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# Generalized Quot-schemesanddeformationsoftorsionsheaves

I'm interested in this, because of the following

Edit: Given an $R$-quotient $E\rightarrow T$ of $R$-length $l$. Now i'm trying to find a deformation of $R$-quotients over a connected base $B$ Thanks to another $R$-quotient $E\rightarrow Q$ of $R$-length $l$, s.t. $supp(Q)$ consists of $l$ distinct points {$p_1,...,p_l$} and $Q_{p_i}$ is a simple $R_{p_i}$-module.

For example assume that if $p\in S$ is a closed point, then the simple $R_p$-modules have length $1$ or $2$ over $O_{S,p}$. Now if $E\rightarrow T$ is an $R$-quotient of $R$-length $l$, then there are $a,b$ with $a+b=l$ and $T$ has length $k:=2a+b$ as an $O_S$-module. That is $E\rightarrow T$ is an element of $Quot(E,k)$. I'm interested in the set of all quotients $E\rightarrow Q$ in $Quot(E,k)$ such that $Q$ is also an $R$-module t3suji and has $R$-length $l$. So in fact it would be enough if Sasha this subscheme(?) of $Quot(E,k)$ question is connected/irreduciblesolved.

Is anything known about this? Is I remoed the approach via rest of the Quot-scheme perhaps not question, so good? Is there another way to get such a i can accept their answers, and i think the deformation of an $R$-quotient i'm looking for? I have no feeling if this even possible or are there some obvious reasons why this can't work at all?problem deserves its own question anyway :-).

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# Generalized Quot-schemes and deformations of torsion sheaves

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 $l$.

Are there any generalizations of this scheme?

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.

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?

I'm interested in this, because of the following: Given an $R$-quotient $E\rightarrow T$ of $R$-length $l$. Now i'm trying to find a deformation of $R$-quotients over a connected base $B$ to another $R$-quotient $E\rightarrow Q$ of $R$-length $l$, s.t. $supp(Q)$ consists of $l$ distinct points {$p_1,...,p_l$} and $Q_{p_i}$ is a simple $R_{p_i}$-module.

For example assume that if $p\in S$ is a closed point, then the simple $R_p$-modules have length $1$ or $2$ over $O_{S,p}$. Now if $E\rightarrow T$ is an $R$-quotient of $R$-length $l$, then there are $a,b$ with $a+b=l$ and $T$ has length $k:=2a+b$ as an $O_S$-module. That is $E\rightarrow T$ is an element of $Quot(E,k)$. I'm interested in the set of all quotients $E\rightarrow Q$ in $Quot(E,k)$ such that $Q$ is also an $R$-module and has $R$-length $l$. So in fact it would be enough if this subscheme(?) of $Quot(E,k)$ is connected/irreducible.

Is anything known about this? Is the approach via the Quot-scheme perhaps not so good? Is there another way to get such a deformation of an $R$-quotient i'm looking for? I have no feeling if this even possible or are there some obvious reasons why this can't work at all?