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Do I need to put any grease on the barrel part that goes into the receiver? I saw one video show with a grease and another video it went in dry.
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That's what I was referring to earlier. That's called the "extension" of the barrel. I put a dab of red loctite on it and put it into the receiver. It sounds nutty and no, red loctite won't cause it to be any harder to remove but it will help with accuracy a little.
It doesn't need grease and depending on your upper it may just be really easy to go in. I know BCM makes their receiver a little tight while others tend to just slide in very easily.
I recommend picking up the The AR15 complete assembly guide. It will help you much better than coming on the forum (or watching videos from unknown persons) as you'll have opinions from a expert builder who does it for a living: http://www.fulton-armory.com/thear-15completeassemblyguidebuildyourownar-15.aspx
Edit: I should also add that grease and chemicals that you use in your build is not as important as the torque values that you apply. I recommend investing in a proper torque wrench, proper vise, proper ar15 wrench that you can connect to the torque wrench, and block for your build.