More Like this

1. Permanence properties of F-injectivity

   https://doi.org/10.4310/MRL.241118233550  

   Datta, Rankeya; Murayama, Takumi (November 2024, Mathematical Research Letters)

   We prove that F-injectivity localizes, descends under faithfully flat homomorphisms, and ascends under flat homomorphisms with Cohen–Macaulay and geometrically F-injective fibers, all for arbitrary Noetherian rings of prime characteristic. As a consequence, we show that the F-injective locus is open on most rings arising in arithmetic and geometry. As a geometric application, we prove that over an algebraically closed field of characteristic p > 3, generic projection hypersurfaces associated to suitably embedded smooth projective varieties of dimension ≤5 are F-pure, and hence F-injective. This geometric result is the positive characteristic analogue of a theorem of Doherty. 

   more » « less
2. The K-theory of perfectoid rings

   https://doi.org/10.4171/DM/X22  

   Antieau, Ben; Mathew, Akhil; Morrow, Matthew (January 2022, Documenta Mathematica)

   We establish various properties of thep-adic algebraic\text{K}-theory of smooth algebras over perfectoid rings living over perfectoid valuation rings. In particular, thep-adic\text{K}-theory of such rings is homotopy invariant, and coincides with thep-adic\text{K}-theory of thep-adic generic fibre in high degrees. In the case of smooth algebras over perfectoid valuation rings of mixed characteristic the latter isomorphism holds in all degrees and generalises a result of Nizioł. 

   more » « less
3. SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE

   https://doi.org/10.1017/fmp.2020.1  

   LE, DANIEL; LE HUNG, BAO V.; LEVIN, BRANDON; MORRA, STEFANO (January 2020, Forum of Mathematics, Pi)

   We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $$p$$ . This is a generalization to $$\text{GL}_{3}$$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $$n$$ -dimensional Galois representations’, Duke Math. J.   149 (1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math.   212 (1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $$\text{GL}_{3}(\mathbb{F}_{q})$$ . 

   more » « less
4. Lefschetz properties through a topological lens

   https://doi.org/10.2478/aupcsm-2025-0001  

   Seceleanu, Alexandra (June 2025, Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica)

   Abstract These notes were prepared for theLefschetz Preparatory School, a graduate summer course held in Krakow, May 6–10, 2024. They present the story of the algebraic Lefschetz properties from their origin in algebraic geometry to some recent developments in commutative algebra. The common thread of the notes is a bias towards topics surrounding the algebraic Lefschetz properties that have a topological flavor. These range from the Hard Lefschetz Theorem for cohomology rings to commutative algebraic analogues of these rings, namely artinian Gorenstein rings, and topologically motivated operations among such rings. 

   more » « less
5. Search problems in algebraic complexity, GCT, and hardness of generator for invariant rings

   Garg, A; Ikenmeyer, C; Makam, V; Oliveira, R; Walter, M; Wigderson, A (January 2020, Proceedings of the Computational Complexity Conference (CCC) 2020)

   We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of SLn(C)-representations. We provide simple examples that disprove this conjecture (under standard complexity assumptions). We develop a general framework, denoted algebraic circuit search problems, that captures many important problems in algebraic complexity and computational invariant theory. This framework encompasses various proof systems in proof complexity and some of the central problems in invariant theory as exposed by the Geometric Complexity Theory (GCT) program, including the aforementioned problem of computing succinct encodings for generators for invariant rings. 

   more » « less