Coherent rings of differential operators

Let $A$ be a finitely generated commutative algebra over a field $k$, and consider the ring $D$ of $k$-linear differential operators on $A$ in the sense of Grothendieck [5]. Any system of linear differential equations defined over $A$ has the form

where $P_{ij}\in D$ are differential operators for $1\leq i\leq m,\;1\leq j\leq n$ and $u_{1},u_{2},\dots,u_{n}$ are unknown functions. To this system, we can associate the finitely presented left $D$-module $M=\operatorname{coker}(\phi)$, given by

where $\phi:D^{m}\to D^{n}$ is given by right multiplication by the $m\times n$ matrix $(P_{ij})$ with coefficients in $D$. It is therefore reasonable to define an algebraic $D$-module to be a finitely presented left $D$-module $M$. With this definition, we may identify $\operatorname{Hom}_{D}(M,\mathcal{S})$ with the set of solutions of the above system of differential equations with values in a left $D$-module $\mathcal{S}$.

If $A$ is a smooth integral domain over a field $k$ of characterstic $0$, then $D$ is a simple Noetherian ring, and a finitely generated $k$-algebra. In this case, any finitely generated left $D$-module is finitely presented. However, $D$ is neither finitely generated nor Noetherian in general; see for instance Smith [6]. In fact, when $A$ is either of the rings

1. (1)

   $A=k[x,y,z]/(x^{3}+y^{3}+z^{3})$ when $k$ has characteristic $0$

2. (2)

   $A=k[x_{1},\dots,x_{n}]$ when $k$ has characteristic $p>0$

then the ring $D$ of differential operators on $A$ is not a Noetherian ring, and not a finitely generated $k$-algebra. The study of algebraic $D$-modules are in these cases considered hopeless, but this is not necessarily so; if $D$ is a coherent ring, then the category of finitely presented left $D$-modules, or coherent left $D$-modules, have good properties.

The main result in this paper is that the ring $D$ of differential operators on a finitely generated, smooth and connected commutative algebra $A$ over a field $k$ of characteristic $p>0$ is coherent. We conjecture that this is also the case for the ring of differential operators on the cubic cone $A=k[x,y,z]/(x^{3}+y^{3}+z^{3})$ when $k$ has characteristic $0$. However, this is an open question, as far as we know.

In Bavula [1], the author studies finitely generated and finitely presented left $D$-modules when $D$ is the ring of differential operators on the polynomial ring $A=k[x_{1},\dots,x_{n}]$ over a field $k$ of characteristic $p>0$. His results show that finitely presented $D$-modules give a far more reasonable theory than finitely generated $D$-modules. Our results show that $D$ is a coherent ring in this case, and a $D$-module is therefore coherent if and only if it is finitely presented. Bavula’s results fit very nicely with our point of view, that coherent modules is the “correct” notion for $D$-modules.

The results in Bavula [1] give a classification of the holonomic $D$-modules when $D$ is the ring of differential operators on $A=k[x_{1},\dots,x_{n}]$ in characteristic $p>0$. The result is that there are very few holonomic $D$-modules, and they are trivial as differential operators since they are given by multiplicative operators of order $0$. As a comment to these results, we show that when $p=2$ and $n=1$, then the coherent left $D$-module $M=D/D\cdot\partial$ is not holonomic, but has dimension $\dim(M)=2$ and multiplicity $e(M)=1/2$.

Let $R$ be an associative ring. A left $R$-module $M$ is coherent if it is a finitely generated $R$-module with the property that any finitely generated submodule of $M$ is finitely presented. We write $\mathsf{Coh}({R})\subseteq\mathsf{Mod}({R})$ for the full subcategory of coherent left $R$-modules.

We recall some fundamental results for coherent modules. Proofs of these results are given in Chaper 2 of Glaz [4] in the commutative case (the same proofs hold when $R$ is any associative ring); see also Exercise I.2.11-12 in Bourbaki [2].

We say that $R$ is a left coherent ring if $R$ is a coherent as a left $R$-module, or equivalently, if any finitely generated left ideal in $R$ is finitely presented. If follows that a left Noetherian ring is left coherent. We recall that $R$ is left semi-hereditary if any finitely generated left ideal in $R$ is projective. It follows from Lemma 1 that a left semi-hereditary ring is left coherent.

