List questions from graded-rings

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List Question

What is $S_d$ in algebraic geometry?

Filtration on commutative algebra

Gradation of the quotient module

homogeneous idempotent element has degree $e$

Nonzero-divisor on the associated graded ring at the maximal ideal implies nonzero-divisor on the ring?

Is it true that every ideal contains some non-zero homogeneous elements?

Writing down the graded pieces of $k[x_1,...,x_n]/(x_1^2,...,x_n^2) $ in terms of the irrelevant maximal ideal

Grading and tensor product

On ideals , in polynomial rings , generated by degree $2$ homogeneous polynomials

Equality of localization of homogeneous ideal by a variable $x_i$.

Question about localisation in Hartshorne

On the $k$-vector space dimension of graded pieces of an Artinian $k$-algebra $k[x,y]/J$

generators of associated graded ring

Are open $\mathbb{G}_m$-invariant subschemes of an affine scheme precisely the homogeneous radical ideals of its coordinate ring?

Ring graded by a non-Abelian monoid

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