Can an algebra be morita equivalent to its dg-extension?

Question

Say we have a DG algebra $A=\bigoplus_{n\geq 0}A_n$, let $B=A_0$, the 0th degree of $A$. Assume we have that the category of DG-modules over $A$ is equivalent to the category of module over $B$. Does this imply that $A_i=0$ for $i\geq 1$?

Answer 1

Yes, but rather vacuously so: it implies that $A_i=0$ for all $i$, not just $i\geq 1$! Indeed, the category of modules over a ring has the following property: if $(M_i)$ is an infinite family of nonzero objects, the canonical map $\bigoplus M_i\to\prod M_i$ is not an isomorphism. On the other hand, if $A$ is a nonzero nonnegatively graded dg-algebra, then the category of dg-modules over $A$ does not have this property, since for instance you can take $M_i$ to be $A$ with its grading shifted up by $i$, and then the direct sum $\bigoplus M_i$ has only finitely many nonzero terms in each degree and so coincides with the product. So the category of dg-modules over a nonzero nonnegatively graded dg-algebra can never be equivalent to the category of modules over any ring.