User fls - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T09:47:05Z http://mathoverflow.net/feeds/user/2286 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7688/the-category-of-finite-locally-free-commutative-group-schemes The category of finite locally-free commutative group schemes fls 2009-12-03T17:27:33Z 2010-01-07T18:59:45Z <p>I'm trying to understand the properties of the category $FL/S$ of finite locally-free commutative group schemes over an arbitrary base-scheme $S$. I know it is not in general an abelian category: Over the integers the morphism $\mathbb{Z}/2\mathbb{Z}\to\mu_2$ given by the ring homomorphism $\mathbb{Z}/(T^2-1)\to\mathbb{Z}\times\mathbb{Z}$ with $T\mapsto (1,-1)$ is a monomorphism and an epimorphism in $FL/\mathbb{Z}$ but it is clearly not an isomorphism.</p> <p>What I don't know: Does every morphism in $FL/S$ have a kernel and/or a cokernel?</p> <p>One has a notion of short exact sequences in $FL/S$: If $f:G'\to G$ and $g:G\to G''$ are morphisms in $FL/S$ we say that the sequence $0\to G'\to G\to G''\to 0$ is exact if its image under the embedding of $FL/S$ into the category of abelian fppf-sheaves is exact. How can we detect in $FL/S$ whether a sequence is exact? Specifically: If $f$ is a kernel of $g$ and if $g$ is a cokernel of $f$ (both in $FL/S$), is the corresponding sequence exact?</p> <p>I'd appreciate (references for) answers to any of these questions.</p> http://mathoverflow.net/questions/7688/the-category-of-finite-locally-free-commutative-group-schemes Comment by fls fls 2009-12-03T19:34:19Z 2009-12-03T19:34:19Z We consider commutative group schemes G over S such that the structural morphism f:G-&gt;S is finite locally free, i.e. it is affine and the direct image f_* O_G is a finitely generated locally free O_S module.