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1. Equivariant resolutions over Veronese rings

   https://doi.org/10.1112/jlms.12848  

   Almousa, Ayah; Perlman, Michael; Pevzner, Alexandra; Reiner, Victor; VandeBogert, Keller (December 2023, Journal of the London Mathematical Society)

   Abstract Working in a polynomial ring , where is an arbitrary commutative ring with 1, we consider the th Veronese subalgebras , as well as natural ‐submodules inside . We develop and use characteristic‐free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple ‐equivariant minimal free ‐resolutions for the quotient ring and for these modules . These also lead to elegant descriptions of for all and for any pair of these modules . 

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2. 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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3. Combinatorial aspects of virtually Cohen-Macaulay sheaves

   https://doi.org/10.1112/tlm3.12036  

   Berkesch, Christine; Klein, Patricia; Loper, Michael C.; Yang, Jay (July 2022, Seminaire lotharingien de combinatoire)

   Bauer, Tomer; Liu, Gaku (Ed.)

   There is an abundance of deep literature on the use of free resolutions to study modules and vector bundle resolutions to study coherent sheaves. When studying a module over the Cox ring of a smooth projective toric variety X, each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman, and Smith introduced virtual resolutions, which are amenable to algebraic and combinatorial study and also capture desirable geometric information. In this extended abstract, we continue this program in the combinatorially-rich Stanley–Reisner setting. In particular, when X is a product of projective spaces, we produce a large new class of virtually Cohen–Macaulay Stanley–Reisner rings. After augmenting the simplicial complexes associated to these Stanley–Reisner rings with a coloring that reflects the product structure on X, our primary tool is Reisner’s criterion, whose conclusion we interpret in the virtual setting. We also provide two constructions of short virtual resolutions for use beyond the Stanley–Reisner case. 

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4. Free resolutions of orbit closures of Dynkin quivers

   https://doi.org/10.1090/tran/7697  

   Lorincz, Andras; Weyman, Jerzy (May 2019, Transactions of the American Mathematical Society)

   We use the Kempf-Lascoux-Weyman geometric technique in order to determine the minimal free resolutions of some orbit closures of quivers. As a consequence, we obtain that for Dynkin quivers orbit closures of 1-step representations are normal with rational singularities. For Dynkin quivers of type $$ \mathbb{A}$$, we describe explicit minimal generators of the defining ideals of orbit closures of 1-step representations. Using this, we provide an algorithm for type $$ \mathbb{A}$$ quivers for describing an efficient set of generators of the defining ideal of the orbit closure of any representation. 

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5. Burch Index, Summands of Syzygies and Linearity in Resolutions

   https://doi.org/10.1007/s41980-023-00755-x  

   Dao, Hailong; Eisenbud, David (February 2023, Bulletin of the Iranian Mathematical Society)

   Abstract The Burch index is a new invariant of a local ringRwhose positivity implies a kind of linearity in resolutions ofR-modules. We show that ifRhas depth zero and Burch index at least 2, then any non-free 7thR-syzygy contains the residue field as a direct summand. We compute the Burch index in various cases of interest. 

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