Solving differential equations on a scheme

Question

Is there a way to define differential equations on a scheme? If so, is there a Galois-like theory for adding solutions to the differential equation to the sections of the Scheme?

Answer 1

Question: "If so, is there a Galois-like theory for adding solutions to the differential equation to the sections of the Scheme?"

Answer: If $A:=k[x]$ is a polynomial ring over a field $k$ it follows $T:=Der_k(A)\cong k[x]\partial_x$ were $\partial_x$ is partial derivative wrto the $x$-variable. Let $E:=A\{e_1,..,e_n\}$ be the free $A$-module of rank $n$. A "connection" on $E$ is a map

$$\nabla: T \rightarrow End_k(E)$$

defined by ($z:=\sum_i a_ie_i$ with $a_i \in A$)

$$\nabla(\partial)(z):=\sum_i \partial(a_i)e_i + A(\partial) z$$

where $A: T \rightarrow Mat(n,A)$ is an $A$-linear map where $Mat(n,A)$ is the ring of $n\times n$-matrices with coefficients in $A$. The kernel of the connection

$$E^{\nabla} \subseteq E$$

is a $k$-vector space - the "solution space" of the connection.

Note: The solution space $E^{\nabla}$ consists of polynomial solutions to the system $(E,\nabla)$, and usually polynomial solutions to systems of linear differential equations are not that interesting.

Example: Complex manifolds: One may do something similar for complex manifolds: If $X \subseteq \mathbb{P}^n_{\mathbb{C}}$ is a compex manifold, $E$ a holomorphic finite rank vector bundle and $\nabla: T_X \rightarrow End(E)$ a flat holomorphic connection, it follows $E^{\nabla}$ is a local system on $X$. There is an equivalence of categories between the category of local systems of complex vector spaces on $X$, and complex representations of the topological fundamental group of $X$, hence to $E^{\nabla}$ we get a finite dimensional complex representation

$$\rho_{\nabla}:\pi_1(X) \rightarrow End_{\mathbb{C}}(W)$$

of the topological fundamental group. This correspondence leads to the famous "Riemann-Hilbert correspondence" - a much studied topic in algebra/topology. Similar constructions exist for schemes. Such questions leads to the study of differential Galois theory, Picard-Vessiot theory and linear algebraic groups. A "linear algebraic group" is an algebraic subgroup of $GL_k(V)$ for some vector space $V$ of finite dimension over a field $k$.

https://en.wikipedia.org/wiki/Riemann%E2%80%93Hilbert_correspondence#Examples

https://en.wikipedia.org/wiki/Picard%E2%80%93Vessiot_theory

https://en.wikipedia.org/wiki/Differential_Galois_theory