The notion of a right coherent module and of a right coherent ring can be defined similarly, and by symmetry, the results in this section also hold for right modules. We say that $R$ is a coherent ring if it is left and right coherent. The polynomial ring $R=k[x_{1},x_{2},\dots]$ in an infinite number of variables $x_{1},x_{2},\dots$ over a field $k$ is an example of a coherent ring that is not Noetherian.

Let $A=k[x_{1},x_{2},\dots,x_{n}]$ be the polynomial ring in $n$ variables over a field $k$. We consider the ring $D=D(A)$ of $k$-linear differential operators on $A$, in the sense of Grothendieck [5]. This is a filtered ring, equipped with the order filtration

Let us write $\mu:A\otimes_{k}A\to A$ for the multiplication map, given by $\mu(a\otimes b)=ab$, and $J=\ker(\mu)$ for its kernel, which acts on $\operatorname{End}_{k}(A)$ in the natural way. The the set of differential operators of order at most $i$ is given by $D^{i}=\{P\in\operatorname{End}_{k}(A):J^{i}\cdot P=0\}$.

Let us describe the ring $D$ in concrete terms. We consider the partial derivations $\partial_{i}=\partial/\partial x_{i}\in\operatorname{Der}_{k}(A)$ for $1\leq i\leq n$, and define their divided powers $\partial_{i}^{[r]}:A\to A$ to be the $k$-linear operators given by

for all multi-indices $m=(m_{1},m_{2},\dots,m_{n})\in\mathbb{N}_{0}^{n}$ and for all integers $r\geq 0$, where we use multi-index notation

and write $m-r\epsilon_{i}=(m_{1},\dots,m_{i}-r,\dots,m_{n})$. Notice that the binomial coefficients in $k$ are the canonical images of the usual integer-valued binomial coefficients. The name divided powers come from the fact that $r!\,\partial_{i}^{[r]}=\partial_{i}^{r}$. The following result is well-known, see for instance Section 4 in Bavula [1]:

If $\operatorname{char}(k)=0$, then $D=A_{n}(k)$ is the $n$’th Weyl algebra, which is a simple Noetherian ring, generated by $\{x_{1},x_{2},\dots,x_{n},\partial_{1},\dots,\partial_{n}\}$. If $\operatorname{char}(k)=p>0$, then it is known that $D$ is not Noetherian and not a finitely generated $k$-algebra. We claim that $D$ is a coherent ring. In fact, we shall prove a more general result in the next section.

In any characteristic, we have a finite dimensional filtration $\{B^{i}\}$ of the ring $D$, given by the $k$-linear spaces

for $i\geq 0$, where $\mathbf{\partial}^{[r]}=\partial_{1}^{[r_{1}]}\cdots\partial_{n}^{[r_{n}]}$. We follow Bavula [1] and call this filtration the canonical filtration of $D$. When $\operatorname{char}(k)=0$, it coincides with the usual Bernstein filtration. Notice that $\dim_{k}B^{i}<\infty$ for all $i\geq 0$, since we have

Moreover, we have that $B_{i}=0$ for $i<0$, that $B^{i}\subseteq B^{i+1}$ and $B^{i}\cdot B^{j}\subseteq B^{i+j}$ for all $i,j\geq 0$, and that $\cup_{i}\,B^{i}=D$.

Let $k$ be a field of characteristic $p>0$, and let $A$ be a finitely generated, smooth and connected commutative algebra over $k$. We shall use the following construction, introduced in Section 3 of Chase [3]: Let $A_{r}\subseteq A$ be the $k$-subalgebra generated by $\{a^{p^{r}}:a\in A\}$ for all $r\geq 0$, and consider the chain

of $k$-algebras. We define $D_{r}=\operatorname{End}_{A_{r}}(A)\subseteq\operatorname{End}_{k}(A)$ for $r\geq 0$, and identify $A$ with $D_{0}=\operatorname{End}_{A}(A)$. From Lemma 3.3 in Chase [3], it follows that

where $D=D(A)$ is the ring of $k$-linear differential operators on $A$.

When $A=\mathcal{O}(X)$ is the coordinate ring of a non-singular affine algebraic variety $X$ of dimension $d$ over an algebraically closed field $k$ of characteristic $p>0$, then $D=D(A)$ has global homological dimension $d$ by Theorem 3.7 in Smith [6]. In case $d\leq 1$, it is therefore known that the ring $D=D(A)$ of differential operators is a coherent ring; by definition, any hereditary ring is semi-hereditary, and therefore coherent. As far as we know, this result is new for $d\geq 2$.

