Cost refers to opportunity cost, i.e., the value of the alternative foregone when taking an action. In production, an entrepreneur uses assets he owns and buys inputs from others. Both owned assets and bought inputs impose costs on the entrepreneur. This is true for any entrepreneur, not just one who owns the entire supply of a resource. Any entrepreneur who has a rising demand for his product will have a rising cost structure for his owned assets. That is how he gains a monetary reward for superior entrepreneurship. The prices of assets he owns rises.
Any competitor who purchased a mine from Alcoa would have to pay the higher price (i.e., the price that incorporates the monopoly price of output) and therefore, would not be able to stay in business by undercutting Alcoa’s price. Because Alcoa can obtain the higher price by selling the mine, the cost of retaining ownership of the mine is also higher.
Consider another example: Government price supports for corn push up prices for corn. An investor who wants to get into farming would find that the price of corn-growing land has risen to the point at which no extra profit can be earned. The competitive bidding of outside investors to obtain higher profits generates this result. (Technically, it generates a tendency toward this result. Actually, all investment is uncertain and therefore, investors with superior foresight can buy assets before their prices fully rise and so earn extra profit.) Farmers who owned land before the price supports were raised do not earn extra profit (at least not after the adjustment of the market). Instead, they earn a capital gain in the market price of their assets.