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Showing 1 to 20 of 24 for “"Koszul"”.
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Generalized Koszul complexes
We introduce Koszul type complexes associated to a linear map from any module to a free module, and vice versa to a linear map from a free module to an arbitrary module, generalizing both the classical Koszul complex and the family of Eagon-Northcott complexes. Given a short complex of finite free …
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Koszul duality for multigraded algebras
Classical Koszul duality sets up an adjoint pair of functors establishing an equivalence of categories. The equivalence is between the bounded derived category of complexes of graded modules over a graded algebra and the bounded derived category of complexes of graded modules over the quadratic …
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Graphs and Noncommutative Koszul Algebras
… class of directed graphs to noncommutative Koszul algebras. The directed graphs in this class are called full graphs and are defined by a set of criteria on the edges. The structural properties of full graphs are studied as they relate to the edge criteria. A method is introduced for …
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Combinatorial Koszul Homology: computations and Applications
… on explicit computations and applications of the Koszul homology and Betti numbers of monomial ideals, the main goals od this thesis are the following: Analyze the Koszul homology of monomial ideals and apply it to describe the structure of monomial ideals. Describe algorithms to perform efficient …
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Generalized Koszul Properties of Commutative Local Rings
… local rings that generalize the notion of Koszul algebra. The properties are expressed in terms of the Ext algebra of the ring, or in terms of homological properties of powers of the maximal ideal of the ring. We analyze relationships between these properties and we identify large classes …
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Koszul Homology and Resolutions over Commutative Rings
<p>The Koszul homology algebra of a commutative local (or graded) ring R tends to reflect important information about the ring R and its properties. We examine Koszul homology and the relationship between Koszul homology and minimal free resolutions in various settings. We provide tools which allow …
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Koszul duality and the bar spectral sequence
… for algebras over a spectral operad relates Koszul duality phenomena in several contexts. In this thesis, we apply this classical tool to the Koszul dual pair given by the (non-unital) E subscript ∞-operad and the spectral Lie operad over Fₚ. The bar spectral sequence for E subscript …
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2D Chiral Algebras from Koszul Duality & Associativity
… thesis, we will explain how the techniques of Koszul duality and associativity can be used to completely determine certain celestial chiral algebras, characterizing collinear splitting functions not just at tree-level but to arbitrary loop order. To apply this technique, and indeed for the …
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Tilting sheaves for real groups and Koszul duality
For a certain class of real analytic varieties with the real Lie group action we define a tstructure on the category of equivariant-monodromic sheaves and develop the theory of tilting sheaves. In case of a quasi-split real form of an algebraic group acting on the flag variety we construct an …
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Generic lines in projective space and the Koszul property
In this thesis, we study the Koszul property of the homogeneous coordinate ring of a generic collection of $m$ lines in $\mathbb{P}^n$ and the homogeneous coordinate ring of a collection of lines in general linear position in $\mathbb{P}^n.$ In particular, we show that if $\mathcal{M}$ is a …
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Koszul Complexes, Local Cohomology, Universal Resolutions, and Cohomological Support Varieties
We will explore three fields which utilize the Koszul complex: local cohomology, Golodity, and the cohomological support variety. First we explore the support of local cohomology modules, showing in particular the Zariski closure of certain local cohomology modules of ideals generated by regular …
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Betti numbers of Koszul algebras and codimension two matrix factorizations
… theorem in Chapter 3 for the defining ideals of Koszul almost complete intersections and, in the process, give an affirmative answer for all such rings to a question of Avramov, Conca, and Iyengar about the Betti numbers of Koszul algebras. In Chapter 4, we study the codimension two matrix …
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The Koszul dual to n-Lie, n-Com algebras, and Young tableaux
… to more than one Hamiltonian. We find the Koszul dual of $n\text{-}Lie_{-d+n-2}$ to be the operad $n\text{-}Com_d$, whose relations come from the Specht module $S^{(n,n-1)}$. We combine the operads $n$-Lie and $m$-Com to construct the operad $(m,n)-Poiss$, where the rewriting rule that …
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A Classification of some Quadratic Algebras
… of the ideal for the resulting algebra to be Koszul. Techniques include the use of Gröbner bases and development of Koszul resolutions. When the quadratic algebra is Koszul, we provide the associated linear resolution of the field. When not Koszul, we describe the maps of the resolution up to …
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The Golod Property on Monomial Rings
… "A non-Golod Ring with a Trivial Product on its Koszul Homology". In this counterexample, Katthan disproved a claim made by Berglund and Jollenbeck which stated that a monomial ring is Golod if and only if the product on its Koszul Homology is trivial. At the end, we find a non-Golod monomial …
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Bar constructions for topological operads and the Goodwillie derivatives of the identity
… the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and …
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Hochschild Cohomology of Short Gorenstein Rings
… we describe an explicit monomial basis for the Koszul dual of sGor(N) for N≥2.</p>
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Hochschild Cohomology Of Short Gorenstein Rings
… we describe an explicit monomial basis for the Koszul dual of sGor(N) for N≥2.</p>
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Finite Generation of Ext-Algebras for Monomial Algebras
… by studying other monomial algebras, including d-Koszul monomial algebras and a new class of monomial algebras which we term "left/right-symmetric" algebras.
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Properties of the Toric Rings of a Chordal Bipartite Family of Graphs
… the classification and study of a group of Koszul algebras coming from the toric ideals of a chordal bipartite infinite family of graphs (alternately, these rings may be interpreted as coming from determinants of certain ladder-like structures). We determine a linear system of parameters for …
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