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### Hilbert functions for affine vs projective varieties and filtered vs graded algebras.

I'm somewhat confused about how to define Hilbert functions for affine varieties and for filtered rings in a compatible way. I'm familiar with how they are defined for projective varieties: Let $X$ be ...
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### Reason to apply the Koszul sign rule everywhere in graded contexts

I'm copy-pasting this question I asked in MO that received no answer. The Koszul sign rule is a sign rule that arises from graded commutative algebras. For instance, let $\bigwedge(x_1,\dots, x_n)$ ...
30 views

### Computation of Associated Graded Module

I am trying to compute $\mathrm{gr}_m(P)$ where $m=\langle X,Y\rangle$ and $P=\langle X^2-Y^3\rangle$ in the power series ring $R=\mathbb C[[X,Y]]$ with the $m$-adic filtration and show that it is ...
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### How to extend to a unique derivation on the graded tensor algebra $TV$?

If $V$ is a finitely generated graded module and $TV$ its graded tensor algebra then, Any degree $k$ linear map $V \to TV$ extends to a unique derivation of $TV$. I found this in page 45 of the book "...
10 views

### Why is this sequence of graded algebras wrong?

I'm studying Hilbert sequences, and when trying to understand the proof of $h_A(t)= \frac{1}{(1-t)^m}$ when $A= k[x_1,\dots,x_m]$ ($m \geq 2)$, I'm kinda stuck. I can prove it using the standard ...
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### Suspension Operator on Graded Algebras

Given a morphism of GAs $f:A\to B$ of degree $-k$, that is, $f(A_n)\subset B_{n-k}$, I want to understand the sign conventions of commutation with the suspension operator. That is, if $s:A\to A$ is ...