like to prove that $I^n=(y)$. Source: The above question comes from the ...
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0answers
44 views

Conjugation in algebraic number theory

Let $K$ be an algebraic number field of deg $n$ over $\mathbb Q$, then given $\alpha \in$ $O_k$ its ring of integers, we can choose a $\mathbb Q$-basis $\omega_1, \omega_2, ...,\omega_n$ of $K$ s.t. ...
2
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1answer
48 views

Is every “prefield” a field?

Definition 0. Call a poset $P$ well-ranked iff it is well-founded, and for all $x \in P$, we have that any two maximal subchains in the lowerset generated by $x$ have the same length. ...
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0answers
28 views

Analytical isomorphism implies same multiplicities [duplicate]

I want to prove the following problem in Robin Hartshorne's Algebraic Geometry Chapter 1 exercise 5.14 If $P\in Y$ and $Q\in Z$ are analytically isomorphic plane curve singularities, show that the ...
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1answer
22 views

Regular element of a Noetherian ring [duplicate]

Let $R$ be a Noetherian ring and $x\in R$ an $R-\mathrm{regular}$ element. Show that $\mathrm{Ass}_R(R/(x^n))=\mathrm{Ass}_R(R/(x))$ for every $n\geqslant 1$. Let $M$ be an $R-\mathrm{module}$. An ...
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0answers
29 views

What came first: pythagoras number or pythagorean fields? [migrated]

Which concept was first introduced: the pythagoras number of a field or pythagorean fields? I have not found anything on this matter, but my gut feeling says the latter. One can more directly link the ...
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0answers
29 views

Characterization of prime homogeneous ideals

Let $R$ be a graded ring and $I$ ideal in $R$ and homogeneous. $I$ is prime if and only if for all $a, b\in R$ homogeneous such that $ab\in I$ then $a\in I$ or $b\in I$. Let $ab\in I$ and $a ...
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1answer
44 views

Existence of homogeneous non-unit non-zero divisor in a particular graded ring.

Let $R$ be a finitely generated $k$-algebra of dimension greater than $1$, let $Q$ be any maximal ideal of $R$. It is claimed by my lecturer that one can find a homogeneous, non-unit, non-zero divisor ...
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1answer
28 views

Exact sequence of graded modules and localization

I know that a sequence of modules is exact iff the localization at each prime ideal is exact What happens in the case we are working with graded modules? Can we say that a sequence is exact iff the ...
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0answers
30 views

Local ring of an affine curve $K$ at a point $p\in K$

I'm reading A Royal Road to Algebraic Geometry by Holme. The book defines the local ring as follows: The local ring of $K$ at $P=(a,b)$ is the ring ...
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1answer
38 views

Flatness of quotient rings

The following is Exercise 2.4, in Chapter 1 of Liu, Algebraic Geometry and Arithmetic Curves: Let $I$ be a finitely generated ideal of $A$: $A/I$ is flat. $I^2 = I$. $I = (e)$ where $e^2=e$. I ...
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1answer
30 views

Hilbert function and Hilbert polynomial

I have largely studied Hilbert function and Hilbert polynomial for polynomial rings over fields of characteristic zero. Is it possible to extend the theory also for polynomial rings over fields of ...
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1answer
76 views

In $A$-Mod, $M\oplus A\cong A\oplus A$ implies $M\cong A$

(Exercise from an introductory course in homological algebra) Whenever $A$ is a commutative ring with unit and $M$ an $A$-module, the following holds: $$M\oplus A\cong A\oplus A \Rightarrow ...
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1answer
50 views

Easy explanation on primary decomposition of ideals. [duplicate]

The primary decomposition of an ideal $(x^2, xy)$ is $$(x^2, xy) = (x) \cap (x, y)^2$$ which can be found on these notes. Could someone explain to me how this can be done? Edited: My question ...
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1answer
60 views

Ideal of 8 general points in $\mathbb{P}^2$

I am working through chapter 3 of Eisenbud's Geometry of Syzygies. In the first example he makes the claim that the ideal of 8 general points in $\mathbb{P}^2$ is generated by two cubics and a ...
2
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1answer
24 views

General procedure to prove something is a tensor product of modules

I'm trying to understand some proofs of statements of the form: Show that some module is the tensor product of two other modules. When I'm looking at these proofs I always see that they start ...
2
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1answer
26 views

Is the localization of an injective cogenerator an injective cogenerator?

We know that in Noetherian rings any localization of an injective module is again an injective module. Is the localization of any injective cogenerator again injective cogenerator?
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1answer
19 views

Regular functions extension to normal points of varieties

I am doing the exercise 3.20 in Robin Hartshorne's Algebraic Geometry, Chapter 1. Let $Y$ be a variety of dimension $\geq2$, and let $P\in Y$ be a normal point. Let $f$ be a regular function on ...
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1answer
38 views

Example of a projective variety that is not projectively normal but normal

I want to prove the following statement: Let $Y$ be the quartic curve in $\mathbb{P}^3$ given parametrically by $(x,y,z,w)=(t^4,t^3u,tu^3,u^4)$. Then $Y$ is normal but not projectively normal. ...
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0answers
33 views

Proof of Theorem 4.2.1 in Herzog-Hibi, “Monomial Ideals”

The Theorem and its proof can be found here. Specifically, i am stuck at the fourth paragraph of the proof. Let me give some context: Let $I$ be a graded ideal over a polynomial ring ...
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0answers
29 views

Localization and completion under a strong hypothesis

This question is closely related to this one, but in my case I think the hypotheses are different. Let $(A,\mathfrak m)$ be a regular, local noetherian domain (the local ring at a smooth point of ...
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1answer
22 views

A possible characterization of divisible modules

According to mathworld: Definition. Let $R$ denote a commutative ring and $M$ denote a module over $R$. Then $M$ is divisible iff for every $a \in R$, if $a$ is not a zero-divisor, then for all $x ...