It's rusted on and won't come off. Some reach for a certain stinky penetrating solvent (WD40) but it doesn't really work well for this kind of thing. Here's what does:

Shock and awe (there are no lubricants or solvents in this particular 'ible).

Shock is provided with a hammer. Awe comes from the awesome power of hellfire... or a little propane torch, pick one.

This technique has resurrected many rusty bikes in salty places like Montreal.

Remember to vote for me if you find this knowledge useful.

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So many people have spoken of their success with various penetrating oils, I can't even keep track of them anymore. Please view the comments at the end.

Some people have kindly spoken in favour of penetrating lubricants as well. Whyteboar, Cross and toad all advise using PB Blaster and letting it soak in repeatedly. mastershake916 is partial "Yield" by Chemsearch and dlaninga1 suggests JB80 metal conditioner. rimar2000 says "You MUST try using liquid phosphating. You can dip the pieces many hours or days into normal liquid phosphating. It does not attack hands or skin."

Also, mastershake916 asked what I wanted the cog for. It was going to go on a Stokemonkey electric bike motor using an adapter but I changed my mind. You see, I'm building a electric/solar/pedal car in an attempt to merge the superb efficiency and light weight of an electric bicycle with some of the features of a car.

Kevanf1 had a neat idea: "...putting the lot into a freezer for 24 hours. Then try heating. But this time you will not have to bring the metal up to red hot and thus not risk changing the crystalline composition of the metal". Someone buy this dude a beer.

Step 1: Needful Thingies

-Hammer
-Punch (if you're freeing something like a stuck, busted off bolt)
-Blowtorch (mixed gas if you're in a hurry, otherwise sometimes even a candle can work fine)
-Water (optional)
-Real gasmask (very nice to have. I use mine daily. Heck, I have two for when a guest visits.)
-Love (you're gonna be hammering, torching, shocking, twisting... it's helpful to love the process)

And regular tools for your part eg. vice, wrench, screwdriver etc.

Step 2: Knock Knock Shock

Knock knock.
Who's there?
Hammer.
Hammer who?
...
Uh oh.

Since you've already tried your regular methods, here's where we apply some shock.

My theory here is that two pieces of metal have rusted into each other under compression so a sudden shockwave passing through the material will cause a sort of whiplash effect allowing the parts to jump apart for an instant. They may even loosen themselves a little.

So give the parts a sharp tap with the hammer. Recessed things may be easier to reach with a punch. BUT, be careful here. The point is to shock the part but not to bend it all out of shape. Place and power your hits such that you don't permanently distort your item; otherwise you may end up having more trouble getting it off. Be especially careful if you're using a punch, you don't want to flare out and expand a bolt that's stuck in a hole.

Hit hard but not too hard.

This step is dedicated to The Lightning Stalker who says "there is just no replacement for the "heat it and beat it" method". Sexy times.

Step 3: Torch Time

Try to undo the thingie. Is it loose? Great!

If not: Apply fire. Put on your gasmask first though; often grease, paint, and little bits of plastic end up getting burt up in the crossfire. Don't breath this.

Heat until red hot. Unless you're working on aluminum. As Esmagamus kindly reminded me, aluminum melts before it glows. You might not want that.

The theory for this step is that the parts will heat up and expand at different rates due to their being different sizes, slightly different materials, unevenly heated, etc. This difference in expansion probably causes little micro-shifts at the rusted/stuck surface. Once you have broken this seal, static friction goes waaay down.

Note: heating up metal can change its properties considerably. I've included a random time-temperature transformation chart (TTT) just to make it look complicated.

Optional: pour on water. I think rapid, uneven cooling might shock the parts apart more. Also I don't have to wait for things to cool. This can make steel more brittle, meh.

Pro bike shops might use an oxy propane or oxy acetylene torch. I just use propane which takes longer. In a pinch I once used a candle for an hour and that worked too.

Step 4: Hammer Time, Take Two

Try unscrewing it now. Usually your part will come off at this point but mine was still solidly stuck. Sometimes a few more taps with the hammer will make all the difference at this stage. This was the case in my example here.

Step 5: Free at Last

Free at last, free at last. Thank God almighty. We are free at last.
-King Jr. Martin Luther

After trial by hammer,
trial by fire,
trial by water,
and trial by hammer again for good measure,
the part comes off like it was never stuck in the first place, if you're lucky.

Epilogue: There were a couple of cracked teeth. This might have happened when I hammered after I poured on water. The rapid cooling may have made the metal brittle. Whatever, this type of sprocket is prone to broken teeth...