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Primary invariants on MAGMA for a graded ring

I have asked this question on mathstacks, but a collegue of mine recommended me to post it here. I am trying to find an optimal system of parameters for a graded ring using Magma. Specifically, I want ...
Rustam T's user avatar
1 vote
0 answers
47 views

Examples of graded subrings of $\mathbb Q(T)$

The following question came up in some discussion on some very unrelated matters. A graded algebra $A$ is an algebra $A$ with a decomposition $A = \oplus_{i \in \mathbb Z} A_i$ such that $A_i A_j \...
user5831's user avatar
  • 2,029
4 votes
1 answer
187 views

Order of pole of Poincaré series

Let $R = \bigoplus_{n \geq 0} R_n$ be a graded Noetherian ring and $M = \bigoplus_{n \geq 0} M_n$ a finitely generated graded $R$-module. Let $\lambda$ be an additive function on the class of all ...
psl2Z's user avatar
  • 331
8 votes
1 answer
191 views

Do graded-commutative rings satisfy the strong rank condition?

Let $R$ be a ring. Recall that $R$ is said to satisfy the strong rank condition if, whenever $R^m \to R^n$ is a monomorphism of right $R$-modules (with $m,n \in \mathbb N$), we have $m \leq n$. It is ...
Tim Campion's user avatar
1 vote
0 answers
106 views

Uniqueness of indecomposable decomposition (Krull–Schmidt) for finitely generated modules over commutative Noetherian standard graded rings

Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently,...
uno's user avatar
  • 412
3 votes
1 answer
372 views

Subalgebras of quadratic algebras that are not quadratic

Suppose $A=k\oplus A_1 \oplus A_2\oplus \cdots$ is a quadratic algebra over a field $k$. Let $B$ be the subalgebra generated by a subspace $V\subseteq A_1$. What are the examples of such subalgebras $...
Lorenzo Del Vecchiopontopolos's user avatar
4 votes
0 answers
84 views

Dimension of a positively graded ring after a suitable localization

Quesion- Let $R=\bigoplus_{i\ge 0} R_i$ be a (non-trivial) positively graded commutative Noetherian ring with $1(\not=0)$ of (Krull) dimension $d\ge 0$. Let $S\subset R_0$ be a multiplicative set such ...
Sourjya Banerjee's user avatar
1 vote
1 answer
182 views

A question about surjective maps between quadratic algebras

Let $V$ be a finite-dimensional vector space and $$ U \subseteq W \subseteq V \otimes V $$ be a proper inclusion of vector subspaces. Then take the tensor algebra $$ T(V) = \bigoplus_{i=1}^{\infty} V^{...
Pierre Dubois's user avatar
12 votes
2 answers
776 views

Hilbert polynomials of graded algebras evaluated at negative numbers

Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus_{n=0}^\infty R_n$ with $R_0=k$ (and $R=k[R_1]$). The Hilbert function $h_R:\mathbb{N}\rightarrow \...
walkar's user avatar
  • 253
2 votes
1 answer
91 views

Is the integral closure of a $\mathbb{Z}/n\mathbb{Z}$-graded noetherian domain in a bigger $\mathbb{Z}/n\mathbb{Z}$-graded domain also graded?

Let $A\subset B$ be an inclusion of $\mathbb{Z}/n\mathbb{Z}$-graded noetherian domains. Is the integral closure of $A$ in $B$ also $\mathbb{Z}/n\mathbb{Z}$-graded? This is true for the $G$-graded case ...
Heer's user avatar
  • 997
1 vote
0 answers
45 views

Lifting module homomorphisms imposing conditions on characteristic polynomials

Suppose that we are in the setting described in the first two paragraphs of this MSE post. My question wants to deal with an instance of the study of the amount of freedom that the choice of the ...
Hvjurthuk's user avatar
  • 573
3 votes
4 answers
807 views

$R$ is a UFD iff $R_{\frak{m}}$ is a UFD?

Let $R$ be a graded ring such that $R_0$ is a field and let $\frak{m}$ be the maximal ideal generated by all the elements of positive degree. Then, is it true that $R$ is a UFD iff $R_{\frak{m}}$ is a ...
It'sMe's user avatar
  • 839
9 votes
2 answers
417 views

Over which (graded) rings are all modules decomposable into indecomposables?

