Continuous dependence of linear differential systems on polynomial modules

Abstract

Linear differential systems, in Willems’ behavioral system theory, are defined to be the solution sets to systems of linear constant coefficient PDEs, and they are naturally parameterized in a bijective way by means of polynomial modules. In this article, introducing appropriate topologies, this parametrization is made continuous in both directions. Moreover, the space of linear differential systems with a given complexity polynomial is embedded into a Grassmannian.

Appendices

Appendix A: Convergence in Grassmannians

Let W be a finite-dimensional linear space of dimension, say, l. The Grassmannian $\text {Gr}(W)$ is a collection of all linear subspaces of W. For $1\le k\le l$, one defines the $\text {Gr}_k(W)$ to be the set of all k-dimensional linear subspaces of W. The set $\text {Gr}(W)$ is the disjoint union of its k-dimensional parts.

Let us denote by $Mon({\mathbb {F}}^k,W)$ the set of all monomorphisms from ${\mathbb {F}}^k$ to W. This is an open subset of $Hom({\mathbb {F}}^k,W)$, which is a finite-dimensional Hausdorff topological space.

There is a surjective map

$$\begin{aligned} \pi _k: Mon({\mathbb {F}}^k,W) \rightarrow \text {Gr}_k(W),\ \ \ f\ \mapsto \ Imf. \end{aligned}$$

The topology in $\text {Gr}_k(W)$ is the quotient topology; namely, U is an open set of $\text {Gr}_k(W)$ if and only if $\pi _k^{-1}(U)$ is open in $Mon({\mathbb {F}}^k,W)$. The topology can be described in terms of convergence as follows. A family $X_\varepsilon$ ($\varepsilon \ne 0$) of k-dimensional subspaces of W converges to a k-dimensional subspace X if and only if there exist injective homomorphisms $f_\varepsilon :{\mathbb {F}}^k\rightarrow W\ (\varepsilon \ne 0)$ and $f:{\mathbb {F}}^k\rightarrow W$ such that

$$\begin{aligned} X_\varepsilon =Im(f_\varepsilon ),\ X=Im(f)\ \ \text{ and }\ \ f_\varepsilon {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} f. \end{aligned}$$

Lemma 9

Let R be a full row rank (scalar) matrix, and assume that $R=[R_1\ | -R_2]$, where $R_1$ is square nonsingular. Then,

$$\begin{aligned} Ker(R)=Im\left[ \begin{array}{c}R_1^{-1}R_2\\ I\end{array}\right] . \end{aligned}$$

Proof

Follows from the theory of linear equations. $\square$

Lemma 10

If $X_\varepsilon \rightarrow X$, then

$$\begin{aligned} X_\varepsilon ^\perp \rightarrow X^\perp . \end{aligned}$$

Proof

We certainly may assume that $W={\mathbb {F}}^l$.

Choose full column rank matrices $M_\varepsilon$ and M so that $M_\varepsilon \rightarrow M$ and $Im(M_\varepsilon )=X_\varepsilon$, $Im(M)=X$. We then have

$$\begin{aligned} M_\varepsilon ^\mathrm{tr} \rightarrow M^\mathrm{tr}, \end{aligned}$$

and

$$\begin{aligned} X_\varepsilon ^\perp =Ker(M_\varepsilon ^\mathrm{tr}),\ \ \ X^\perp =Ker(M^\mathrm{tr}). \end{aligned}$$

(It is a standard fact that if R is a (scalar) matrix, then $Im(R)^\perp =Ker(R^\mathrm{tr})$.) Without loss of generality, we may assume that M and $M_\varepsilon$ have the forms

$$\begin{aligned} M=\left[ \begin{array}{r}M_1\\ -M_2\end{array}\right] \ \ \ \text{ and }\ \ \ M_\varepsilon =\left[ \begin{array}{r}M_{1,\varepsilon }\\ -M_{2,\varepsilon }\end{array}\right] , \end{aligned}$$

where $M_1$ and $M_{1,\varepsilon }$ are square nonsingular. By the above lemma,

$$\begin{aligned} X_\varepsilon ^\perp =Im\left[ \begin{array}{c}(M_{1,\varepsilon }^\mathrm{tr}) ^{-1}M_{2,\varepsilon }^\mathrm{tr}\\ I\end{array}\right] , \ X^\perp =Im\left[ \begin{array}{c}(M_1^\mathrm{tr})^{-1}M_2^\mathrm{tr}\\ I\end{array}\right] . \end{aligned}$$

The inverse of a nonsingular matrix is a continuous function of the entries of the matrix, and so

$$\begin{aligned} \left[ \begin{array}{c}(M_{1,\varepsilon }^\mathrm{tr})^{-1}M_{2,\varepsilon }^\mathrm{tr}\\ I\end{array}\right] {\mathop {\longrightarrow }\limits ^{\varepsilon \rightarrow 0}} \left[ \begin{array}{c}(M_1^\mathrm{tr})^{-1}M_2^\mathrm{tr}\\ I\end{array}\right] . \end{aligned}$$

This completes the proof. $\square$

Lemma 11

Assume that $X_\varepsilon \rightarrow X$ and $Y_\varepsilon \rightarrow Y$.

