Are static members of a generic class tied to the specific instance?

A static field is shared across all instances of the same type. Foo<int> and Foo<string> are two different types. This can be proven by the following line of code:

// this prints "False"
Console.WriteLine(typeof(Foo<int>) == typeof(Foo<string>));

As for where this is documented, the following is found in section 1.6.5 Fields of the C# Language Specification (for C# 3):

A static field identifies exactly one storage location. No matter how many instances of a class are created, there is only ever one copy of a static field.

As stated before; Foo<int> and Foo<string> are not the same class; they are two different classes constructed from the same generic class. How this happens is outlined in section 4.4 of the above mentioned document:

A generic type declaration, by itself, denotes an unbound generic type that is used as a “blueprint” to form many different types, by way of applying type arguments.

The problem here is actually the fact that "generic classes" are not classes at all.

Generic class definitions are just templates for classes, and until their type parameters are specified, they are just a piece of text (or a handful of bytes).

At runtime, one can specify a type parameter for the template, thus bringing it to life, and creating a class of the, now, fully specified type. That's why static properties are not template-wide, and that's why you cannot cast between List<string> and List<int>.

That relationship kinda mirrors the class-object relationship. Just like classes do not exist* until you instantiate an object from them, generic classes do not exist, until you make a class based on the template.

P.S. It's quite possible to declare

class Foo<T> {
    public static T Member;
}

From this is kinda obvious that the static members cannot be shared, as T is different for different specializations.

They are not shared. Not sure where it's documented but analysis warning CA1000 (Do not declare static members on generic types) warns against just this due to the risk of making the code more complicated.

C# implementation of generics is more closer to C++. In both of these languages MyClass<Foo> and MyClass<Bar> don't share static members but in Java they do. In C# and C++ MyClass<Foo> internally creates entirely new type at compile time as if generics are kind of macros. You can usually see their generated names in stack trace, like MyClass'1 and MyClass'2. This is why they don't share static variables. In Java, generics are implemented by more simpler method of compiler generating code using non-generic types and adding type casts all over. So MyClass<Foo> and MyClass<Bar> don't generate two entirely new class in Java, instead they both are same class MyClass underneath and that's why they share static variables.