The graded-modules tag has no wiki summary.
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votes
1answer
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Questions on (subring/ submodule) of a graded (ring/ module)
I have a question which seems a bit silly...
If we have $R$ a graded ring, does it follow that every subring of $R$ is also graded?
Because I have a problem here as such: I have a graded ring $R$ ...
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votes
2answers
114 views
Path Algebra for Categories
For a while I had been thinking that the path algebra of a quiver $Q$ over a commutative ring $R$ is the same as the "category ring" $R[P]$ (analogous to "group ring", "monoid ring", "semigroup ring", ...
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A good description of $M^{\vee\vee}$?
Let $R$ be a f.g. $\mathbb{N}$-graded non-commutative $\mathbb{C}$-algebra. Assume $R$ is connected, i.e. $R_0=\mathbb{C}$. Let $M$ be a f.g. torsion free graded right $R$-module of rank $1$. Is there ...
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Construction of graded rings and modules
In Algebraic Geometry and Homological Algebra - as far as I know - we often consider graded rings and modules so as to encode more information, say, some sort of (computational) complexity. For ...
2
votes
1answer
91 views
Compute Hilbert function of a monomial ideal
I'd like to know whether there exist easy methods that compute the Hilbert
function of a graded $k$-algebra, without computer programs. My homework
asks to me to compute the Hilbert function of ...
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0answers
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Why do we have $\operatorname{Proj}(\oplus_{i=0}^\infty t^i \left< x\right>^i)\cong \operatorname{Proj}(\oplus_{i=0}^\infty t^i \left< x^2\right>^i)$?
I'm just curious but why is it that
$$
\operatorname{Proj}\left(\oplus_{i=0}^\infty t^i \left< x\right>^i\right)
$$
isomorphic to
$$
\operatorname{Proj} \left(\oplus_{i=0}^\infty t^i \left< ...
1
vote
1answer
55 views
Graded Vector Spaces and the Interchange Law
I'm a little confused about how to correctly interchange factors in tensor
products on graded vector spaces.
In particular let $V:= \bigoplus_{n \in \mathbb{N}} V_n$ be a $\mathbb{N}$-graded
vector ...
1
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0answers
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Understanding $Bl_{\mathcal{I}}(k^4)/S_2$ where $\mathcal{I}$ is defined by $(x_1-x_2,x_3-x_4)$
Let $k_4=Spec(k[x_1, x_2, x_3,x_4])$ and $\mathcal{I}$ is the ideal sheaf defined by $(x_1-x_2,x_3-x_4)$. Then
$$
Bl_{\mathcal{I}}(k^4) = Proj (\oplus_{i\geq 0} I^i t^i)
$$
where ...
2
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0answers
70 views
Degree 1 elements in a graded ring from a blow-up perspective
This may be an elementary question but I hope this question will benefit others as much as myself.
Let $k^4 = Spec \; k[x_1, x_2, x_3, x_4]$.
Writing $Bl_{\mathcal{I}}(k^4)$ as $R = ...
2
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1answer
95 views
Blow-up along an ideal sheaf
Let $k^2=\operatorname{Spec} \; k[x,y]$ where $k$ is an algebraically closed field. Let $\mathcal{I}$ be the ideal sheaf defined by $(x,y)$. Then
$$
Bl_{\mathcal{I}}k^2
$$
is covered by two open ...
0
votes
1answer
47 views
The variety associated to a polynomial ring with a particular grading
We impose various grading on an algebra or a module to understand its homological properties. So I made up the following problem and I would like to understand its correspondence as a variety.
...
0
votes
2answers
170 views
Radicals of homogeneous ideals over semigroup-graded rings.
In this post I set out to prove an equation holding in a ring $R$ graded over $\mathbb{Z}$ or $\mathbb{N}$.
The equation was: $\sqrt{J^\ast}=(\sqrt{J})^\ast$, where $J$ is an ideal of $R$, $J^\ast$ ...
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0answers
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geometrical interpretation of $\mathbb{Z}/2\mathbb{Z}$ graded space
According to wikipedia, a $\mathbb{Z}/2\mathbb{Z}$ graded space (super vector space) $V$ is a a vector space which can be decomposed in a direct sum $V=V_0 \oplus V_1$ where elements of $V_0$ are ...
2
votes
1answer
86 views
Should Hom(M,N) be also graded ??
If $M,N$ are graded modules over graded ring $R$, my question: is $\operatorname{Hom}_{R}(M,N)$ also a graded module and how ?
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vote
2answers
154 views
question about direct sum and direct product in Graded Modules?
let M, N be R- Graded Modules, say: $$M= \bigoplus_{i\in Z} M_{i}, N= \bigoplus_{j\in Z} N_{j}$$
then
$$Hom(M,N) \cong \prod_{i \in Z} \bigoplus_{j\in Z}Hom(M_{i},N_{j})$$
and if $\phi:M\to N $ then ...
0
votes
1answer
95 views
Isomorphism of ($\mathbb{Z}/{(n)}$-graded) Rings
Let $A=\bigoplus_{d=0}^n A_g$ and $B=\bigoplus_{d=0}^n B_h$ be $\mathbb{Z}/{(n)}$-graded rings. In particular, we assume $A_n\ne 0$ and $B_n\ne 0$. Let $\phi:A\to B$ be an isomorphism of rings. My ...
2
votes
1answer
110 views
Notes on dga's with a look towards Rational Homotopy Theory?
What is the best set of notes that give an introduction to differential graded algebras, preferably with ample examples and calculations, for someone that is ultimately interested in doing ...
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vote
2answers
96 views
Showing: If $w\in C\ell^1(V,Q)$ anticommutes with all $v\in V$, then $w=0$
Show that if an element of the odd part of the Clifford Algebra anticommutes with everything in the vector space, then it is 0.
Been having a really hard time making any progress with this one.
2
votes
1answer
81 views
Does the category of graded rings have limits?
Let $\mathfrak{C}$ be the category of ($\mathbb{Z}$)-graded-commutative rings. Does this category have limits in it?
I am particulary interested in power series rings over a field. Is there a ...
3
votes
3answers
239 views
Hilbert-Poincaré Series of Finite-Dimensional Graded Algebras
Suppose I have two finite-dimensional $\mathbb{Z}_{\geq 0}$-graded $\mathbb{C}$-algebras $A = \bigoplus_{k \geq 0} A_{k}$ and $B = \bigoplus_{k \geq 0} B_{k}$ with Hilbert-Poincaré series, $P_{A}(t) = ...
2
votes
1answer
185 views
Some questions on graded rings
At the moment I'm mostly interested in commutative graded rings (and in particular those graded over $\mathbb{N}$), but any comments or references about more general graded rings would also be ...
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votes
2answers
395 views
Homomorphisms of graded modules
Let $M$ and $N$ be graded $R$-modules (with $R$ a graded ring). Let $\varphi:M\rightarrow N$ be an homogeneous homomorphism of degree $i$ (that is $\varphi(M_n)\subset N_{n+i}$). Denote by ...
2
votes
1answer
50 views
The zero component of tensor products of graded mods
I guarantee there is an easy reference on this, but for some reason I cannot find it. If you can point me to a reference or just write a short proof for me, I would be appreciative.
Given a graded ...
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votes
0answers
133 views
An elegant description for graded-module morphisms with non-zero zero component
In an example I have worked out for my work, I have constructed a category whose objects are graded $R$-modules (where $R$ is a graded ring), and with morphisms the usual morphisms quotient the ...