(a) If $dim(X_\varepsilon \cap Y_\varepsilon )=dim(X\cap Y)$ for all sufficiently small $\varepsilon$, then

$$\begin{aligned} X_\varepsilon \cap Y_\varepsilon \ \ \rightarrow \ \ X\cap Y. \end{aligned}$$

(b) If $dim(X_\varepsilon + Y_\varepsilon )=dim(X + Y)$ for all sufficiently small $\varepsilon$, then

$$\begin{aligned} X_\varepsilon + Y_\varepsilon \ \ \rightarrow \ \ X + Y. \end{aligned}$$

Proof

Left to the reader. $\square$

Appendix B: F.g. graded modules and coherent sheaves

A graded module (over T) is a T-module M together with a decomposition into ${\mathbb {F}}$-linear spaces

$$\begin{aligned} M=\bigoplus _{d\in {\mathbb {Z}}} M_d \end{aligned}$$

such that $s_iM_d\subseteq M_{d+1}$ for $i\in [0,n]$ and $d\in {\mathbb {Z}}$. An element $x\in M$ is called homogeneous of degree d if $x\in M_d$. A submodule $N\subseteq M$ is graded if $N=\bigoplus (N\cap M_d)$.

A homomorphism of graded modules $M\rightarrow N$ is a module homomorphism $u:M\rightarrow N$ such that $u(M_{d})\subseteq N_{d}$ for all d.

For a graded T-module M and an integer a, one denotes by M(a) the graded T-module whose homogeneous components are defined by

$$\begin{aligned} M(a)_d = M_{a+d}. \end{aligned}$$

The module $T(-a)$ is a free graded module of rank 1 generated by an element of degree a. A f.g. graded free module is a one that is isomorphic to a graded module of the form

$$\begin{aligned} \bigoplus _{j=1}^pT(-a_j). \end{aligned}$$

If M is a f.g. graded module, then by Hilbert’s syzygy theorem, there exists an exact sequence

$$\begin{aligned} 0\rightarrow F_l{\mathop {\rightarrow }\limits ^{\phi _l}} F_{l-1}{\mathop {\rightarrow }\limits ^{\phi _{l-1}}} \cdots {\mathop {\rightarrow }\limits ^{\phi _2}} F_1{\mathop {\rightarrow }\limits ^{\phi _1}} F_0, \end{aligned}$$

where $0\le l\le n$, $F_0, \ldots ,F_l$ are graded free T-modules of finite rank, and $F_0/Im\phi _1\simeq M$. Such a sequence is called a free graded resolution. The number l is called the length of the resolution, and the ranks of the free modules are called the Betti numbers. A free graded resolution is said to be minimal if, for each $i\ge 1$,

$$\begin{aligned} \phi _iF_i\subseteq s_0F_{i-1}+\cdots + s_nF_{i-1}. \end{aligned}$$

A minimal free graded resolution exists and is unique up to isomorphism. The length of the minimal free resolution is called the projective dimension of M; the ranks of the free modules are called the Betti numbers. Supposing that $F_i\simeq \oplus _jT(-a_{i,j})$ in the minimal resolution of M, the number

$$\begin{aligned} reg(M) = \max \{a_{i,j} - i\ | \ i,j\} \end{aligned}$$

is called the Castelnuovo–Mumford regularity (see Ch.4 in Eisenbud [1]). The projective dimension and the Betti numbers measure the size of the minimal resolution; the Castelnuovo–Mumford regularity measures its complexity.

Free graded resolutions were invented by Hilbert in order to compute the dimensions of the components of a graded module. The Hilbert function of a graded module $M=\bigoplus M_d$ is defined by the formula

$$\begin{aligned} HF(M,d)=\dim _{\mathbb {F}}(M_d), \ \ \ d\in {\mathbb {Z}}. \end{aligned}$$

It is easily seen that HF(M, d) becomes a polynomial, when $d\ge reg(M)$. This polynomial is called the Hilbert polynomial and is denoted by HP(M, d).