A module is decomposable if it is the direct sum of two modules. The process of splitting summands off of a decomposable module does not need to terminate, so infinitely generated modules do not ...
Tilman's user avatar
  • 6,162
1 vote
0 answers
119 views

Commutative monoid gradings via group scheme actions

$\newcommand{\Spec}{\mathrm{Spec}}$Recall the following result, proved in Section 2.9 of Neil Strickland's Formal schemes and formal groups, and in Lemma 1.3.2 of Eric Peterson's Formal Geometry and ...
Emily's user avatar
  • 11.8k
2 votes
2 answers
411 views

Dimension of the associated graded module at an ideal

Let $I$ be an ideal of a Noetherian local ring $(R, \mathfrak m)$. Define the associated graded ring $G_I(R):=\bigoplus_{n=0}^\infty I^n/I^{n+1}$. Then $G_I(R)$ is a Noetherian ring of the same ...
Louis 's user avatar
  • 279
3 votes
1 answer
178 views

On the degree of the Hilbert polynomial of a graded module over the Rees algebra

If $A=\oplus_{n=0}^\infty A_n$ is a Noetherian graded ring of finite dimension such that $A_0$ is local and $A=A_0[A_1]$, and if $M=\oplus_{n=0}^\infty M_n$ is a finitely generated graded $A$-module ...
Louis 's user avatar
  • 279
1 vote
0 answers
33 views

Making a generating set of a section of a graded polynomial $R$-module coming from a quotient into a basis of a quotient by higher degree polynomial

Denote the graded rings $R:=\mathbb{R}[x_{1},\dots x_{n}]$ and $S:=R[x_{0}]$ adding the homogenizing variable $x_{0}.$ Consider $h\in S$ a homogenous polynomial of degree $d$ with leading coefficient $...
Hvjurthuk's user avatar
  • 573
1 vote
0 answers
212 views

Surjection from finite rank free $R$-module to finitely generated $R$-module and basis associated to generator set

Suppose the we have an epimorphism $s\colon M\to N,$ where $M$ is a free $R$-module of rank $r$ and $N$ is a finitely generated $R$-module, such that there exists a basis $B:=\{m_{1},\dots, m_{r}\}$ ...
Hvjurthuk's user avatar
  • 573
1 vote
0 answers
210 views

Strongly graded rings

In Theorem 3.1 of Graded rings over arithmetical orders, the authors prove that for a strongly $\mathbb{Z}$-graded ring $R$, if $R_0$ is left and right Goldie and a maximal order in its (classical) ...
a196884's user avatar
  • 323
1 vote
0 answers
106 views

Homomorphisms and indecomposable decompositions of finite modules over polynomial rings [closed]

I am studying $\mathbb{N}^n$-graded, finitely generated modules $M$ over the $\mathbb{N}^n$-graded polynomial ring $K[X_1,X_2,X_3...,X_n]$. I know every such $M$ has an indecomposable decomposition $M\...
SCM's user avatar
  • 11
4 votes
1 answer
195 views

Weighted blowup of a Cohen-Macaulay ring along a regular sequence is Cohen-Macaulay?

Let $(R, \mathfrak{m})$ be a Cohen-Macaulay ring, let $f_1, \dotsc, f_d \in \mathfrak{m}$ be a regular sequence, and let $n_1, \dotsc, n_d > 0$ be weights (feel free to assume that $n_1 = 1$ if it ...
One More Question's user avatar
6 votes
0 answers
159 views

Ring with different graded and ungraded global dimensions

Let $A$ be a $\mathbb N$-graded ring. One can consider the two categories $M_A^g$ and $M_A^u$ of graded and ungraded modules over $A$. Both have, say, enough projectives, hence one can compute various ...
Andrea Ferretti's user avatar
1 vote
1 answer
111 views

non-archimedean valuations on graded rings

Let $R$ be a commutative (non-trivially) graded ring. By a non-archimedean valuation I mean a map $v: R \to \Gamma \cup {0}$ such that for all $x,y \in R$, we have $v(x+y) \leq \max\{v(x),v(y)\}$, $v(...
Leray Jenkins's user avatar
8 votes
3 answers
691 views

Are open $\mathbb{G}_m$-invariant subschemes of an affine scheme precisely the homogeneous radical ideals of its coordinate ring?

This question is certainly not research level and in fact quite elementary which is why I asked it on math.stackexchange before: math.stackexchange. However it doesn't seem to get much attention there ...
Jakob Werner's user avatar
  • 1,153
3 votes
0 answers
62 views

Asymptotic stability of associated primes of graded local cohomology modules

This question concerns the asymptotic behaviour of associated primes of graded components of local cohomology modules. A survey of this can be found in M. Brodmann, Asymptotic behaviour of cohomology: ...
Fred Rohrer's user avatar
  • 6,700
1 vote
1 answer
646 views

Affine cone example

Consider the complete intersection ideal ${\displaystyle (f,g_{1},g_{2},g_{3})\subset \mathbb {C} [x_{0},\ldots ,x_{n}]}$ and let $X$ be a projective variety defined by the (product) ideal sheaf ${\...
user267839's user avatar
  • 6,038
6 votes
0 answers
407 views

Homogeneous regular sequence

Consider a $\mathbb{Z}$-graded polynomial ring $R = k[x_1,\cdots,x_n]$ over a field $k$, where the elements $x_i$ are homogeneous (they may have negative degree). Let $I$ be a homogeneous ideal that ...
kcnitin's user avatar
  • 71
6 votes
1 answer
463 views

If a commutative graded algebra is free over a graded subalgebra, then must it have a graded basis?

