Семинар НУЛ АГП "Non-existence of negative weight derivations on a class of graded Artinian algebras"

Let P = C[x₁,…,x_n] be the polynomial algebra in n weighted variables with positive integer weights w₁,…,w_n, and consider the ideal I = (f₁,…,f_m) generated by weighted homogeneous polynomials f₁,…,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 f_j are bounded below by a suitable constant. The approach allows to prove the following theorem:

Let R be as above. Suppose that w₁ >= … >= w_n and deg f_j > 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 w_i are pairwise coprime and deg f_j > 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 and is held within the framework of the project “International academic cooperation” HSE University.