We pass now to coherent sheaves on the projective space ${\mathbb {P}}^n$. One can give a purely algebraic definition of these objects. (We omit the definition of the projective space itself as it is not necessary for the purposes of this article.)

For each nonempty subset $\alpha \subseteq [0, n]$ and each $d\ge 0$, write $s_\alpha ^d$ for $\prod _{i\in \alpha }s_i^d$ and define $T_\alpha$ to be the ring

$$\begin{aligned} T_\alpha = \left\{ \frac{f}{s_\alpha ^d} \ | \ f\in T \ \text{ is } \text{ homogeneous }, \ d\ge 0, \ degf=d|\alpha | \right\} . \end{aligned}$$

If $\alpha$ and $\beta$ are nonempty subsets of [0, n] and if $\alpha \subseteq \beta$, then there is a canonical homomorphism

$$\begin{aligned} T_\alpha \rightarrow T_\beta . \end{aligned}$$

It should be noted that, for each $k=0,\ldots ,n$, the ring $T_{\{k\}}$ is a polynomial ring; namely,

$$\begin{aligned} T_{\{k\}}={\mathbb {F}} \left[ \frac{s_0}{s_k},\ldots , \widehat{\frac{s_k}{s_k}},\ldots ,\frac{s_n}{s_k} \right] . \end{aligned}$$

(The “hat” = “omission.”) Notice that $T_{\{0\}}$ is canonically isomorphic to S.

A presheaf (of modules) is a map ${{{\mathcal {F}}}}$ assigning to every $\alpha$ a $T_\alpha$-module ${{{\mathcal {F}}}}(\alpha )$ and to each inclusion $\alpha \subseteq \beta$ a $T_\alpha$-homomorphism

$$\begin{aligned} {{{\mathcal {F}}}}(\alpha )\rightarrow {{{\mathcal {F}}}}(\beta ) : \ \ \ m \mapsto m|^\beta . \end{aligned}$$

If $\alpha =\beta$, the homomorphism is required to be the identity map, and if $\alpha \subseteq \beta \subseteq \gamma$, the composition ${{{\mathcal {F}}}}(\alpha )\rightarrow {{{\mathcal {F}}}}(\beta )\rightarrow {{{\mathcal {F}}}}(\gamma )$ is required to be equal to the homomorphism ${{{\mathcal {F}}}}(\alpha )\rightarrow \mathcal{F}(\gamma )$. (In short, a presheaf is a covariant functor from the ordered set of nonempty subsets of [0, n] to the category of modules.)

A presheaf ${{{\mathcal {F}}}}$ is called a sheaf if the canonical homomorphism

$$\begin{aligned} {{{\mathcal {F}}}}(\alpha )\otimes _{T_\alpha } T_\beta \rightarrow {{{\mathcal {F}}}}(\beta ) \end{aligned}$$

is bijective for each inclusion $\alpha \subseteq \beta$. A sheaf whose modules are finitely generated is called coherent.

A homomorphism of sheaves $\phi :{{{\mathcal {F}}}}\rightarrow {{{\mathcal {G}}}}$ is a collection of homomorphisms

$$\begin{aligned} \phi (\alpha ):{{{\mathcal {F}}}}(\alpha )\rightarrow {{{\mathcal {G}}}}(\alpha ),\ \ \ \emptyset \ne \alpha \subseteq [0,n] \end{aligned}$$

commuting with the “extension” homomorphisms.

One defines in an obvious way subsheaves and quotient sheaves, tensor products of sheaves. One defines as well, the kernels and the images of homomorphisms of sheaves, and hence, one can speak about exact sequences of sheaves.

Given a sheaf ${{{\mathcal {F}}}}$, for each $p=0,1,\ldots ,n$, set

$$\begin{aligned} C^p({{{\mathcal {F}}}})=\bigoplus _{|\alpha |=p+1}{{{\mathcal {F}}}}(\alpha ). \end{aligned}$$

The $\check{\mathrm{C}}\hbox {ech}$ complex of ${{{\mathcal {F}}}}$ is the sequence

$$\begin{aligned} 0\rightarrow C^0({{{\mathcal {F}}}}){\mathop {\rightarrow }\limits ^{\delta }} C^1({{{\mathcal {F}}}}){\mathop {\rightarrow }\limits ^{\delta }} \cdots \rightarrow C^{n-1}({{{\mathcal {F}}}}){\mathop {\rightarrow }\limits ^{\delta }} C^n({{{\mathcal {F}}}})\rightarrow 0, \end{aligned}$$

where the map $C^{p-1}({{{\mathcal {F}}}}){\mathop {\rightarrow }\limits ^{\delta }} C^p({{{\mathcal {F}}}})$ is defined by the formula

$$\begin{aligned} d(f_\alpha )=\sum _{k\notin \alpha }(-1)^{\circ (\alpha ,k)}f_\alpha |^{\alpha \cup \{k\}}. \end{aligned}$$

