A Uniform Identification of Stable Sheaf Cohomology

TL;DR

This work addresses the problem of uniformly identifying stable sheaf cohomology for Schur functors applied to the cotangent sheaf on projective spaces. It introduces arithmetic complexes $C_\bullet(\underline{w})$ over $R=\mathbb{Z}\langle \binom{x}{n}\rangle$ and proves the conjectured isomorphism $C_\bullet(x,1^d)\cong C_\bullet(-x-2d,1^d)$, yielding a uniform stability description for ribbon and two-column Schur functors and extending results to $\mathbb{P}^n$ defined over $\mathbb{Z}$. The authors develop explicit combinatorial formulas for the chain maps and show invertibility via upper-triangular matrices with diagonal entries $\pm1$, enabling a robust integral version of stability analogous to Raicu–Keller. By relating the stable cohomology $H_{st}^i(\mathbb{S}_{\lambda/\mu}\Omega)$ to the homology of $C_\bullet(\underline{w})$ (with shifts) and employing hypercohomology, the paper provides a comprehensive integral framework that unifies and extends prior characteristic-zero and modular insights, with clear directions for broader Schur functors and arithmetic settings.

Abstract

This paper considers generalizations of certain arithmetic complexes appearing in the work of Raicu and VandeBogert in connection with the study of stable sheaf cohomology on flag varieties. Defined over the ring of integer valued polynomials, we prove an isomorphism of these complexes as conjectured by Gao, Raicu, and VandeBogert. In particular, this shows that a previously made identification between the stable sheaf cohomology of hook and two column partition Schur functors applied to the cotangent sheaf of projective space can be made to be uniform with respect to these complexes. These results are extended to the projective space defined over the integers.

Figures (2)

• Figure 1: Undirected Weighted Path Graph with Vertex Weights $\underline{w}$
• Figure 2: A decomposition for $d=6$, $t=4$, and $(\lambda_4,\lambda_3,\lambda_2,\lambda_1)=(0,2,0,1)$.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Remark 2.2
• Example 2.3
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• proof : Proof of \ref{['main']}