1. Yes, that’s right. Equivalently, this category is Morita equivalent to the one-object category with endomorphisms . This is because both of them have the same Cauchy completion, namely all finitely presented projective -modules.

2. I don’t know anything about this, unfortunately.

3. Nope.

]]>I have some things I want clarified, as well as some questions:

1. You mention that given a ‘basis’ for an abelian category, one may consider the ‘higher vector space’ or ‘higher module’ $\text{Mod}(C)$. In the case my ‘basis’ consists of the finitely presented projective generators over an algebra $A$ (over a commutative ground ring), I assume this construction would return the module category $\text{Mod}(A)$?

2. You mention certain trace maps and zeroth Hochschild (co)homology. There is currently a conjecture regarding the invariance of degree zero Hochschild (co)homology under stable equivalences of Morita type, where one relaxes the condition of a Morita equivalence by adding a projective bimodule factor to the regular bimodules in the Morita theorem. (That is, $P \otimes Q \cong A \oplus P’$ for a projective bimodule $P’$ and vice-versa.) Can similar questions be asked in this setting? (These questions are addressed in quite some detail A. Zimmermann’s book on representation theory, and trace methods are one of the key ingredients for the theory developed in this direction.)

3. Do you know of any references in which this material is treated?

Thanks for a fun read!

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