Does the dreamhost servers support the .net programming technology?
Nope DH uses Debian Linux with absolutely no MS technology - at least as far as I know. So you can use things like perl or php or ruby but I’m certain you can’t use .net on DreamHost.
Every so often, someone will mention mono, but I don’t think anyone has successfully run it here and I suspect you’d need a Dreamhost PS account to even try.
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I visit the mono website. It is a great tool. But (I agree with you) I don’t think we can install it on a shared server. PS will be a choice but I’ll suggest to double check with DH support first.