# Higher derived complete intersections

This question is about a class of commutative algebras that is (potentially) a little wider than locally complete intersection, but should still have reasonable properties.

Fix a ground field $\Bbbk$ of characteristic zero (might as well assume it to be algebraically closed) and let $A$ be a finitely generated commutative $\Bbbk$-algebra. Alternatively, we can think of the scheme $X=Spec(A)$.

Motivation. Suppose $A$ is a locally complete intersection. Then after localization (which I suppress in the notation) we can write $A=B/(f_1,...,f_n)$, where $B$ is a regular ring and $f_1,...,f_n\in B$ is a regular sequence. Many properties of $A$ can be derived using the Koszul resolution $K$ of $A$. This $K$ is a free commutative graded algebra over $B$ with generators $t_1,...,t_n$ of degree $-1$ and differential $d$ such that $d(t_i)=f_i$.

Reformulation. $A$ is locally quasi-isomorphic to a differential graded algebra over a regular ring which, as a graded algebra, is free commutative and finitely generated in degree $-1$ (we can also add some generators in degree $0$ if we feel like it).

Now it seems that for some applications, it is only important that the generators have odd degree (this ensures that the free algebra is cohomologically bounded). This leads to the following notion:

Generalization. Consider the following condition on $A$: $A$ is locally quasi-isomorphic to a dg-algebra $K$ over a regular algebra $B$ such that if we forget the differential, $K$ is a free commutative graded $B$-algebra that is finitely generated in negative odd degrees.

My (vague) question is what is known about this kind of condition (a couple of concrete questions are below). Are there any references?

Comments. Obviously, any lci algebra satisfies this condition. If I am not mistaken, any algebra satisfying this condition is Gorenstein (essentially for the same reason that lci is Gorenstein). But is this class of algebras closer to lci algebras or closer to Gorestein algebras? In fact, does it coincide with either? (I.e.: are there any non-lci examples? Or maybe all Gorenstein algebras have this kind of Koszul resolution?)

It also seems that this condition is equivalent to the condition that the cotangent complex of $A$ (at any point) sits in finitely many odd degrees and zero.

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This is not exactly what you are looking for, but if one considers connected dgas and graded complete intersections, there is a related but slightly more general notion of a pure Sullivan algebra. A paper that I found interesting on related issues is "Complete intersections in Rational Homotopy theory" by Hess et al. If I am remembering right they prove that being a complete intersection is somehow intrinsic to the dga but they only consider connected dgas. I understand you are interested in the non-connected case. – Daniel Pomerleano Jan 5 '11 at 18:53
@Daniel Pomerleano: Thanks a lot! While going through the paper you mentioned, I found a reference to Avramov's paper which seems to solve the question (the class coincides with lci, which is a weak variant of the Quillen Conjecture). – t3suji Jan 5 '11 at 19:34
Could one of you actually put that as an answer? (I think folks will agree that, in this case, it is perfectly acceptable to answer your own question.) – Alexander Woo Jan 5 '11 at 21:47