Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 8 / 3 / Solution

Work over $\mathbb{C}$ in characteristic zero. The second Weyl algebra has generators $x_1,x_2,{\partial }_1,{\partial }_2$ and relations $[{\partial }_i,x_j]={\delta }_{ij}$, $[x_i,x_j]=[{\partial }_i,{\partial }_j]=0$. Its Bernstein filtration gives every generator degree one. The ordered monomials $x_1^{a_1}{x}_2^{a_2}{\partial }_1^{b_1}{\partial }_2^{b_2}$ form a basis, soFor a nonzero finitely generated module, choose a finite-dimensional generating subspace $M_0$ and put $M_r=F_r{A}_2{M}_0$. Its graded module is finite over this polynomial ring. The Hilbert polynomial theorem gives an eventual polynomial $\dim M_r$, whose degree $d(M)$ is the Bernstein growth dimension of a Weyl algebra module, equal to its Gelfand–Kirillov dimension of a module. Two choices of finite generators bound each other after fixed filtration shifts, proving independence of the choice. The characteristic variety of a Weyl algebra module is the support of this graded module; its algebraic dimension is $d(M)$. In particular $d(M)$ is an integer at most four.

Here is a growth proof of the lower bound, rather than just quoting the Bernstein inequality for Weyl algebra modules. A nonzero $u\in F_r{A}_2$ can be turned into a nonzero scalar by at most $r$ successive commutators with the four generators: choose a monomial of maximal total degree, differentiate it to a constant using $[{\partial }_i,\cdot ]$ and $[x_i,\cdot ]$, and all different monomials are killed by that derivative multi-index. This uses the ordered-monomial expressions and characteristic zero. Expanding the iterated commutator expresses that scalar as ${\sum }_j{v}_{j}u{w}_j$ with $v_j,w_j\in F_r{A}_2$.

If $u$ acted as zero on $M_r$, every $u{w}_j{M}_0$ would be zero, so the scalar would kill the nonzero $M_0$, a contradiction. Therefore the action mapis injective. This is the faithful finite-step action of a Weyl algebra. Comparing growth ingives $4\le 2d(M)$. Thus the only possible dimensions for nonzero modules are $\boxed{d(M)\in \{2,3,4\}}$.

All three values occur in cyclic modules. On $\mathbb{C}[x_1,x_2]$, let $x_i$ multiply and ${\partial }_i$ differentiate. The module is generated by $1$, its filtered pieces are precisely polynomials of degree at most $r$, and $\dim M_r=\left(\genfrac{}{}{0ex}{}{r+2}{2}\right)$. Hence $\boxed{d(\mathbb{C}[x_1,x_2])=2}$; equivalently it is $A_2/(A_2{\partial }_1+A_2{\partial }_2)$.

For $M=A_2/A_2{\partial }_2$, the ordered basis shows that the left ideal $A_2{\partial }_2$ is spanned by monomials with positive ${\partial }_2$ exponent. The remaining basis is $x_1^{a_1}{x}_2^{a_2}{\partial }_1^{b_1}$. Its degree-$\le r$ piece has dimension $\left(\genfrac{}{}{0ex}{}{r+3}{3}\right)$, so $\boxed{d(A_2/A_2{\partial }_2)=3}$. Finally the regular module has the four-variable ordered basis above, giving $\boxed{d(A_2)=4}$.

The dimension-two modules are the holonomic Weyl algebra modules. Exact sequences satisfy $d(M)=\max \{d(M^{\prime }),d(M^{\prime \prime })\}$, by using compatible good filtrations and the resulting Hilbert polynomials. Therefore nonzero subquotients of a holonomic module remain holonomic. Their positive integer multiplicities, namely $2!$ times the leading Hilbert-polynomial coefficient, add in exact sequences. Every nonzero successive factor consumes at least one unit of multiplicity, bounding the length of a submodule chain; hence holonomic modules have finite composition length. The polynomial module above is even simple: differentiating a nonzero polynomial to a constant and then multiplying by arbitrary polynomials generates the whole module. In contrast the regular and dimension-three examples exhibit the larger growth allowed for nonholonomic modules. These examples and the growth bound explain both the range and its representation-theoretic significance.