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1answer
16 views

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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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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0answers
21 views

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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0answers
142 views

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
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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
34 views

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 ...
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0answers
29 views

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
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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
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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 $R$, $J^\ast$ ...
0
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0answers
51 views

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
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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 ?
1
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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
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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
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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 ...
1
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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
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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
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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
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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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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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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 ...