# Tagged Questions

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

38 views

### Image of element is square of an element, precisely two maximal ideals satisfying condition.

Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us look at the ring $\mathbb{F}_q[x, \sqrt{f}]$....
38 views

### The content of a polynomial vs the ideal of its values

Let $f(x) = \sum_i a_i x^i$ be a degree $d$ polynomial over some ring $A$. Define the content of $f$ to be the ideal: $$c(f) = (a_0,\dots,a_d).$$ One can ask for the relation of the above ideal to the ...
Let $A$ be an affine $K$-algebra and $f$ be a non-zero divisor of $A$ then can one say that $\dim A=\dim A_f$ ? What I proved that if $A$ is an affine domain and $f$ is a non-zero element in $... 1answer 18 views ### Sum of Hilbert functions of a finite exact sequence of finitely generated graded modules Let$A = \bigoplus_{n\geq 0} A_n$be a graded ring that is generated as an$A_0$-algebra by a finite collection of elements of$A_1$, where$A_0\$ is artinian. I wish to show that if $$0 \to M(1) \... 2answers 43 views ### The localization is localization of some affine domain. Let A be a finitely generated K-algebra, and let \mathfrak p be a prime ideal of A such that A_{\mathfrak p} is an integral domain. Then have to show that A_{\mathfrak p} is a localization ... 0answers 47 views ### Question about the rational normal curve and different representations of it. I know the rational normal curve as the image of a polynomial map \begin{gather} \phi:K\rightarrow K^n\\ \phi(t)=(t,t^2,\dots,t^n) \end{gather} My question is proving the variety defined by the set ... 1answer 54 views ### Unramified primes of splitting field I would like to show the following: Theorem: Let K be a number field and and L be the splitting field of a polynomial f over K. If f is separable modulo a prime \lambda of K, then L ... 1answer 124 views ### anillo noetheriano y de generación finita. [closed] Sea A=\oplus{A_i}, i\geq{0} anillo graduado. Si A es anillo noetheriano entonces A_0 es noetheriano y A es de generación finita como A_0-álgebra. Dm: Defino I=\oplus{A_i} donde i\geq{... 0answers 21 views ### Nullstellensatz theorem- find vector not orthogonal to a given set of vectors in \Bbb{Z}^n Let v_1,v_2,...,v_r \in \Bbb{Z}^n-\{0\}. Show there exists w\in \Bbb{Z}^n such that w_1=\langle w,e_1\rangle=0 and \langle w,v_i\rangle \neq 0 for all 1 \le i \le r. I've tried to ... 2answers 41 views ### Question about S.Lang's proof of Kummer's Lemma I have a question about the proof of Kummer's Lemma in Serge Lang's Cyclotomic fields (i.e. Theorem 6.1). Let K = \mathbf{Q}(\xi_p) the p-th cyclotomic field extension of \mathbf{Q}. Let u be ... 1answer 66 views ### When does a f.g. algebra over a field F make it “look like F is algebraically closed?” Let F be a field, and let A be a finitely generated algebra over F. If \mathfrak m is a maximal ideal of A, then A/\mathfrak m is an algebraic extension of F, although it is in general ... 1answer 35 views ### Subgroup of idele class group is open On page 380 of Neukirch's Algebraic Number Theory the author states that the subgroup$$\prod_{\mathfrak{p} \nmid \infty} U_\mathfrak{p} \times \prod_{\mathfrak{p} \mid \infty} K_\mathfrak{p}^\times$$... 1answer 57 views ### localized at associated prime of an ideal [duplicate] The problem is as follows: Let I\subseteq J be ideals in a Noetherian ring. Show that if I_{p}=J_{p} for every associated prime p of I,then I=J. It seems reasonable to consider J/I\... 1answer 83 views ### To prove that an ideal cannot generated by two elements [duplicate] Let k be an algebraically closed field and let \ Y\subset \mathbb{A}^n(k) be the curve given parametrically by x=t^3, y=t^4,z=t^5 I want to show (i) I(Y) is a prime ideal of height 2 (ii) ... 1answer 26 views ### Dimension of an Artin K-algebra and cardinal of its spectrum Let A be an Artin ring that is also a finitely generated K-algebra. In particular, the krull dimension of A is 0. By Noether's Normalisation Lemma we have that A is a K-vector space of ... 1answer 35 views ### Is there an adjoint functor to the contravariant hom functor in the category of A-modules. I should start by saying that I don't know any category theory. However, I am reading Atiyah-MacDonald and have just learned that in the category of A-modules (where here A is a commutative unital ... 1answer 42 views ### A counterexample to a statement Question Give an example that z\in\mathbb{Z}[\sqrt{-d}], d\geq1, |z|^2 is a prime number in \mathbb{Z} but z is not prime in \mathbb{Z}[\sqrt{-d}]. Problem I understand that if \mathbb{... 