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Saturday, 21 April 2007

Depth Zero Ideals in the Homogenized Weyl Algebra

  • Let mathcalD be the nth Weyl algebra mathcalD:=k[x1,...,xn,partial1,...,partialn], where partialixixipartiali=1.

  • Let widetildemathcalD be its Rees algebra, which is mathcalD:=k[t,x1,...,xn,partial1,...,partialn], where partialixixipartiali=t, and t is central.

  • Let mathcalOX denote the polynomial algebra k[x1,...,xn], which is a left widetildemathcalD-module, where t and all the partiali act by zero. NOTE: this is different than the homogenization of the standard mathcalD-module structure on mathcalOX.

The question I am interested in is, how many generators does a left ideal M in widetildemathcalD need before HomwidetildemathcalD(mathcalOX,widetildemathcalD/M) can be non-zero? My conjecture is that M needs at least n+1 generators. NOTE: Savvy Weyl algebra veterans will know every left ideal in mathcalD can be generated by two elements; however, this is not true in widetildemathcalD. There can be ideals generated by n+1 elements and no fewer.



The functor HomwidetildemathcalD(mathcalOX,) acts as a relative analog of the more familiar functor HomR(k,) (where R=k[t,y1,...yn]). Therefore, the above question is analogous to asking "how many generators must an ideal IsubseteqR have before R/I can have depth zero?" The answer here is n+1, which follows from Thm 13.4, pg 98 of Matsumura (essentially a souped up version of the Hauptidealsatz).



In the noncommutative case, if you try to make this work with the widetildemathcalD-module k (where t, xi and partiali all act by zero), it doesn't work. The natural conjecture would be that HomwidetildeD(k,widetildemathcalD/M)neq0 implies M had at least 2n+1 generators, except this fails even for the first Weyl algebra and M=widetildemathcalDx1+widetildemathcalDpartial1 (since widetildemathcalD/M=k).



However, it seems that things might work right for the relative module OX, based on a fair amount of experimentation. It is easily true in the first Weyl algebra. Oh, and equivalent condition is to ask when HomoverlinemathcalD(mathcalOX,overlineM)neq0, where overlinemathcalD=widetildemathcalD/t, and overlineM is M/Mt.

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