Minimal models in algebra, combinatorics and topology

Detta är en avhandling från Stockholm : Matematiska institutionen

Sammanfattning: The thesis consists of seven papers.In Paper I, II, III, IV and V, we study homological invariants of monomial rings — rings of the form R = k[x1, . . . , xn]/I where I is a monomial ideal in the polynomial ring k[x1, . . . , xn] over a field k — and we study combinatorial aspects of these invariants. Specifically, we study the Poincaré series PR(z), the homotopy Lie algebra of the Koszul complex ?'(KR), and the following question: When is R a Golod ring? We find a formula for the Poincaré series in terms of homology groups of lower intervals in a finite lattice KI , and we relate the Golod property of R with the Cohen-Macaulay property of KI . A description of the homotopy Lie algebra ?'(KR) in terms of the cohomology of a certain combinatorially defined ?-algebra is given, and it is used to prove that R is Golod if and only if the homology algebra H'(KR) has trivial multiplication. We prove that a certain combinatorial criterion, called the strong gcd-condition, implies Golodness of R and we relate this criterion to (non-pure) shellability of simplicial complexes.In Paper VI, we lay the foundations of a cohomology theory for associative algebras, called Cofinite Hochschild cohomology, which may be seen as a continuous version of Hochschild cohomology. We prove that under reasonable hypotheses, the natural map from cofinite cohomology to Hochschild cohomology is an isomorphism for commutative noetherian algebras.In Paper VII, we extend homological perturbation theory of chain complexes to encompass (co)algebra structures over (co)operads. This is done by ‘thickening’ the category of (co)algebras over a (co)operad, and the functorial properties of the thick categories are carefully worked out. As an application, this theory provides means of proving transfer theorems for algebras over a large class of operads.

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