阿贝尔概形的有限扩张与交换群层的对偶性
Finite extensions of abelian schemes and duality for commutative group stacks
- Western University（韦仕敦大学）
- McMaster University（麦克马斯特大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究在离散赋值环及任意基上，证明交换群概形与群层的结构定理，推导其对偶的性质，应用Fourier-Mukai论证得到导出范畴的等价，证明了Brochard对偶猜想。
AI中文摘要：
设R为离散赋值环。我们证明，每个真平坦有限表现的交换R-群概形P都可嵌入正合序列0→E→P→B→0，其中E是有限平坦的，B是阿贝尔概形。对于拟阿贝尔模型——即具有阿贝尔一般纤维的真平坦有限表现R-群概形——我们证明了一个更精细的结构定理：其正规化是阿贝尔概形，且该模型通过涉及有限平坦群概形的推出构造得到。对于具有有限平坦惯性的真平坦交换群层，存在类似的阿贝尔商。这些结构定理给出了对偶的显式商表示。在任意基上，我们证明真平坦有限表现的交换群代数空间的对偶是代数的、真的、平坦的且有限表现的，且双对偶成立。当2可逆时，群层的对应结果证明了Brochard的对偶猜想。作为离散赋值环上的应用，Polishchuk的核-代数Fourier-Mukai论证给出了每个真平坦有限表现的交换群概形与其对偶的拟凝聚层无界导出范畴之间的等价，以及它们的凝聚层有界导出范畴之间的等价，且无需驯顺性或剩余特征假设。对于拟阿贝尔模型，对偶范畴是其阿贝尔正规化上的有限平坦等变范畴。
英文摘要：
Let $R$ be a discrete valuation ring. We prove that every proper flat finitely presented commutative $R$-group scheme $P$ fits into an exact sequence \[ 0\longrightarrow E\longrightarrow P\longrightarrow B\longrightarrow0, \] where $E$ is finite flat and $B$ is an abelian scheme. For a quasiabelian model---that is, a proper flat finitely presented $R$-group scheme with abelian generic fibre---we prove a finer structure theorem: its normalization is an abelian scheme, and the model is obtained from it by a pushout involving finite flat group schemes. An analogous abelian quotient exists for proper flat commutative group stacks with finite flat inertia. These structure theorems give explicit quotient presentations for duals. Over an arbitrary base, we show that the dual of a proper flat finitely presented commutative group algebraic space is algebraic, proper, flat, and finitely presented, and biduality holds. When $2$ is invertible, the corresponding result for group stacks proves Brochard's duality conjecture. As an application over a discrete valuation ring, Polishchuk's kernel-algebra Fourier--Mukai argument gives equivalences between the unbounded derived categories of quasi-coherent sheaves on every proper flat finitely presented commutative group scheme and its dual, and between their bounded derived categories of coherent sheaves, with no tameness or residue-characteristic hypothesis. For a quasiabelian model, the dual category is a finite-flat equivariant category on its abelian normalization.
