Seminars: D. A. Shunin, Non-existence of negative weight derivations on a class of graded Artinian algebras

SEMINARS


Seminar of the Laboratory on Algebraic Transformation Groups HSE University December 17, 2025 18:00, Moscow, Pokrovsky b-d 11, D109

Non-existence of negative weight derivations on a class of graded Artinian algebras D. A. Shunin National Research University Higher School of Economics, Moscow
https://vkvideo.ru/video-227316419_456239161 Abstract: Let $P = \mathbb{C}[x_1, \ldots ,x_n]$ be the polynomial algebra in $n$ weighted variables with positive integer weights $w_1,\ldots ,w_n$, and consider the ideal $I = (f_1,\ldots ,f_m)$ generated by weighted homogeneous polynomials $f_1,\ldots ,f_m$. As $P$ is graded and $I$ is homogeneous, the quotient algebra $R = P / I$ is also graded, and so is the algebra of derivations $Der(R)$. Suppose that $R$ is Artinian. There is a number of conjectures concerning the existence of negative weight derivations on different classes of graded Artinian algebras $R$. For example, Aleksandrov Conjecture claims that if $R$ is a complete intersection algebra then $R$ has no negative weight derivations. A more specific Halperin Conjecture claims the same in the case when $m = n$, i.e. the complete intersection is zero-dimensional. The latter has a topological interpretation assuming $R$ is the cohomology ring of a good enough space $X$. Although these questions remain open, there exists a general approach providing a way to prove the non-existence of negative weight derivations when the degrees of fj are bounded below by a suitable constant. The approach allows to prove the following theorem: Let R be as above. Suppose that w1 >= … >= wn and deg fj > c = (m-1) (w1w2)n-1. Then there are no negative weight derivations on R. In the talk, a version of this theorem with an extra condition will be proved. Namely, we will prove that if the weights wi are pairwise coprime and deg fj > c = (m-1) w1w2 then there are no negative weight derivations, too. The talk is based on the paper of H. Chen, S. S.-T. Yau, and H. Zuo «Non-existence of negative weight derivations on positively graded Artinian algebras.