The tag has no wiki summary.

learn more… | top users | synonyms

2
votes
1answer
106 views

Endomorphisms preserving bilinear form

Let $V=V_0 \oplus V_1$ be a $\mathbb{Z_2}$ finite dimensional graded vector space of dimension $2n$. Let $x_1,..,x_n,y_1,...,y_n$ be a basis of $V$ such that the $x_i$ form a basis for $V_0$ and the ...
0
votes
0answers
84 views

Extension of multiplication to the tensor algebra.

In this wikipedia article http://en.wikipedia.org/wiki/Tensor_algebra#Construction We construct $T(V)$ as the direct sum of vector spaces $T^kV$ for $k=0,1,2,…$ $$ T(V)= ...
0
votes
0answers
83 views

Question about initial forms

I am working through Eisenbud's Commutative Algebra, and in Chapter $5$ he defines the following map. Say we have a filtration of modules ${\cal F}:M=M_0\supset M_1\supset\cdots$. Then for $f\in M$, ...
1
vote
1answer
48 views

How to think of a chain complex as a module?

I just started learning the subject, so the question should be basic. A complex $\mathbf{F}$ over a ring $R$ is a sequence of homomorphisms of $R$-modules $$\mathbf{F}: \cdots \to ...
2
votes
0answers
153 views

Computing ext over graded rings

This question came up as I was reading Beilinson, Ginzburg, Soergel paper Koszul Duality Patterns in Representation Theory. Suppose that $A$ is a Koszul ring (for the definition of Koszul ring ...
4
votes
1answer
112 views

What does it mean for the coordinate ring of an affine variety to be graded?

My question is relatively simple, assume that $X$ is an affine variety such that its coordinate ring $A:=\Bbbk[X]:=H^0(\mathcal O_X,X)$ is $\Lambda$-graded for some monoid $\Lambda$. Now if ...
4
votes
1answer
240 views

Proof details of Theorem 11.1 in Atiyah-Macdonald

I have some trouble filling in the details of this proof from Atiyah-Macdonald. In this result, the authors assume what follows: 1) $A = \oplus_{n=0}^\infty A_n$ is a Noetherian graded ring, and ...
2
votes
2answers
131 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, ...
1
vote
4answers
448 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
139 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! :)
0
votes
1answer
86 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
457 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", ...
4
votes
0answers
209 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
247 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
vote
0answers
39 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
77 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
vote
0answers
32 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
81 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 = ...
4
votes
1answer
226 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
56 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. ...
1
vote
2answers
235 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 $R$, $J^\ast$ ...
2
votes
1answer
149 views

Should Hom(M,N) be also graded ??

If $M,N$ are graded modules over a graded ring $R$, then $\operatorname{Hom}_{R}(M,N)$ is also a graded module and how ?
1
vote
1answer
317 views

Direct sum and direct product of graded modules

Let $M$, $N$ be $R$-graded modules, say: $$M= \bigoplus_{i\in\mathbb Z} M_{i}, N= \bigoplus_{j\in\mathbb Z} N_{j}.$$ Then $$\operatorname{Hom}(M,N) \cong \prod_{i\in\mathbb Z} \bigoplus_{j\in\mathbb ...
0
votes
1answer
186 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 ...
3
votes
1answer
183 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
vote
2answers
101 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
101 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 ...
5
votes
2answers
438 views

Localizing and taking degree zero commutes with tensor product

Let $S$ be a graded ring ($S_n=0$ for $n<0$), $f\in S$ a homogeneous element, and $M, N$ two graded $S$-modules. I'm trying to prove that $$(M\otimes_S N)_{(f)}\simeq ...
4
votes
3answers
466 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) = ...
3
votes
1answer
439 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 ...
12
votes
3answers
941 views

Homomorphisms of graded modules

Let $M$ and $N$ be graded $R$-modules (with $R$ a graded ring). $\varphi:M\rightarrow N$ is a homogeneous homomorphism of degree $i$ if $\varphi(M_n)\subset N_{n+i}$. Denote by $\mathrm{Hom}_i(M,N)$ ...
5
votes
2answers
186 views

Graded modules over $k[t,t^{-1}]$

If $R=k[t,t^{-1}]$ is a graded ring where $R_0=k$ is a field and $t\in R$ is a homogeneous element of positive degree which is transcendental over $k$, how can I prove that every graded $R$-module ...
2
votes
1answer
55 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 ...
8
votes
0answers
250 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 ...