1answer 46 views ### F-rationality of a ring. Given R = \dfrac{k[x,y,z]}{(x^2 - y^3 -z^5)}  where \operatorname{char}k>5. Check whether R is F-rational or not. (F = Frobenius map) I know, by the theorem of Karen Smith, we have to ... 1answer 29 views ### A uniqueness theorem for primary decomposition "Let R be an arbitrary ring and \mathfrak a an ideal of R admitting an irredundant primary representation \mathfrak a=\bigcap_{i}\mathfrak q_{i} and let \mathfrak p_i=\sqrt{\mathfrak q_i}. ... 1answer 133 views ### Defining the set \{(t^3,t^4,t^5) : t \in \mathbb{C}\}\subset \mathbb{C}^3 by two polynomial equations What are two polynomials f,g \in \mathbb{C}[x,y,z] such that$$\{(x,y,z): f(x,y,z)=g(x,y,z)=0\}\;=\;\{(t^3,t^4,t^5): t \in \mathbb{C}\}$$holds as an equality of subset of \mathbb{C}^2? This ... 0answers 63 views ### Separable morphism and smooth fibers Let f:X \to Y be a separable, dominant morphism of finite type between noetherian k-schemes for k algebraically closed. Does it mean that For a closed point x \in X, f^{-1}(f(x)) is smooth ... 1answer 39 views ### Necessary and sufficient condition for a regular sequence. f_1, \ldots, f_r is a regular sequence in S/I (where S is a polynomial ring in n variables, and I its ideal) iff$$(I, f_1, \ldots, f_{i-1}): (f_i)= (I, f_1, \ldots, f_{i-1}) \quad i \ge 2.$$... 0answers 35 views ### Difference between parameters and system of parameters in a local commutative ring Can you please tell me the difference between the 'parameters' and the 'system of parameters' of a commutative local ring? Also, is there any relation between parameters and associated primes of the ... 0answers 33 views ### Buchberger algorithm and ideals I'm working on Groebner bases using the book Ideals, Varieties and Algorithms. I'm interested in this problem : Let \mathbb{Q}[x,y,z] with the graded lexicographic order with x>y>z. For ... 1answer 56 views ### Projective module with non-zero annihilator [closed] Let M be a projective module. Suppose \operatorname{Ann}_{R} \left(M \right) \neq 0, where \operatorname{Ann}_{R} \left( M \right) =\{r\in R : mr = 0, \ \forall m \in M \}. Then there exists an ... 1answer 13 views ### Find g\in I such that LT(g)\notin \langle LT(g_1),LT(g_2),LT(g_3)\rangle. Let I=\langle g_1,g_2,g_3\rangle\subset \Bbb R[x,y,z] where$$g_1=xy^2-xy+y,\qquad g_2=xy-z^2, \text{ and } g_3=x-yz^4$$Using lexicographic order find g\in I such that LT(g)\notin \langle LT(... 1answer 40 views ### Criterion for the integral closure of an domain in a finite field extension being a finitely generated algebra A is an integral domain, K=\operatorname{Frac}A, L/K finite field extension (not necessarily separable), B is the integral closure of A in L. Question: with some extra conditions on A, ... 1answer 102 views ### Line bundle trivial on fibers then isomorphic to the pullback of a line bundle \require{AMScd} I'm currently reading Milne's notes about Abelian varieties. On page 26 he proves the following theorem: Let V and T be varieties over k with V complete, and let \... 1answer 27 views ### M_{1} \oplus M_{2} is a cyclic A-module \iff \rm{Ann}(M_1)+\rm{Ann}(M_2)=A [duplicate] Let A be a commutative ring with an identity element 1. An element x in an A-module M is called cyclic if Ax=M. An A-module which has a cyclic element is called cyclic A-module. Let ... 1answer 45 views ### do formal group laws induce group structures on schemes (as opposed to formal schemes) Let R be a ring and f \in R[[x]] a commutative formal group law over R, meaning f(f(x, y), z)=f(x, f(y, z)), \ f(x, y)=f(y, x) and f(x, y)=x+y + \text{higher order terms}. Let G=\... 0answers 32 views ### An algebraic set is called defined over k if its ideal can be generated by polynomials in k[x]. [duplicate] I find this definition in Silverman's book, The Arithmetic of Elliptic Curves. An algebraic set (in A^n(\bar{K})) is called defined over K if its ideal can be generated by polynomials in K[X]=K[... 1answer 60 views ### Same kernels for homomorphisms of free modules Let f: R^n \rightarrow R^m be an isomorphism of free R-modules (R commutative with unity) and \pi_1: R^n \rightarrow R^n/\mathfrak m^n, \pi_2: R^m \rightarrow R^m/\mathfrak m^m the canonical ... 2answers 58 views ### Triviality of \mathrm{Ann}(\mathfrak m) This question is regarding the first paragraph of the proof of Proposition 2.4 from this paper. QUESTION: Is it true that if (0) is irreducible, then \mathrm{Soc}(R)=\mathrm{Ann}(\mathfrak m)=(0:... 