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Register a new userThe structure of the minimal free resolution of semigroup rings obtained by gluing
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We construct a minimal free resolution of the semigroup ring k[C] in terms of minimal resolutions of k[A] and k[B] when <C> is a numerical semigroup obtained by gluing two numerical semigroups <A> and <B>. Using our explicit construction, we compute the Betti numbers, graded Betti numbers, regularity and Hilbert series of k[C], and prove that the minimal free resolution of k[C] has a differential graded algebra structure provided the resolutions of k[A] and k[B] possess them. We discuss the consequences of our results in small embedding dimensions. Finally, we give an extension of our main result to semigroups in N^n
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In a paper in 1962, Golod proved that the Betti sequence of the residue field of a local ring attains an upper bound given by Serre if and only if the homology algebra of the Koszul complex of the ring has trivial multiplications and trivial Massey operations. This is the origin of the notion of Golod ring. Using the Koszul complex components he also constructed a minimal free resolution of the residue field. In this article, we extend this construction up to degree five for any local ring. We describe how the multiplicative structure and the triple Massey products of the homology of the Koszul algebra are involved in this construction. As a consequence, we provide explicit formulas for the first six terms of a sequence that measures how far the ring is from being Golod.
Numerical invariants of a minimal free resolution of a module $M$ over a regular local ring $(R, )$ can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable $ $-stable filtrations ${mathbb M} $ of $M $ and to compare the Betti numbers of $M$ with those of the associated graded module $ gr_{mathbb M}(M). $ This approach has the advantage that the same module $M$ can be detected by using different filtrations on it. It provides interesting upper bounds for the Betti numbers and we study the modules for which the extremal values are attained. Among others, the Koszul modules have this behavior. As a consequence of the main result, we extend some results by Aramova, Conca, Herzog and Hibi on the rigidity of the resolution of standard graded algebras to the local setting.
We consider simplicial complexes admitting a free action by an abelian group. Specifically, we establish a refinement of the classic result of Hochster describing the local cohomology modules of the associated Stanley--Reisner ring, demonstrating that the topological structure of the free action extends to the algebraic setting. If the complex in question is also Buchsbaum, this new description allows for a specialization of Schenzels calculation of the Hilbert series of some of the rings Artinian reductions. In further application, we generalize to the Buchsbaum case the results of Stanley and Adin that provide a lower bound on the $h$-vector of a Cohen-Macaulay complex admitting a free action by a cyclic group of prime order.
We show how the minimal free resolution of a set of $n$ points in general position in projective space of dimension $n-2$ explicitly determines structure constants for a ring of rank $n$. This generalises previously known constructions of Levi-Delone-Faddeev and Bhargava in the cases $n=3,4,5$.
Let $ R $ be a $ d $-dimensional Cohen-Macaulay (CM) local ring of minimal multiplicity. Set $ S := R/({bf f}) $, where $ {bf f} := f_1,ldots,f_c $ is an $ R $-regular sequence. Suppose $ M $ and $ N $ are maximal CM $ S $-modules. It is shown that if $ mathrm{Ext}_S^i(M,N) = 0 $ for some $ (d+c+1) $ consecutive values of $ i geqslant 2 $, then $ mathrm{Ext}_S^i(M,N) = 0 $ for all $ i geqslant 1 $. Moreover, if this holds true, then either $ mathrm{projdim}_R(M) $ or $ mathrm{injdim}_R(N) $ is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.
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