Higher derived complete intersections - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:27:53Z http://mathoverflow.net/feeds/question/51222 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51222/higher-derived-complete-intersections Higher derived complete intersections t3suji 2011-01-05T17:42:56Z 2011-01-05T17:42:56Z <p>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.</p> <p>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 $X=Spec(A)$.</p> <p><strong>Motivation.</strong> 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 $d(t_i)=f_i$.</p> <p><em>Reformulation.</em> $A$ 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). </p> <p>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:</p> <p><strong>Generalization.</strong> 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.</p> <p>My (vague) question is what is known about this kind of condition (a couple of concrete questions are below). Are there any references? </p> <p><em>Comments.</em> 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?)</p> <p>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. </p>