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1. Lim Ulrich sequences and Boij-Söderberg cones

   https://doi.org/10.1017/fms.2023.108  

   Iyengar, Srikanth B; Ma, Linquan; Walker, Mark E (January 2023, Forum of Mathematics, Sigma)

   Abstract This paper extends the results of Boij, Eisenbud, Erman, Schreyer and Söderberg on the structure of Betti cones of finitely generated graded modules and finite free complexes over polynomial rings, to all finitely generated graded rings admitting linear Noether normalizations. The key new input is the existence of lim Ulrich sequences of graded modules over such rings. 

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2. A cocharge formula for the ∆-Springer modules

   Gillespie, Maria; Griffin, Sean T (April 2023, Séminaire Lotharingien de Combinatoire)

   We conjecture a simple combinatorial formula for the Schur expansion of the Frobenius series of the Sn-modules Rn,λ,s, which appear as the cohomology rings of the “∆-Springer” varieties. These modules interpolate between the Garsia-Procesi modules Rµ (which are the type A Springer fiber cohomology rings) and the rings Rn,k defined by Haglund, Rhoades, and Shimozono in the context of the Delta Conjecture. Our formula directly generalizes the known cocharge formula for Garsia-Procesi modules and gives a new cocharge formula for the Delta Conjecture at t = 0, by introducing battery-powered tableaux that “store” extra charge in their battery. Our conjecture has been verified by computer for all n ≤ 10 and s ≤ ℓ(λ)+2, as well as for n ≤ 8 and s ≤ ℓ(λ)+7. We prove it holds for several infinite families of n,λ,s. 

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3. Relative injective modules, superstability and noetherian categories

   https://doi.org/10.1142/S0219061324500272  

   Mazari-Armida, Marcos; Rosický, Jiří (April 2026, Journal of Mathematical Logic)

   We study classes of modules closed under direct sums, [Formula: see text]-submodules and [Formula: see text]-epimorphic images where [Formula: see text] is either the class of embeddings, RD-embeddings or pure embeddings. We show that the [Formula: see text]-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and [Formula: see text]-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah’s non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability. 

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4. Constructing nonproxy small test modules for the complete intersection property

   https://doi.org/10.1017/nmj.2021.7  

   Briggs, Benjamin; Grifo, Eloísa; Pollitz, Josh (June 2021, Nagoya Mathematical Journal)

   Abstract A local ring R is regular if and only if every finitely generated R -module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category $$\mathsf {D}^{\mathsf f}(R)$$ , which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer–Greenlees–Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in $$\mathsf {D}^{\mathsf f}(R)$$ is proxy small. In this paper, we study a return to the world of R -modules, and search for finitely generated R -modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley–Reisner rings. 

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5. Non-existence of Ulrich modules over Cohen-Macaulay local rings

   https://doi.org/10.1090/cams/44  

   Iyengar, Srikanth; Ma, Linquan; Walker, Mark; Zhuang, Ziquan (January 2025, Communications of the American Mathematical Society)

   Over a Cohen-Macaulay local ring, the minimal number of generators of a maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984 Ulrich [Math. Z. 188 (1984), pp. 23–32] asked whether there always exist modules for which equality holds; such modules are known nowadays as Ulrich modules. We answer this question in the negative by constructing families of two dimensional Cohen-Macaulay local rings that have no Ulrich modules. Some of these examples are Gorenstein normal domains; others are even complete intersection domains, though not normal. 

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