In this paper, we present a comprehensive study of the categorical structures inherent in the theory of $ \mathcal{D} $-modules on a smooth algebraic variety $X$ over a field of characteristic zero. We established that the category $\mathbb{Q}\mathrm{Coh}(\mathcal{D}_{X})$ of quasicoherent $\mathcal{D}_{X}$-modules, while not Cartesian closed, naturally admits the structure of a closed monoidal category $(\mathbb{Q}\mathrm{Coh}(\mathcal{D}_{X}), \otimes_{\mathcal{O}_{X}}, \mathcal{O}_{X})$. The monoidal structure is given by the $\mathcal{O}_{X}$-tensor product, and the closure is exhibited by an internal Hom functor $\mathscr{H}om_{\mathcal{D}_{X}}(-, -)$ which is proven to be a quasicoherent $\mathcal{D}_{X}$-module. We then systematically lift this structure to the bounded derived category $\mathsf{D}^{b}(\mathbb{Q}\mathrm{Coh}(\mathcal{D}_{X}))$, introducing the derived tensor product $\otimes_{\mathcal{O}_{X}}^{\mathcal{L}}$ and the derived internal Hom $\mathbf{R}\mathcal{H}om_{\mathcal{D}_{X}}$. This foundational framework enables us to articulate and prove powerful duality theorems in this context. A central result is a new and detailed proof of the derived Grothendieck duality for proper morphisms of smooth varieties, formulated within the $\mathcal{D}$-module setting. Furthermore, we explicated the profound connection between this abstract categorical duality and the concrete, more familiar Verdier duality via the Riemann-Hilbert correspondence for regular holonomic $ \mathcal{D} $-modules. Our work clarifies the intricate interplay between the algebraic structure of $\mathcal{D}_{X}$, the homological algebra of its module category, and the topological nature of solutions to differential systems. Several applications in geometric representation theory are also discussed, highlighting the utility of this categorical perspective.
Citation: Jian-Gang Tang, Miao Liu, Huang-Rui Lei. The closed monoidal structure and derived dualities of the category of $ \mathcal{D} $-modules on a smooth variety[J]. AIMS Mathematics, 2026, 11(3): 7304-7329. doi: 10.3934/math.2026301
In this paper, we present a comprehensive study of the categorical structures inherent in the theory of $ \mathcal{D} $-modules on a smooth algebraic variety $X$ over a field of characteristic zero. We established that the category $\mathbb{Q}\mathrm{Coh}(\mathcal{D}_{X})$ of quasicoherent $\mathcal{D}_{X}$-modules, while not Cartesian closed, naturally admits the structure of a closed monoidal category $(\mathbb{Q}\mathrm{Coh}(\mathcal{D}_{X}), \otimes_{\mathcal{O}_{X}}, \mathcal{O}_{X})$. The monoidal structure is given by the $\mathcal{O}_{X}$-tensor product, and the closure is exhibited by an internal Hom functor $\mathscr{H}om_{\mathcal{D}_{X}}(-, -)$ which is proven to be a quasicoherent $\mathcal{D}_{X}$-module. We then systematically lift this structure to the bounded derived category $\mathsf{D}^{b}(\mathbb{Q}\mathrm{Coh}(\mathcal{D}_{X}))$, introducing the derived tensor product $\otimes_{\mathcal{O}_{X}}^{\mathcal{L}}$ and the derived internal Hom $\mathbf{R}\mathcal{H}om_{\mathcal{D}_{X}}$. This foundational framework enables us to articulate and prove powerful duality theorems in this context. A central result is a new and detailed proof of the derived Grothendieck duality for proper morphisms of smooth varieties, formulated within the $\mathcal{D}$-module setting. Furthermore, we explicated the profound connection between this abstract categorical duality and the concrete, more familiar Verdier duality via the Riemann-Hilbert correspondence for regular holonomic $ \mathcal{D} $-modules. Our work clarifies the intricate interplay between the algebraic structure of $\mathcal{D}_{X}$, the homological algebra of its module category, and the topological nature of solutions to differential systems. Several applications in geometric representation theory are also discussed, highlighting the utility of this categorical perspective.
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Jian-Gang Tang, Miao Liu, Huang-Rui Lei. The closed monoidal structure and derived dualities of the category of $ \mathcal{D} $-modules on a smooth variety[J]. AIMS Mathematics, 2026, 11(3): 7304-7329. doi: 10.3934/math.2026301
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