Carbon emission reduction has become a research hot spot in energy sector due to the growing environmental crisis. With the improvement of the carbon emission quota, different entities in the energy market should consider the cost when exceeding the quota. In this paper, we propose a local energy market with Double-side Auction Mechanism considering carbon costs. The bidding/offering prices of market entities (i.e., electricity producers and consumers) are decided by their generation cost functions or utility functions, respectively. These prices form the supply and demand curves to determine the market clearing price and quantity. Market-related information is collected and distributed by the independent system operator (ISO) to all entities through the blockchain network. The ISO assigns and settles the carbon quotas according to the clearing results of the market. Several case studies are presented to demonstrate the proposed local energy market mechanism. We find that introducing carbon costs benefits the renewable energy sources and the enterprises with low-carbon technologies.