Remove a file from a Git repository without deleting it from the local filesystem

From the man file:

When --cached is given, the staged content has to match either the tip of the branch or the file on disk, allowing the file to be removed from just the index.

So, for a single file:

git rm --cached mylogfile.log

and for a single directory:

git rm --cached -r mydirectory

To remove an entire folder from the repo (like Resharper files), do this:

git rm -r --cached folderName

I had committed some resharper files, and did not want those to persist for other project users.

You can also remove files from the repository based on your .gitignore without deleting them from the local file system :

git rm --cached `git ls-files -i -X .gitignore`

Or, alternatively, on Windows Powershell:

git rm --cached $(git ls-files -i -X .gitignore)