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I had originally asked this question on math stack exchange but I think maybe it's more appropriate to ask it here.

In the paper of Beilinson, Ginzburg and Soergel entitled "Koszul Duality Patterns..." After stating the following proposition, they make the statement: " a Koszul ring is a positively graded ring that is "as close to being semisimple as it can possibly be.""

Proposition: Let $A = \bigoplus_{j\geq 0} A_j$ be a positively graded ring and suppose that $A_0$ is semisimple. The following are equivalent.

1) $A$ is Koszul

2) For any two pure $A$-modules $M$, $N$ of weights $m$, $n$ respectively we have $ext^i_A (M,N) = 0$ unless $i = m-n$.

3) $ext_A ^i (A_0, A_0 \langle n \rangle) = 0$ unless $i=n$.

To clarify some things: A graded module $M$ over a graded ring, is pure of weight $m$ iff $M = M_{-m}$. Also the $ext$'s we are using are in the category of graded modules.

I am very new to all this and I don't even have a vague idea of what they mean. Any thoughts at all would be super helpful.

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  • $\begingroup$ Why did you delete your question on math stack ? $\endgroup$ Aug 2, 2013 at 13:23
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    $\begingroup$ Let $M$ and $N$ be pure. In the semisimple case $ext^i_A (M,N) = 0$ for all $i$. In the Koszul case $ext^i_A (M,N) = 0$ for all $i$ except possibly one value. I guess this is what they mean by the phrase in the title. $\endgroup$ Aug 2, 2013 at 13:41
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    $\begingroup$ @Dietrich Thanks for your responses. As to why I deleted the question on math stack, when I deleted it there were no comments or answers to the question and I thought it would be easier to just have one version of the question up. $\endgroup$
    – Anette
    Aug 2, 2013 at 14:10
  • $\begingroup$ To make my previous comment more precise, I should add that if $M$ and $N$ are pure of the same weight, then you can have $\mathrm{ext}^0_A (M,N) \neq 0$ also in the semisimple case. $\endgroup$ Aug 2, 2013 at 18:26

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In the semisimple case it is really easy to calculate $ext_A^i(M,N)$, $i\ge 1$, with the above assumptions. It is zero. For Koszul rings this is almost true, i.e., $ext^i(M,N)$ is concentrated in degree $i$ (that is, if $S$ is the direct sum of all simples, then the two natural gradings on the algebra $A^{!}=ext^{\bullet}(S,S)$ coincide). It seems reasonable that the authors had this in mind as they wrote: "Morally a Koszul ring is a graded ring that is as close to semisimple as a $\mathbb{Z}$-graded ring possibly can be”.

This was already said in the comments. I can just add another reference here, a discussion on Koszul algebras and Koszul duality, which explains several things and refers also to the paper of Beilinson, Ginzburg and Soergel, here: http://sbseminar.wordpress.com/2007/11/01/koszul-algebras-and-koszul-duality/.

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