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If you consider $S$ as you propose, that is $\bigoplus\limits_{n_i,n_j=0}T_{\underline{n}}$, then $T$ is a bi-graded ring over $S$: $$T_{p,q}=\bigoplus\limits_{n_i=p,n_j=q}T_{\underline{n}}.$$ By th …

Additively, the integral cohomology is the same as for the ordinary projective space (multiplication in cohomology is different though), so there is no torsion. This is Theorem 1 in the paper of Kawas …

Mathscinet mentions some 30 papers citing the paper you mention, among which the following looks like potentially relevant: Kapur, Deepak (1-NM-C); Sun, Yao (PRC-ASBJ-MML); Wang, Dingkang (PRC-ASBJ-MM …

You might want to look at the paper Groebner bases of ideals invariant under endomorphisms by Drensky and La Scala.

The answer (already given in comments, with a small misprint/mistake) is: (a) for even $r=2k+2$, it is $q^{m-1}+(q-1)q^{m-k-2}\eta((-1)^{k+1})$, where $\eta$ is the quadratic character of $\mathbb{F} …

The original reference for the general result is A.A. Beilinson, Coherent sheaves on $P^n$ and problems of linear algebra, Func. Anal. Appl. 12 (1978), pp. 214-216. Google search brings many relate …

FWIW, for $n=d=2$, this polynomial is irreducible, as I just checked in Magma. The naive code for this (which even the online calculator http://magma.maths.usyd.edu.au/calc/ can handle) is S<x1,x2,x3 …

Yes there is such a general statement, except for the fact that (as Nicolas suggests in his answer) you need to consider sequences of blow-ups with centers at different subspaces, not just points. The …

I believe what is meant by this is the usual external tensor product: if $p_1$ and $p_2$ are projections from $V_1\times V_2$ to $V_1$ and $V_2$ respectively, the external tensor product $\mathcal{F}_ …

This is a bit too long for a comment. If you take the relation $$ \sum_{i,j\in S;k,l\notin S}\delta_{\{S,S^c\}}=\sum_{i,k\in S;j,l\notin S}\delta_{\{S,S^c\}} $$ and add to both sides $\sum\limits_{i,j …

Some ideas of references that come to my mind right away: Fuks, Cohomology of infinite-dimensional Lie algebras. - contains lots of things that are relevant to your question Gelfand and Kazhdan, Cer …

To prove that the Hilbert series (the generating function of the sequence of dimensions of homogeneous components) of a finitely generated commutative graded algebra is a rational function, the easies …

It is: https://arxiv.org/abs/1902.06318 - this paper also explains how to use the Koszul dual algebra for something, where something is estimating Betti numbers of the free loop spaces of $\overline{M …

Superalgebras have been used in various questions of algebra in a very striking way. To give some instances: Kemer's proof of the fact that, over a field of zero characteristic, every system of ident …