EDIT
Why is this problem extremely painful ? While I was debugging / testing my web application, I just edited .ts in the VS IDE, saved, and then reloaded the page and tested the changes. Now I need to build. However, the assembly is not activated when a debugging session is running in the VS IDE. So now I have to do a full review: edit.ts -> save -> stop debugging -> build -> start debug. Again, this is very painful. End edit
I have not checked / saved / checked the option on the TypeScript Build tab in my project settings.

The corresponding <PropertyGroup> been generated and saved in my .csproj file.
(note without changing any TypeScript Build parameter, this <PropertyGroup> does not exist (when using an existing project that was previously created using VS 2013), and the default values for the assembly do not match the default values for the IDE UI, so that you were misled.)
In any case, after this change / save, my .csproj file is synchronized with the IDE interface, here it is:
<PropertyGroup Condition="'$(Configuration)' == 'Debug'"> <TypeScriptRemoveComments>True</TypeScriptRemoveComments> <TypeScriptTarget>ES5</TypeScriptTarget> <TypeScriptCompileOnSaveEnabled>True</TypeScriptCompileOnSaveEnabled> <TypeScriptNoImplicitAny>False</TypeScriptNoImplicitAny> <TypeScriptModuleKind>None</TypeScriptModuleKind> <TypeScriptOutFile /> <TypeScriptOutDir /> <TypeScriptGeneratesDeclarations>False</TypeScriptGeneratesDeclarations> <TypeScriptNoEmitOnError>True</TypeScriptNoEmitOnError> <TypeScriptSourceMap>True</TypeScriptSourceMap> <TypeScriptMapRoot /> <TypeScriptSourceRoot /> </PropertyGroup>
As shown, the Compile on Save function is definitely enabled. However, the .js file is only generated during build, and not when the .ts file is saved.
Btw: Regarding the fact that TypeScript compilation is the msbuild task in VS 2015 implementation, I wonder how this Compile on save setting even works ...
Additional diagnostics: It is interesting that if I delete the entire group of properties, then it will actually compile when saved. (The user interface remains unchanged, and .map files are not generated when saved, only during assembly.)