User:Alodyne/Koszul algebra

In homological algebra, a branch of mathematics, Koszul algebras are a certain kind of graded algebra whose trivial modules admit a particular kind of canonical projective resolution. They also enjoy various important duality relations, referred to as Koszul duality. They are named after Jean-Louis Koszul, and were introduced by Stewart Priddy.

Koszul algebras have a particularly nice kind of presentation by generators and relations, and a simple relational structure. The presentation is engineered so that the homological algebra of the ring, in particular a minimal projective resolution of the trivial module, is easier to understand.

Motivation
Koszul algebras arose originally from an attempt to isolate a class of algebras for which simplified calculations involving Ext were possible via what Priddy called Koszul resolutions. More precisely, if A is an augmented graded algebra over the field k, then k is an A-module for which A acts trivially, by the augmentation map A &rarr; k. One is very frequently interested in the structure of the Yoneda algebra $$\mathrm{Ext}^*_{A}(k,k)$$ (see Ext functor). This can always be calculated as the homology of the bar complex, but in practice the bar complex is very large and becomes unwieldy. Stewart Priddy introduced a new class of canonical resolutions, called Koszul resolutions, which were much smaller than the bar complex but had isomorphic homology. Koszul algebras are the simplest class of algebras for which the Koszul resolution of the trivial module is a minimal free resolution.

Definition
There are several equivalent definitions of what it means for a graded ring to be Koszul. It is possible to define Koszulity for algebras over quite general rings, but for brevity we will take a more naive approach. Let A be a connected graded algebra over a field k. Since A is graded, the groups Extp(M,N) are also graded, if M and N are allowed to be graded A-modules. If M is a left graded A-module, we say that M is concentrated in degree j if Mi = 0 for all i &ne; j.

The ring A is said to be (left) Koszul if k admits a free resolution by graded left A-modules Fp such that Fp is concentrated in degree p, for each p &ge; 0.

The other characterizations are more homological in nature. The assumption that k is a field implies that any simple graded A-module is concentrated in some degree, and that any graded module concentrated in a single degree is semisimple.

This allows one to derive a vanishing condition for the groups Extp(M,N) as follows. Suppose that the graded A-modules M and N are concentrated in degrees i and j respectively. Then


 * $$\mathrm{Ext}^p_A(M,N) = 0 \quad \mathrm{if} \quad p > j - i.$$

This leads to another definition of a Koszul algebra. If M and N are any graded A-modules as above, then A is said to be Koszul if


 * $$\mathrm{Ext}^p_A(M,N) = 0 \quad \mathrm{unless} \quad p = j - i$$.

This turns out to be equivalent to the following condition, by standard techniques of homological algebra:


 * $$\mathrm{Ext}^p_A(k,k[n]) = 0 \quad \mathrm{unless} \quad p = -n$$,

for all nonnegative integers n.

Examples
Let A be an exterior algebra over k, generated by one element x in degree 1, so that A is isomorphic to k[x]/(x2). Then A is a Koszul algebra, because k admits the following free resolution:


 * $$\text{put in the resolution}$$