1answer 35 views ### Example of associated ideal in primary decomposition Let I be a decomposable ideal of a commutative ring R with minimal primary decomposition I=\bigcap_{i=1}^n\mathfrak q_i. The first uniqueness theorem shows that \{\sqrt {\mathfrak q_i}:1\le i\... 0answers 34 views ### Let a,b have the same divisor (content) in an integral domain A. When can I deduce a/b\in A^\times? Given a Noetherian integral domain A and a finitely generated torsion A-module M, we can define the divisor, or content, of M to be div(M)= \sum_{P, ht(P)=1} \ell(M_P) [P], where the sum ranges ... 1answer 65 views ### Almost-invariant polynomials under dihedral group action Think about the dihedral group D_4 acting on the polynomial algebra \mathbb C[x_1, \cdots, x_4] via generating permutations (x_1\ x_2), (x_3\ x_4), and (x_1\ x_3)(x_2\ x_4). I'd like to ... 1answer 22 views ### Reference on a result about integral closures. Could you please give a reference or a sketch of a proof for the following proposition? Proposition: The integral closure of a complete local Noetherian domain R is module-finite over R You ... 0answers 49 views ### Independent set of variables modulo ideal and Krull dimension Let \mathfrak{a}\subseteq \Bbbk[x_1,\ldots,x_n] be an ideal, where \Bbbk is a field. Let the maximal set of indeterminates independent modulo \mathfrak{a} be of cardinality k. There is a ... 1answer 45 views ### Fiber of morphism induced by map on stalks Given a morphism of schemes f\colon X\to Y and a point x\in X, the map on the stalks induces a morphism \operatorname{Spec}\mathcal{O}_{X,x} \to \operatorname{Spec}\mathcal{O}_{Y,f(x)} . Is it ... 1answer 83 views ### Exercise 2 from chapter 5 of Eisenbud's Geometry of Syzygies book I am trying to solve exercise 2 from chapter 5 of Eisenbud's The Geometry of Syzygies book.The problem is as follows: Let X be the union of two disjoint lines in \mathbb P^3, or a conic ... 2answers 53 views ### Is the ideal of a variety the annihilator of a subspace of the symmetric algebra? Let V be a vector space over an algebraically closed field K. Let \mathrm{Sym}(V^*)=\mathrm{Sym}(V)^* be the symmetric algebra on V, i.e. if we give a basis e_1,...,e_n of V and let x_1,... 1answer 96 views ### Annihilator of a flat ideal Let R be a commutative ring and let I be a finitely generated flat ideal of R. Let J=\mathrm{Ann}(I). How can one prove that I\cap J=0? This can be found as a remark in the paper of ... 1answer 36 views ### Characterization of Groebner Bases in terms of uniqueness of remainders Let I be an ideal of a polynomial ring R=k[x_1,\ldots,x_n] over a field k. A Groebner basis of I is a finite generating set \{g_1,\ldots,g_m\} such that every leading monomial (according to ... 1answer 24 views ### Fraction rings ideals members Let R be a ring with fraction ring R_S and ideal I. I saw in arguments that when a/s is in I_S they dont say a is in I. Instead they say a/s=b/t with b \in I. Why? Many thanks. 1answer 32 views ### System of parameters in Noetherian local rings I'm trying to understand the theorem for systems of parameters in Noetherian local rings, which says: Let R be a Noetherian local ring with maximal ideal m. Then there exists an m-primary ideal ... 1answer 40 views ### Regularity and Short Exact Sequence Suppose  0 \to M_1 \to M_2 \to M_3 \to 0 is a short exact sequence of finitely generated graded k[x_0,...,x_r]-modules. Then show that \mathrm{reg}(M_1) \leq\max(\mathrm{reg}(M_2),\mathrm{reg}(... 0answers 29 views ### What is the ring of integers in \mathbb Q^c\otimes_K K_\mathfrak p? Let K be a number field with ring of integers \mathcal O_K and \mathfrak p a prime of K. Let \mathbb Q^c be the algebraic closure of \mathbb Q in \mathbb C. If L is a number field ... 0answers 35 views ### Trying to Compute Regularity and degree Definition: For a finite subset X \subset \mathbb P^r,the Hilbert function H_X(d) is constant for large d and its value is the number of points in X,usually called the degree of X. Let ... 1answer 55 views ### Filling in Proof: Well-definedness of depth(I,M). From Eisenbud's Commutative Algebra with A View Toward Algebraic Geometry (Theorem 17.4): Let M be a finitely generated R-module, where R is Noetherian. If$$r= \min \{i : H^i(M\otimes K(x_1,...
If we change the ideal $$(X_1,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ to $$(X_1^2,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ in this problem, what is the answer to the raised question? Again, the new local ring ...