Finite Compression of Derived Quot Presentations via Homogeneity and Rank Propagation
Volume 4 (2025) Issue 2,
DOI:
https://doi.org/ 10.71448/jcm2025v4i22
Submission Received: 16 Jul 2025
Revised: 30 Sep 2025
Accepted: 17 Oct 2025
Published: 30 Dec 2025
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Abstract
The Derived Quot schemes retain the obstructive information which is lost in a classical Quot scheme. Their finite stretch construction is typically written using products of Grassmannians, multilinear Hom-spaces, Maurer--Cartan equation, and quotient by products of general linear groups. In this paper, we give a finite criterion under which the same bounded derived Quot data is captured by a shorter Grassmannian stretch taking into account at once both the homogeneity span of each operation and the rank propagation of admissible submodules. This criterion, called rank-sealed span audit, is based on filtering the differential graded Lie algebra whose elements are
\[
\Hom_0(R_+^{\otimes k}\otimes N_{\geq a},N_s)\oplus
\Hom_0(R_+^{\otimes(k-1)}\otimes N_{\geq a},M_s),
\]
by the difference of homogeneity degrees of input and output and detecting its acyclic coordinate pairs by means of the associated spectral sequence; after that one removes the Koszul dual coordinates that do not contribute to the tangent cohomology up to degree k. The resulting affine dg Lie algebra is then further reduced by imposing on it the rank condition ker(Ns→Ms)=0 and the generation condition \coker(Rs−s′⊗Ns′→Ns)=0 that allows forming a quotient over the shorter base ∏s=ab\Gr(h(s),Ms). We prove that the reduced presentation has the same classical admissible points and the same tangent cohomology up to degree k as the corresponding finite stretch derived Quot model; the finite-generation argument gives a weakly equivalent dg-manifold of finite type. The construction provides an algebraic record that is finite, a concrete audit process, and a theorem-based criterion that determines whether a bounded derived Quot presentation is already known through a brief period of homogeneity.
Keywords
derived Quot schemes,dg manifolds,Maurer--Cartan equations,Grassmannian quotients,obstruction theory,invariant theory,cotangent complexes