Generalized Quot-schemes - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T08:46:49Z http://mathoverflow.net/feeds/question/65253 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/65253/generalized-quot-schemes Generalized Quot-schemes TonyS 2011-05-17T16:23:00Z 2011-05-18T15:17:03Z <p>Given $S=\mathbb{P}^2$ and a locally free $O_S$-module $E$ of rank r and an integer $l\geq 1$. Then it is known that the scheme $Quot(E,l)$ is irreducible, due to Ellingsrud and Lehn. Here $Quot(E,l)$ parametrizes zero dimensional quotients $E\rightarrow T$ of length $l$.</p> <p>Are there any generalizations of this scheme?</p> <p>I'm thinking for example: Given a locally free sheaf $R$ of associative $O_S$-algebras, not necessarily commuative, of finite rank and a locally projective $R$-module $E$, which is locally free and of finite rank as an $O_S$-module.</p> <p>Is there a scheme $Quot_R(E,l)$ parametrizing zero dimensional $R$-quotients $E\rightarrow T$ of $R$-length $l$? Does such a scheme have similar properties, i.e. it is irreducible or connected?</p> <p>Edit: Thanks to t3suji and Sasha this question is solved. I remoed the rest of the question, so i can accept their answers, and i think the deformation problem deserves its own question anyway :-).</p> http://mathoverflow.net/questions/65253/generalized-quot-schemes/65268#65268 Answer by Sasha for Generalized Quot-schemes Sasha 2011-05-17T19:26:11Z 2011-05-17T19:26:11Z <p>Of course there is a scheme $Quot_R(E,l)$. Indeed, each $R$-module quotient $E \to T$ is an $O_S$-module quotient, so $Q_R(E,l)$ is a closed subscheme of $Quot(E,l)$ consisting of all surjections $E \to T$ of $O_S$-modules such that the kernel is invariant under the action of $R$.</p> http://mathoverflow.net/questions/65253/generalized-quot-schemes/65273#65273 Answer by t3suji for Generalized Quot-schemes t3suji 2011-05-17T21:01:40Z 2011-05-17T21:01:40Z <p>This can be thought of as a response to your comment to Sasha's answer.</p> <p>Until you try fixing length, there is no issue. Indeed, consider the scheme $$Quot(E)=\coprod Quot(E,k)$$ of all quotients of $E$. Then, as Sasha says, $Quot_R(E)$ is obviously a closed subscheme. </p> <p>You can now consider $Quot_R(E,l)$ as subsets of $Quot_R(E)$. It is not hard to see that you get a stratification of $Quot_R(E)$ in this way. This is the main difference between $Quot_R(E)$ and $Quot(E)$: we get a family of locally closed subsets of $Quot_R(E)$, while for $Quot(E)$, the subsets are both open and close. </p> <p>To me, it seems that now you run into a bit of trouble. Namely, $Quot_R(E,l)$, being a locally closed subset of $Quot_R(E)$, can be equipped with a scheme structure. However, it is not unique (because you do not know the nilpotents in the structure sheaf). You can trace the issue to the definition of $Quot_R(E,l)$: if you want it to parametrize $R$-modules of given length, you have to define what it means to have certain length, not just for $R$-modules, but for their families. The problem is non-existent for $O$-modules because length is invariant in the family (which by the way is only true if the ground field is algebraically closed!)</p> <p>On the other hand, questions like irreducibility or connectivity do not depend on the scheme structure, so maybe you don't need to worry about this.</p>