What are the SEO implications of a CNAME?
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(please ignore ridiculousness of hypothetical situation)
Lets say Amazon had a food division which was at food.amazon.com. I partnered with Amazon's food division and now food.amazon.com will point to my website (food.com). Amazon adds a CNAME record so food.amazon.com resolves to food.com. If food.amazon.com has built up significant page rank / domain authority, will food.com be getting those benefits? Also, lets say food.amazon.com/rice has a lot of PR / authority -- will food.com benefit from the value of those internal pages?
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The subdomain is like a completely different domain and has no impact on the tld.
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In addition to what Tyler said about the subdomain having no impact, the issue is that CNAME isn't actually redirecting anyone. So the traffic that is coming to food.amazon.com isn't really directing to food.com. It's merely renaming that page url.
In the event that you set it up that way, then sometime later removed the CNAME, and started food.com off on it's own, you would not inherit any of the domain authority from food.amazon.com. You'd essentially be starting from scratch with food.com.