# Questions tagged [graded-rings-modules]

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**6**

votes

**1**answer

115 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 ...

**15**

votes

**3**answers

408 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 ...

**3**

votes

**1**answer

131 views

### Global dimension of a graded algebra

Let $A= \bigoplus\limits_{n=0}^{\infty}{A_n}$ be an $\mathbb{N}$-graded algebra with semisimple $A_0$.
Question: Do we have that the global dimension of $A$ is equal to $\sup \{i \geq 0 | Ext_A^i(...

**3**

votes

**1**answer

91 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 $R$ ...

**9**

votes

**2**answers

310 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 ...

**8**

votes

**1**answer

185 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 $\mathcal{G}_A := \{ \mathrm{Proj}\Big((A,g)\Big)\ \vert\ g \}$? ...

**6**

votes

**1**answer

171 views

### Is every (left) graded-Noetherian graded ring (left) Noetherian?

I call a $\mathbb{Z}$-graded (non-commutative, associative, unital) ring $A$ (left) graded-Noetherian if every homogeneous (left) ideal is finitely generated, and (left) Noetherian if it is (left) ...

**1**

vote

**0**answers

31 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 ...

**5**

votes

**0**answers

191 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 $I(\cap_{k=1}^{\infty}I^k M)=\cap_{k=1}^{\infty}I^k M$ . ...

**0**

votes

**0**answers

84 views

### Hochster-Roberts theorem

I am trying to understand this theorem, and as an example I try this case: Let $R=\mathbb{C}[z_1^{\pm},\ldots,z_n^{\pm}]^{S_n}$ be the algebra of Laurent polynomials. Now $R$ should be somehow ...

**1**

vote

**1**answer

171 views

### If $A\subset B$, what to say about their $\operatorname{Proj}$?

Let $k$ be a field and $A$ and $B$ be two graded $k$-algebras satisfying $A\subset B$. The $\operatorname{Proj}$ construction is not functorial but is there nothing to say about $\operatorname{Proj}(A)...

**2**

votes

**1**answer

228 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 ...

**1**

vote

**0**answers

51 views

### References on infinite dimensional graded algebras

I am currently working on some generalisations of known results in 2-category theory, and I have been stymied trying to find references that discuss `graded-finite dimensional' algebras over a field - ...

**1**

vote

**1**answer

139 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 $S$ , $a \in S$ ...

**7**

votes

**1**answer

153 views

### Graded and projective (but not bounded below) module that is not graded-projective?

Let $A = \bigoplus_{n = 0}^\infty A_n$ be a graded algebra over a field $k$ that is locally finite: each $A_n$ is a finite-dimensional $k$-vector space. We say that a graded left $A$-module $P = \...

**0**

votes

**0**answers

26 views

### Can a ideal-like subset that is prime in low degrees be extended to a prime ideal ?

Let $A=\bigoplus_{n \ge 0}A_n$ be a positively graded commutative ring. Let $J_n \le A_n$ be subgroups for $j=0,\ldots,N$, satisfying
$\,\,\,A_n \cdot J_m \subseteq J_{n+m}$ for $n+m \le N$
$\,\,\,...

**1**

vote

**0**answers

78 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 ...

**0**

votes

**0**answers

162 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). ...

**3**

votes

**1**answer

79 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 ...

**4**

votes

**0**answers

315 views

### Inverse limit of graded rings

Let $(I,\le)$ be a directed set and let $(\rho^{\beta\alpha}: R^\beta \to R^\alpha)_{\alpha \le \beta}$ be an $I$-directed system of $\mathbb{Z}$-graded rings whose multiplication is denoted by
$$\...

**5**

votes

**0**answers

543 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 ...

**12**

votes

**2**answers

1k views

### “Spec” of graded rings?

From the discussion here, it seems that general Hochschild cohomology classes correspond to deformations where the deformation parameter can have nonzero degree.
So I have some naive and maybe stupid ...

**4**

votes

**0**answers

405 views

### Tor over graded rings

Let $R$ be a graded ring (concentrated in nonnegative dimensions and maybe bounded from above). For every positive natural number $n$, denote by $R\to\tau_{\leq n}R$ the $n$-truncation and by $\tau_{\...