(The symbol $\circ (\alpha ,k)$ is taken from Theorem 10.2 in Eisenbud [1]; it denotes the number of elements of $\alpha$ less than k.) The $\check{C}\hbox {ech}$ complex is indeed a complex, i.e., the composition of two consecutive arrows is 0. Hence, for each $p=0,1,\ldots ,n$, one defines

$$\begin{aligned} H^p{{{\mathcal {F}}}}=\frac{Ker(C^p({{{\mathcal {F}}}}){\mathop {\rightarrow }\limits ^{\delta }} C^{p+1}({{{\mathcal {F}}}}))}{Im(C^{p-1}({{{\mathcal {F}}}}){\mathop {\rightarrow }\limits ^{\delta }} C^p({{{\mathcal {F}}}})}. \end{aligned}$$

The space $H^0{{{\mathcal {F}}}}$, the space of 0-dimensional cohomologies or global sections, is especially important. By definition, a global section of a sheaf ${{{\mathcal {F}}}}$ is a family $(f_k)_{0\le k\le n}$ with $f_k\in {{{\mathcal {F}}}}({\{k\}})$ such that for all i, j

$$\begin{aligned} f_i|^{\{i,j\}}=f_j|^{\{i,j\}}. \end{aligned}$$

Example

$H^0{{{\mathcal {O}}}}(d)=T_d$.

If $M=\bigoplus _{d\ge 0}M_d$ is a graded module over T, one defines its sheafification ${\widetilde{M}}$ by setting for each $\alpha$

$$\begin{aligned} {\widetilde{M}}(\alpha )= \left\{ \frac{m}{s_\alpha ^d} \ | \ \ d\ge 0, \ m\in M_{d|\alpha |} \right\} . \end{aligned}$$

Notice that if M is finitely generated, then ${\widetilde{M}}$ is a coherent sheaf.

One defines the structure sheaf ${{{\mathcal {O}}}}$ as the sheafification of T. More generally, for every $d\in {\mathbb {Z}}$, one sets

$$\begin{aligned} {{{\mathcal {O}}}}(d)=\widetilde{T(d)}. \end{aligned}$$

The sheaves ${{{\mathcal {O}}}}(d)$ are called the twisting sheaves.

Given a sheaf ${{{\mathcal {F}}}}$, the twist ${{{\mathcal {F}}}}(k)$ of ${{{\mathcal {F}}}}$ by k is defined to be ${{{\mathcal {F}}}}\otimes {{{\mathcal {O}}}}(k)$.

The great thing is that every sheaf can be obtained by the tilde construction. Namely, as the following lemma states

$$\begin{aligned} M\ \mapsto \ {\widetilde{M}} \end{aligned}$$

is a “surjective” functor.

Lemma 12

(Serre) Let ${{{\mathcal {F}}}}$ be a sheaf. Then, there is a canonical isomorphism

$$\begin{aligned} \widetilde{H({{{\mathcal {F}}}})} \simeq {{{\mathcal {F}}}}, \end{aligned}$$

where $H({{{\mathcal {F}}}})=\bigoplus _k H^0{{{\mathcal {F}}}}(k)$.

Proof

See Proposition 5.15 in Ch. II of Hartshorne [3]. $\square$

The sheafification functor does not yield an equivalence between the category of graded modules and the category of sheaves. It is easily seen that a graded module generates the same sheaf as its “tails.” If M is a graded module and $d_0$ an integer, one denotes by $M_{\ge d_0}$ the graded module whose component in the degree d is $M_d$ if $d\ge d_0$ and is zero otherwise. It is clear that

$$\begin{aligned} \widetilde{M_{\ge d_0}} = {\widetilde{M}}. \end{aligned}$$

The following lemma says that this is the only reason why the “tilde” fails to be an equivalence.

Lemma 13

(Serre) Let M and N be two graded modules. Then

$$\begin{aligned} {\widetilde{M}}\simeq {\widetilde{N}} \ \Leftrightarrow \ \exists \ d_0 \ \text{ such } \text{ that }\ M_{\ge d_0}\simeq N_{\ge d_0}. \end{aligned}$$

Proof

See Exercise 5.9 in Ch. II of Hartshorne [3]. $\square$

The following statement is known as Serre’s vanishing theorem.

