0MXX Proposition 3.5. The category DTM(k,ℚ)w=0\operatorname{DTM}(k,\mathbb{Q})^{w=0} is the heart of a weight structure ww on DTM(k,ℚ).\operatorname{DTM}(k,\mathbb{Q}).
0MXY Proof. By (3.2) the collection of objects in the DTM(k,ℚ)w=0\operatorname{DTM}(k,\mathbb{Q})^{w=0} is negative. Moreover DTM(k,ℚ)\operatorname{DTM}(k,\mathbb{Q}) is idempotent closed and generated by DTM(k,ℚ)w=0\operatorname{DTM}(k,\mathbb{Q})^{w=0}. Now the statement follows from Proposition 2.4. ∎