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Let $A$ be a positively graded algebra. This means that $A$ is a $k$-algebra graded non-negatively and $A_0 \cong k \times \dots \times k$ such that each degree is finite-dimensional.

From here, we say $A$ is Koszul if for all simple $A$-modules $L(i)$, $L(i)$ admits a minimal linear projective resolution.

One can also think of an algebra such that for all simple $A$-modules $L(i)$, $L(i)$ admits a minimal linear injective resolution.

Are these definitions equivalent?

If so, why do we not introduce this definition in textbooks as well?

If not, why do we not care about this definition?

Molang
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  • Firstly, in the definition you should require only that graded simple A-modules admit linear minimal resolutions (otherwise what do you mean by minimal?) – Stephen Oct 03 '23 at 20:13
  • Secondly, have you thought about what happens for some of the simplest examples, say, the ring of polynomials in one variable over a field? – Stephen Oct 03 '23 at 20:14
  • @Stephen it should be graded since it is a linear resolution (i.e., homological i-th position is generated by its i-th degree), but could you explain why minimal resolution can only defined for graded modules? – Molang Oct 04 '23 at 20:43

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