Lemma 14

(Serre) Let ${{{\mathcal {F}}}}$ be a coherent sheaf. Then,

$$\begin{aligned} H^i{{{\mathcal {F}}}}(k)=0\ \ \ \forall \ i\ge 1, \ k\gg 0. \end{aligned}$$

Proof

See Theorem 5.2 in Ch. III of Hartshorne [3]. $\square$

The Euler characteristic $\chi ({{{\mathcal {F}}}})$ of a coherent sheaf ${{{\mathcal {F}}}}$ is an integer defined by the formula

$$\begin{aligned} \chi ({{{\mathcal {F}}}})=\sum _{i=0}^{i=n}(-1)^i\dim H^i{{{\mathcal {F}}}}. \end{aligned}$$

The function

$$\begin{aligned} HP({{{\mathcal {F}}}},d)=\chi ({{{\mathcal {F}}}}(d)),\ \ \ d\in {\mathbb {Z}} \end{aligned}$$

is a polynomial, which is called the Hilbert polynomial of ${{{\mathcal {F}}}}$. By Serre’s vanishing theorem, we have

$$\begin{aligned} HP({{{\mathcal {F}}}},d)=\dim H^0{{{\mathcal {F}}}}(d)\ \ \ \forall d \gg 0. \end{aligned}$$

If $M=\bigoplus _{d\ge 0}M_d$ is a f.g. graded module, then the canonical linear map

$$\begin{aligned} M_d\rightarrow H^0({\widetilde{M}}(d) \end{aligned}$$

is bijective for all large enough d. (This is immediate from Lemmas 11 and 12.) It follows that

$$\begin{aligned} HP(M,d)=HP({\widetilde{M}},d). \end{aligned}$$

Appendix C: Mumford’s bounding result

The following “strange” notion is extremely important and useful. It is due to Mumford [6].

Given a coherent sheaf ${{{\mathcal {F}}}}$ and an integer m, one says that ${{{\mathcal {F}}}}$ is m-regular if

$$\begin{aligned} \forall i\ge 1, \ \ \ H^i{{{\mathcal {F}}}}(m-i)=0. \end{aligned}$$

The Castelnuovo–Mumford regularity of ${{{\mathcal {F}}}}$, denoted by $reg({{{\mathcal {F}}}})$, is the smallest integer m such that it is m-regular.

Lemma 15

If ${{{\mathcal {F}}}}$ is an m-regular sheaf, then the following statements hold:

(a) The canonical map

$$\begin{aligned} H^0{{{\mathcal {O}}}}(1)\otimes H^0{{{\mathcal {F}}}}(k) \rightarrow H^0{{{\mathcal {F}}}}(k + 1) \end{aligned}$$

is surjective for $k\ge m$;

(b) Whenever $i \ge 1$ and $k \ge m - i$,

$$\begin{aligned} H^i{{{\mathcal {F}}}}(k)=0. \end{aligned}$$

In other words, if ${{{\mathcal {F}}}}$ is m-regular, then it is $m'$-regular for all $m'\ge m$.

Proof

See Lemma 2.1 in Nitsure [7]. $\square$

The following corollary provides an example where the notion of Castelnuovo–Mumford regularity is useful.

Corollary 3

If ${{{\mathcal {F}}}}$ is a coherent sheaf, then

$$\begin{aligned} \forall k\ge reg({{{\mathcal {F}}}}), \ \ \ HP({{{\mathcal {F}}}},k)=\dim H^0{{{\mathcal {F}}}}(k). \end{aligned}$$

Proof

By (b), when $k\ge reg({{{\mathcal {F}}}})$, $H^i({{{\mathcal {F}}}}(k)) = 0$ for every $i\ge 1$. $\square$

The following result of Mumford played a key role in Sect. 4.

Lemma 16

Let $\theta \in {\mathbb {Q}}[t]$. There exists a sufficiently big integer m, which has the following property: If ${{\mathcal {F}}}$ is a subsheaf of ${{{\mathcal {O}}}}^q$ with the Hilbert polynomial $\theta$, then

$$\begin{aligned} reg({{{\mathcal {F}}}})\le m. \end{aligned}$$

Proof

See Theorem 2.3 in Nitsure [7]. $\square$

Remark

Mumford’s result led to a significant simplification in the proof of Grothendieck’s theorem on existence of Hilbert and Quot schemes [2].