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1. Higher resonance schemes and Koszul modules of simplicial complexes

   https://doi.org/10.1007/s10801-024-01313-2  

   Aprodu, Marian; Farkas, Gavril; Raicu, Claudiu; Sammartano, Alessio; Suciu, Alexander_I (March 2024, Journal of Algebraic Combinatorics)

   Abstract Each connected graded, graded-commutative algebraAof finite type over a field$$\Bbbk $$ $k$of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the(higher) Koszul modulesofA. In this note, we investigate the geometry of the support loci of these modules, called theresonance schemesof the algebra. When$$A=\Bbbk \langle \Delta \rangle $$ $A=k⟨\Delta ⟩$is the exterior Stanley–Reisner algebra associated to a finite simplicial complex$$\Delta $$ $\Delta$, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group. 

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2. Essentially finite generation of valuation rings in terms of classical invariants

   https://doi.org/10.1002/mana.201900287  

   Cutkosky, Steven Dale; Novacoski, Josnei (October 2020, Mathematische Nachrichten)

   Abstract The main goal of this paper is to study some properties of an extension of valuations from classical invariants. More specifically, we consider a valued field and an extension ω of ν to a finite extensionLofK. Then we study when the valuation ring of ω is essentially finitely generated over the valuation ring of ν. We present a necessary condition in terms of classic invariants of the extension by Hagen Knaf and show that in some particular cases, this condition is also sufficient. We also study when the corresponding extension of graded algebras is finitely generated. For this problem we present an equivalent condition (which is weaker than the one for the finite generation of the valuation rings). 

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3. 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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4. Positivity and nonstandard graded Betti numbers

   https://doi.org/10.1112/blms.12917  

   Brown, Michael K; Erman, Daniel (January 2024, Bulletin of the London Mathematical Society)

   Abstract A foundational principle in the study of modules over standard graded polynomial rings is that geometric positivity conditions imply vanishing of Betti numbers. The main goal of this paper is to determine the extent to which this principle extends to the nonstandard ‐graded case. In this setting, the classical arguments break down, and the results become much more nuanced. We introduce a new notion of Castelnuovo–Mumford regularity and employ exterior algebra techniques to control the shapes of nonstandard ‐graded minimal free resolutions. Our main result reveals a unique feature in the nonstandard ‐graded case: the possible degrees of the syzygies of a graded module in this setting are controlled not only by its regularity, but also by its depth. As an application of our main result, we show that given a simplicial projective toric variety and a module over its coordinate ring, the multigraded Betti numbers of are contained in a particular polytope when satisfies an appropriate positivity condition. 

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5. A Type B Analogue of the Category of Finite Sets with Surjections

   https://doi.org/10.37236/11186  

   Proudfoot, Nicholas (July 2022, The Electronic Journal of Combinatorics)

   We define a type B analogue of the category of finite sets with surjections, and we study the representation theory of this category. We show that the opposite category is quasi-Gröbner, which implies that submodules of finitely generated modules are again finitely generated. We prove that the generating functions of finitely generated modules have certain prescribed poles, and we obtain restrictions on the representations of type B Coxeter groups that can appear in such modules. Our main example is a module that categorifies the degree i Kazhdan–Lusztig coefficients of type B Coxeter arrangements. 

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