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### Order of pole of Poincaré series

Let $R = \bigoplus_{n \geq 0} R_n$ be a graded Noetherian ring and $M = \bigoplus_{n \geq 0} M_n$ a finitely generated graded $R$-module. Let $\lambda$ be an additive function on the class of all ...
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### Twisting a graded algebra by an automorphism (Transitivity)

Definition: Let $A=A=\bigoplus_{j=0}^{\infty} A_j$ be a connected $\mathbb{N}$-graded $k$-graded algebra and let $\phi\in\text{Aut($A$)}$ be a graded automorphism of degree zero. A new graded algebra ...
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### $R$ is a UFD iff $R_{\frak{m}}$ is a UFD?

Let $R$ be a graded ring such that $R_0$ is a field and let $\frak{m}$ be the maximal ideal generated by all the elements of positive degree. Then, is it true that $R$ is a UFD iff $R_{\frak{m}}$ is a ...
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### Over which (graded) rings are all modules decomposable into indecomposables?

A module is decomposable if it is the direct sum of two modules. The process of splitting summands off of a decomposable module does not need to terminate, so infinitely generated modules do not ...
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### Proving the graded structure of the tensor algebra from only the universal property

When the tensor algebra is presented, it is usually constructed as the direct sum of all tensor powers of the space. By this construction, the graded structure of the tensor algebra is easy to prove. ...
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### Is there an operad homotopifying the Koszul rule?

In homotopy theory one has the idea of a homotopy-commutative multiplication, in which one replaces the relation $$ab=ba$$ in a commutative monoid/group/ring/etc. for an unspecified homotopy. One ...
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### Commutative monoid gradings via group scheme actions

$\newcommand{\Spec}{\mathrm{Spec}}$Recall the following result, proved in Section 2.9 of Neil Strickland's Formal schemes and formal groups, and in Lemma 1.3.2 of Eric Peterson's Formal Geometry and ...
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### Connection between the classifications of group extensions and group-graded algebras in terms of non-abelian cohomology

First, consider group extensions with non-abelian kernel $$1\to K\to G \to Q \to 1$$ It is well-known that these are classified by certain cohomological objects, specifically: Any such extension ...
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### Rational points of weighted projective spaces

[EDIT (Feb. 27, 2024): No answer to the reference question yet, but I explain a more general statement at the end.] Let $k$ be a field and let $\underline{a}=(a_0,\dots,a_n)$ be a tuple of positive ...
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### Dimension of the associated graded module at an ideal

Let $I$ be an ideal of a Noetherian local ring $(R, \mathfrak m)$. Define the associated graded ring $G_I(R):=\bigoplus_{n=0}^\infty I^n/I^{n+1}$. Then $G_I(R)$ is a Noetherian ring of the same ...
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### On the degree of the Hilbert polynomial of a graded module over the Rees algebra

If $A=\oplus_{n=0}^\infty A_n$ is a Noetherian graded ring of finite dimension such that $A_0$ is local and $A=A_0[A_1]$, and if $M=\oplus_{n=0}^\infty M_n$ is a finitely generated graded $A$-module ...
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### $\mathbb{Z}$-graded algebras and tensor products

Let $A = \bigoplus_{k \in \mathbb{Z}} A_k$ be a not necessarily commutative $\mathbb{Z}$-graded unital algebra over a field $\mathbb{K}$, and assume that it is strongly graded: $$A_kA_l = A_{k+l}.$$ ...
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I am studying $\mathbb{N}^n$-graded, finitely generated modules $M$ over the $\mathbb{N}^n$-graded polynomial ring $K[X_1,X_2,X_3...,X_n]$. I know every such $M$ has an indecomposable decomposition $M\... • 11 4 votes 1 answer 190 views ### Weighted blowup of a Cohen-Macaulay ring along a regular sequence is Cohen-Macaulay? Let$(R, \mathfrak{m})$be a Cohen-Macaulay ring, let$f_1, \dotsc, f_d \in \mathfrak{m}$be a regular sequence, and let$n_1, \dotsc, n_d > 0$be weights (feel free to assume that$n_1 = 1$if it ... 6 votes 0 answers 155 views ### Ring with different graded and ungraded global dimensions Let$A$be a$\mathbb N$-graded ring. One can consider the two categories$M_A^g$and$M_A^u$of graded and ungraded modules over$A$. Both have, say, enough projectives, hence one can compute various ... • 14.7k 8 votes 2 answers 408 views ### Conceptual explanation for the sign in front of some binary operations In several situations, I've seen that given a binary operation on a graded module$m:A\otimes A\to A$, a new operation$M(x,y)=(-1)^{|x|}m(x,y)$is defined so that it satisfies some properties. One ... • 489 5 votes 2 answers 579 views ### Representation theory in braided monoidal categories The crux of what I wish to know is what results from representation theory, a subject usually framed within the category$\text{Vect}_\mathbb{k}$, follow in more general braided monoidal categories? I ... • 191 2 votes 1 answer 194 views ### Detailed proof of$\mathfrak{s}^{-1}\mathrm{End}_V\cong \mathrm{End}_{\Sigma V}$I asked this question on MSE but I want to ask it again here with some more context sine it received no answers. In Chapter 3 (Algebra) of the book Operads in Algebra, Topology and Physics by Markl, ... • 489 0 votes 0 answers 155 views ### Is the free algebra over an operad an algebra over that operad? I'm asking here this question I asked on MSE that got no answers. Let$V$be a dg-module and$P$an operad. The free$P$-algebra on$V$is defined by$P(V)=\bigoplus_{r=0}^\infty (P(r)\otimes V^{\...
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Given a graded vector space $V$ over a field $k$, consider it's suspension $\Sigma V$ such that $(\Sigma V)^i=V^{i-1}$. For an operad of graded vector spaces over a field $\mathcal{O}$, the operadic ...
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### Reason to apply the Koszul sign rule everywhere in graded contexts

The Koszul sign rule is a sign rule that arises from graded-commutative algebras. For instance, let $\bigwedge(x_1,\dots, x_n)$ be the free graded-commutative algebra generated by $n$ elements of ...
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### Are open $\mathbb{G}_m$-invariant subschemes of an affine scheme precisely the homogeneous radical ideals of its coordinate ring?

This question is certainly not research level and in fact quite elementary which is why I asked it on math.stackexchange before: math.stackexchange. However it doesn't seem to get much attention there ...
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Let $A = \bigoplus_{i \in \mathbb{Z}} A_i$ be a strongly graded unital algebra over $\mathbb{C}$, with no zero divisors. Is it always true that $$m: A_i \otimes_{A_0} A_j \to A_{i+j}$$ is an ...