# Picard-surjectivity and Morita-equivalence

Let us say that an algebra $$A$$ over a field $$k$$ is Picard-surjective if the canonical map $$\mathrm{Aut}(A) \rightarrow \mathrm{Pic}(A)$$ is surjective. Here $$\mathrm{Pic}(A)$$ denotes the group of isomorphism classes of invertible $$A$$-$$A$$-bimodules and the map sends an automorphism $$\alpha$$ to the $$A$$-$$A$$-bimodule $$A_\alpha$$, where the left action is the usual one and the right action is via $$\alpha$$.

Q: For any given finite-dimensional $$k$$-algebra $$A$$, does there exist a Morita-equivalent one that is Picard-surjective?

If not, I am interested in conditions under which this is true. I am mainly interested in the case $$k=\mathbb{R}$$ or $$\mathbb{C}$$, and for all the examples that I came up with so far, this seems to be correct, as far as I can tell.

Yes, the basic algebra of $$A$$ will be Picard-surjective.

The basic algebra is the endomorphism algebra $$\operatorname{End}_A(\bigoplus_{i=1}^{n}P_i)$$ of the direct sum of indecomposable projective (right) modules, one from each isomorphism class. It is Morita equivalent to $$A$$.

Suppose $$A$$ is basic. Then as a left or right module, $$A$$ is the direct sum of indecomposable projective modules, one from each isomorphism class. Suppose $$M$$ is an invertible bimodule. Since $$S\otimes_AM$$ is nonzero for every simple module $$S$$, a direct sum decomposition of $$M$$ as a left module must contain at least one copy of each indecomposable projective. Let $$X=\bigoplus_{i=1}^nS_i$$ be the direct sum of simple (right) $$A$$-modules, one from each isomorphism class. Since $$X\otimes_AM\cong X\cong X\otimes_AA$$, as a left module $$X$$ must contain exactly one copy of each indecomposable projective in a direct sum decomposition. So $$M\cong A$$ as left modules. The right $$A$$-module structure of $$M$$ is then given by an injective algebra homomorphism $$A^{op}\to\operatorname{End}_A(_AA)\cong A^{op}$$, which is an isomorphism by finite dimensionality: i.e., the right action is induced by an automorphism of $$A$$.