The graded-modules tag has no wiki summary.
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2answers
20 views
Reference request: Tensor product of DG-modules
Let $A$ be a (skew-) commutative DG-algebra, and let $M,N$ be two DG $A$-modules.
I am looking for a reference which describes the functor $-\otimes_A -$ and its basic properties (associtivity, ...
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vote
5answers
64 views
Split-Lemma for chain complexes
Suppose $k$ is a field and $A$, $B$ and $C$ are chain complexes of $k$-vector spaces, i.e., objects in $\mathbf{Ch}(k\text{-}\mathbf{Vect})$. Is there
are chain complex version of the split lemma, ...
2
votes
3answers
79 views
What does $p+q=k$ mean in the index of summation?
I need help solving something I don't understand. OK so the problem is this:
$$H^k(X,C)=\bigoplus_{p+q=k} H^{p,q}(X),$$
What does the $\;p+q=k\;$ mean?
Thank you anybody that helps! :)
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votes
1answer
21 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 $R$ ...
10
votes
2answers
124 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
22 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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votes
0answers
146 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
votes
1answer
96 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 ...
1
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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
$$
\operatorname{Proj} \left(\oplus_{i=0}^\infty t^i \left< ...
1
vote
1answer
57 views
Graded Vector Spaces and the Interchange Law
I'm a little confused about how to correctly interchange factors in tensor
products on graded vector spaces.
In particular let $V:= \bigoplus_{n \in \mathbb{N}} V_n$ be a $\mathbb{N}$-graded
vector ...
1
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0answers
30 views
Understanding $Bl_{\mathcal{I}}(k^4)/S_2$ where $\mathcal{I}$ is defined by $(x_1-x_2,x_3-x_4)$
Let $k_4=Spec(k[x_1, x_2, x_3,x_4])$ and $\mathcal{I}$ is the ideal sheaf defined by $(x_1-x_2,x_3-x_4)$. Then
$$
Bl_{\mathcal{I}}(k^4) = Proj (\oplus_{i\geq 0} I^i t^i)
$$
where ...
2
votes
0answers
71 views
Degree 1 elements in a graded ring from a blow-up perspective
This may be an elementary question but I hope this question will benefit others as much as myself.
Let $k^4 = Spec \; k[x_1, x_2, x_3, x_4]$.
Writing $Bl_{\mathcal{I}}(k^4)$ as $R = ...
2
votes
1answer
96 views
Blow-up along an ideal sheaf
Let $k^2=\operatorname{Spec} \; k[x,y]$ where $k$ is an algebraically closed field. Let $\mathcal{I}$ be the ideal sheaf defined by $(x,y)$. Then
$$
Bl_{\mathcal{I}}k^2
$$
is covered by two open ...
0
votes
1answer
49 views
The variety associated to a polynomial ring with a particular grading
We impose various grading on an algebra or a module to understand its homological properties. So I made up the following problem and I would like to understand its correspondence as a variety.
...
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2answers
177 views
Radicals of homogeneous ideals over semigroup-graded rings.
In this post I set out to prove an equation holding in a ring $R$ graded over $\mathbb{Z}$ or $\mathbb{N}$.
The equation was: $\sqrt{J^\ast}=(\sqrt{J})^\ast$, where $J$ is an ideal of $R$, $J^\ast$ ...
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0answers
51 views
geometrical interpretation of $\mathbb{Z}/2\mathbb{Z}$ graded space
According to wikipedia, a $\mathbb{Z}/2\mathbb{Z}$ graded space (super vector space) $V$ is a a vector space which can be decomposed in a direct sum $V=V_0 \oplus V_1$ where elements of $V_0$ are ...
2
votes
1answer
86 views
Should Hom(M,N) be also graded ??
If $M,N$ are graded modules over graded ring $R$, my question: is $\operatorname{Hom}_{R}(M,N)$ also a graded module and how ?
1
vote
2answers
155 views
question about direct sum and direct product in Graded Modules?
let M, N be R- Graded Modules, say: $$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
96 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
112 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
82 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
242 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
190 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 ...
8
votes
2answers
407 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
51 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
136 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 ...
