= Grassmannian of locally free quotients
{c}
{title2=$\operatorname{Gr}_r(M)$}
For a finite <free module> $M$ of rank $n$, this <scheme> represents rank-$r$ <locally free module> quotients of $M$ after <base change>, with quotient maps identified up to commuting <isomorphism>. It carries a universal <exact sequence> $0\to\mathcal S\to M\otimes\mathcal O\to\mathcal Q\to0$, of ranks $n-r,n,r$. Its <standard affine charts of the quotient Grassmannian> prove representability over any base <ring>. Over a field, taking kernels gives the usual <Grassmannian> of $(n-r)$-planes.
// Target: ringed-space.bigb
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