In this section, we assume that $A=k[x_{1},\dots,x_{n}]$ is a polynomial algebra over a field $k$ of characteristic $p>0$, and that $D=D(A)$ is the ring of differential operators on $A$. Then it follows from Theorem 7 that $D$ is a coherent ring, and we consider the category $\mathsf{Coh}({D})$ of coherent left $D$-modules.

Let $M$ be a left $D$-module. If $M$ is finitely generated, then there is a finite dimensional $k$-linear subspace $M_{0}\subseteq M$ such that $M=D\cdot M_{0}$. We consider the finite dimensional filtration $\{M_{i}\}$ of $M$ given by

for $i\geq 0$, and define $\dim(M)$ to be the growth of the function $i\mapsto\dim_{k}M_{i}$. We recall that for a function $f:\mathbb{N}_{0}\to\mathbb{N}_{0}$, the growth $\gamma(f)$ is defined as

This definition of $\dim(M)$ appears in Bavula [1]. It does not depend on the choice of generating set $M_{0}$, but may depend on the choice of finite dimensional filtration $\{B^{i}\}$ of $D$. We shall therefore fix the canonical filtration of $D$. The definition of $\dim(M)$ given by Bavula resembles the definition of Gelfand-Kirillov dimension over finite dimensional algebras, but is better suited for coherent $D$-modules in positive characteristic.

Let $M$ be a coherent left $D$-module. We say that $M$ is holonomic if $\dim(M)=n$, and define the category $\mathsf{Hol}({D})$ of holonomic $D$-modules to be the full subcategory of $\mathsf{Coh}({D})$ consisting of holonomic $D$-modules. This definition is different than the one used by Bavula, since he does not require that $M$ is coherent. However, the definitions coincide for coherent $D$-modules.

To require that holonomic modules are coherent, as we do, have consequences. This means that $A$ considered as a left $D$-module is not holonomic. For instance, if $A=k[x]$, then the left $D$-module $A$ can be written as $A=D/I$, where $I$ is the left ideal

Since $I$ is not finitely generated, $A=D/I$ is not finitely presented and therefore not coherent.

Let $M$ be a coherent left $D$-module, and let $\{M_{i}\}$ be a finite dimensional filtration given by $M_{i}=B^{i}\cdot M_{0}$, where $M_{0}\subseteq M$ is a finite dimensional linear subspace with $D\cdot M_{0}=M$. Then there is an integer $k\geq 0$ and polynomials $p_{i}(t)\in\mathbb{Q}[t]$ for $0\leq i<p^{k}$ such that $\dim_{k}\left(M_{mp^{k}+i}\right)=p_{i}(m)$ for all $m\gg 0$. Moreover, the polynomials $p_{i}(t)$ all have the same leading term, and are given by

where $d=\dim(M)$ and $e=e(M)>0$ is the multiplicity of $M$; see Theorem 5.5 of Bavula [1]. The function $mp^{k}+i\mapsto p_{i}(m)$ for $0\leq i<p^{k}$ and $m\geq 0$ is called a quasi-polynomial of period $p^{k}$. In general, we have that $p^{kn}e(M)\in\mathbb{Z}$. If $M$ is holonomic, then $e(M)$ is a positive integer by Theorem 8.7 in Bavula [1]. By Theorem 9.6 and Corollary 6.8, we have the following results:

This means that the simple holonomic $D$-modules are given by multiplication operators of order zero, and therefore trivial as systems of differential equations. Iterated extensions of these simple modules do not give more interesting holonomic modules. For example, when $n=1$, we have that $\operatorname{Ext}^{1}_{D}(M(\underline{\alpha}),M(\underline{\beta}))=0$ for all $\underline{\alpha},\underline{\beta}\in\mathbb{A}^{1}_{k}$.

Hence, any linear system of “interesting” differential equations, given by a matrix $(P_{ij})$ of differential operators as explained in the introduction, corresponds to a coherent left $D$-module that is not holonomic. The following example is instructive: Let $n=1$, and let $M=D/D\cdot\partial$. Then there is an exact sequence

In characteristic $p>0$, the kernel of the map $D\xrightarrow{\cdot\partial}D$ is non-zero. For instance, if the characteristic $p=2$, then the kernel is $D\cdot\partial$ since we have that

Therefore, there is a short exact sequence of coherent left $D$-modules

Since $\dim(D)=2$, we must have $\dim(M)=2$, and $M$ is not holonomic. In fact, $\dim(M)=2$ implies that the multiplicity $e(M)=e(D)/2=1/2$.