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. 



"as close to being semisimple as it can possibly be."
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/>. 


