1. **State the problem:** Simplify the expression $$\frac{(2m^2)^{-1}}{m^2}$$. 2. **Recall the rules:** - Negative exponent rule: $$a^{-n} = \frac{1}{a^n}$$. - Division of powers with the same base: $$\frac{a^m}{a^n} = a^{m-n}$$. 3. **Apply the negative exponent:** $$ (2m^2)^{-1} = \frac{1}{2m^2} $$ 4. **Rewrite the original expression:** $$ \frac{\frac{1}{2m^2}}{m^2} = \frac{1}{2m^2} \times \frac{1}{m^2} $$ 5. **Multiply the denominators:** $$ \frac{1}{2m^2 \times m^2} = \frac{1}{2m^{2+2}} = \frac{1}{2m^4} $$ 6. **Final answer:** $$ \boxed{\frac{1}{2m^4}} $$