Closed Model Category Structure on Chain Complexes Related to A Left-exact Functor - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T04:53:25Z http://mathoverflow.net/feeds/question/69058 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69058/closed-model-category-structure-on-chain-complexes-related-to-a-left-exact-functo Closed Model Category Structure on Chain Complexes Related to A Left-exact Functor Mikhail Gudim 2011-06-28T22:02:48Z 2011-06-29T03:11:07Z <p>Let $F:A \to B$ be an additive left-exact functor of abelian categories (Do not assume that they have enough injectives / projectives.) Suppose we are given a class of objects $R$ adapted to $F$ (see below for definition). Is there a closed model category structure on $Com^+(A)$ such that weak equivalences are quasi-isomorphisms and $R$ is (or contains) the class of cofibrant objects?</p> <p>Here is a definition of class of objects adapted to a left exact functor $F: A \to B$ (from Gelfand and Manin): it is a class of objects $R$ stable under direct sums, $F$ maps acyclic complexes from $Com^+(R)$ to acyclic ones and each object is in $A$ is a subobject of an object in $R$</p> <p>Notes: </p> <p>(1) This question originated from this one: <a href="http://mathoverflow.net/search?q=adapted+class+of+objects+model+categories" rel="nofollow">http://mathoverflow.net/search?q=adapted+class+of+objects+model+categories</a>. </p> <p>(2) This question does not really have anything to do with algebraic geometry, but I thought that algebraic geometers may have something to say about this.</p>