Representation type of local and two-vertex bocses

Let k be an algebraically closed field. It has been proved by Zhang and Xu that if a bocs is of tame representation type, then the degree of the differential of the first solid arrow must be less than or equal to 3. We will prove in the present paper that: The bocs is still wild when the degree of the differential of the first arrow is equal to 3. Especially, the bocs with only one solid arrow is of tame type if and only if the degree of the differential of the arrow is less than or equal to 2. Moreover, we classify in this case the growth problems of the representation category of the bocs and layout the sufficient and necessary conditions when the bocs is of finite representation type, tame domestic and tame exponential growth respectively. This work was supported by National Natural Science Foundation of China (Grant No. 10731070)