Dear Matthew, 

 An increase in nutrient  availability will decreases diversity through competition. More nutrient allow strong competitor to develop rapidly and will excluded some specialist microbes. This is what you can observe with cyanobacterial bloom when there is a fertilization event and one dominant species will take over.

The second statement might be explained with niches theory. The more complex an environment, the more niches you will have  i.e. microenvironment with specific physico-chemical property and biological organization. Only specialist microbes can acclimate to those niches, modifying the micro environmental conditions through their activity and creating new niches for other bugs to come, thus creating new relation and rising diversity. 

Thus, (while I am not completely sure about the connection that you are making and its universality), if you have more nutrient, you might merge niches that are linked to this nutrient limitation, and thus this might reduces the overall complexity of your environment. This will allow competitor to take over those new niches that they where previously excluded from because of the limitation of the nutrient, excluding the specialist that have develop different strategy to cope with this limitation, and so reducing diversity. 

I would be careful with the " A imply B, B imply C so A imply C" kind of reasoning that is applied here. I think this might be more complex than that depending of which environment and which nutrient is considered. I would also argue that at some point, lack of nutrient will reduce the diversity (look at a bare rock, there is very few species able to survive on it), and that it might be (as often) a modal relation more than a monotonous relation (meaning you will have an optimum). Another way to see the connection will be "Is nutrient limitation rising the environmental complexicity"

Hope that help a bit

Best