Fix a field $\mathbf{k}$ and an $\mathbb{N}$-graded commutative $\mathbf{k}$-algebra $A = \bigoplus\limits_{n = 0}^{\infty} A_n$ of finite type. ("Finite type" means that each $A_n$ is a ...
darij grinberg's user avatar
6 votes
1 answer
164 views

Is $[Im:(x)][Im:(y,z)]\subseteq Im$ in $k[x,y,z]$?

Let $k$ be a field and $S=k[x,y,z]$. Let $m=(x,y,z)$ and $I\subseteq m$ a proper homogeneous ideal in $S$. Is this true that we always have: $$[Im:(x)][Im:(y,z)]\subseteq Im \ ?$$ In a paper we ...
Hailong Dao's user avatar
  • 30.5k
27 votes
3 answers
1k views

Graded analogues of theorems in commutative algebra

Many theorems in commutative algebra hold true in a ($\mathbb{Z}$-)graded context. More precisely, we can take any theorem in commutative algebra and replace every occurrence of the word commutative ...
6 votes
1 answer
598 views

Projective dimension of graded modules

Short version: Why is the projective dimension of a graded module the same as the projective dimension of its underlying ungraded module? Longer version: Let $G$ be a commutative group, let $R$ ...
Fred Rohrer's user avatar
  • 6,700
9 votes
2 answers
513 views

Divided power algebra is artinian as a module over the polynomial ring

I already asked this on math.stackexchange.com, but did not receive much responses. I hope this is also appropriate for mathoverflow. In the paper Homological algebra on a complete intersection, with ...
red_trumpet's user avatar
  • 1,286
8 votes
1 answer
399 views

The different gradings of a graded ring, and their schemes

Let $(A,g)$ be a graded commutative ring, where $A$ denotes the commutative ring, and $g$ its grading. What can be said about the set $\mathcal{G}_A := \{ \mathrm{Proj}\Big((A,g)\Big)\ \vert\ g \}$? ...
THC's user avatar
  • 4,545
1 vote
0 answers
40 views

Hilbert functions of graded modules generated by mapped generators

I need to prove the following claim for my work. Intuitively, the claim should hold as it is analogous to the concept of an initial module, but a rigorous approach for the same I cannot find. Any help ...
diff2's user avatar
  • 11
5 votes
0 answers
337 views

Can the Artin-Rees lemma be derived from Krull Intersection theorem?

The Krull Intersection theorem states that : For a finitely generated module $M$ over a Noetherian ring $R$ and any ideal $I$ of $R$, we have $I(\cap_{k=1}^{\infty}I^k M)=\cap_{k=1}^{\infty}I^k M$ . ...
user avatar
3 votes
1 answer
701 views

On graded projective modules

If $R$ is a PID, then we know that any projective module over $R$ is free. Is there any graded version of this? That is, let us call a graded commutative ring $R=\oplus _{g \in G} R_g$ a graded PID if ...
user avatar
1 vote
1 answer
266 views

Providing a grading for the polynomial ring over a commutative unital graded ring

Let $R$ be a commutative unital $G$-graded ring , where $G$ is a monoid ; then does there exist a $G$-grading on $R[X]$ such that whenever we have a commutative unital $G$-graded ring $S$ , $a \in S$ ...
user avatar
1 vote
0 answers
86 views

Characterization of a finitely graded (almost) domain

Let $A= \bigoplus_{i=0}^{N}A_i$ be a finitely graded ring with the following property: if $x \in A_i$ and $y \in A_j$ and $i+j \leq N$, then $$xy = 0 \text{ implies } x = 0 \text{ or } y = 0.$$ Hence ...
Exit path's user avatar
  • 3,019
0 votes
1 answer
372 views

The growth of the Hilbert function of a graded ring

Let $A=\bigoplus A_i$ be a finitely generated commutative unital graded algebra over a field $k$. Let $d(i)=\dim A_i$. In general $d(i)$ is not a polynomial in $i$ (even not eventually polynomial). ...
Rami's user avatar
  • 2,649
3 votes
1 answer
133 views

Derivations annihilated by powers of the augmentation ideal

Consider an augmented commutative ring $R$, with augmentation ideal $\varpi$. Let $\delta$ be a derivation of $R$. The example I have in mind is $R=\mathbb F_p[x]/(x^{p^i})$ and $\delta=d/dx$, though ...
grok's user avatar
  • 2,519
5 votes
0 answers
917 views

Height of maximal homogeneous ideals

Let $R= \oplus_{n \ge 0} R_n$ be a graded Noetherian commutative ring and suppose $R_0$ is Artinian. Do all maximal homogeneous ideals of $R$ have the same height ? Let $R_{>0}$ be the ideal ...
Ralph's user avatar
  • 16.2k