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10. Generalising a result of Schmidt-van den Dries

    1.     A result of Schmidt–van den Dries says the following:

      Suppose that $R = k[x_1,\dots, x_d]/(r_1, \dots, r_s)$ and there exists $n$ such that $\deg(r_i) \leq n$ for all $i$. Then there exists $N = N(d,s,n)$ such that if $R$ is not a domain, then there exist nonzero $p,q \in R$ of degree less than $N$ such that $pq = 0$.

      Add remark

      Problem 10.1.

      [Jason Bell] Does this result have a noncommutative analogue? More precisely, does a similar result hold if we replace $R$ by $k\langle x_1,\dots, x_d \rangle /(r_1, \dots, r_s)$?
        1. Remark. This might be too strong, so what about if $R$ is just PI?


            • Add remark

              Problem 10.2.

              [Michel van den Bergh] Does a version of the above result hold for finitely presented modules $M$ over a finitely presented $k$-algebra $A$? That is, is torsion in $M$ detected in bounded degree?

                  Cite this as: AimPL: Noncommutative surfaces and Artin's conjecture, available at http://aimpl.org/ncsurfaceartin.