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We study the category of S p \mathbf {Sp} -equivariant modules over the infinite variable polynomial ring, where S p \mathbf {Sp} denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely generated module M M fits into an exact triangle T → M → F → T \to M \to F \to where T T is a finite length complex of torsion modules and F F is a finite length complex of “free” modules; we determine the Grothendieck group; and we (partially) determine the structure of injective modules. We apply these results to show that the twisted commutative algebras Sym ( C ∞ ⊕ ⋀ 2 C ∞ ) \operatorname {Sym}(\mathbf {C}^{\infty } \oplus \bigwedge ^2{\mathbf {C}^{\infty }}) and Sym ( C ∞ ⊕ Sym 2 C ∞ ) \operatorname {Sym}(\mathbf {C}^{\infty } \oplus \operatorname {Sym}^2{\mathbf {C}^{\infty }}) are noetherian, which are the strongest results to date of this kind. We also show that the free 2-step nilpotent twisted Lie algebra and Lie superalgebra are noetherian.
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Finite $F$-representation type for homogeneous coordinate rings of non-Fano varieties
Finite $$F$$-representation type is an important notion in characteristic-$$p$$commutative algebra, but explicit examples of varieties with or without thisproperty are few. We prove that a large class of homogeneous coordinate ringsin positive characteristic will fail to have finite $$F$$-representation type. Todo so, we prove a connection between differential operators on the homogeneouscoordinate ring of $$X$$ and the existence of global sections of a twist of$$(\mathrm{Sym}^m \Omega_X)^\vee$$. By results of Takagi and Takahashi, thisallows us to rule out FFRT for coordinate rings of varieties with$$(\mathrm{Sym}^m \Omega_X)^\vee$$ not ``positive''. By using results positivityand semistability conditions for the (co)tangent sheaves, we show that severalclasses of varieties fail to have finite $$F$$-representation type, includingabelian varieties, most Calabi--Yau varieties, and complete intersections ofgeneral type. Our work also provides examples of the structure of the ring ofdifferential operators for non-$$F$$-pure varieties, which to this point havelargely been unexplored.
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- Award ID(s):
- 1840190
- PAR ID:
- 10501633
- Publisher / Repository:
- episciences
- Date Published:
- Journal Name:
- Épijournal de Géométrie Algébrique
- Volume:
- Volume 7
- ISSN:
- 2491-6765
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.46298/epiga.2023.10868
