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"The pdf contains a draft title page, draft copyright page, and a draft manuscript"--
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This is the second of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains surveys on aspects of closure operations, finiteness conditions and factorization. Closure operations on ideals and modules are a bridge between noetherian and nonnoetherian commutative algebra. It contains a nice guide to closure operations by Epstein, but also contains an article on test ideals by Schwede and Tucker and one by Enescu which discusses the action of the Frobenius on finite dimensional vector spaces both of which are related to tight closure. Finiteness properties of rings and modules or the lack of them come up in all aspects of commutative algebra. However, in the study of non-noetherian rings it is much easier to find a ring having a finite number of prime ideals. The editors have included papers by Boynton and Sather-Wagstaff and by Watkins that discuss the relationship of rings with finite Krull dimension and their finite extensions. Finiteness properties in commutative group rings are discussed in Glaz and Schwarz's paper. And Olberding's selection presents us with constructions that produce rings whose integral closure in their field of fractions is not finitely generated. The final three papers in this volume investigate factorization in a broad sense. The first paper by Celikbas and Eubanks-Turner discusses the partially ordered set of prime ideals of the projective line over the integers. The editors have also included a paper on zero divisor graphs by Coykendall, Sather-Wagstaff, Sheppardson and Spiroff. The final paper, by Chapman and Krause, concerns non-unique factorization.
Algebra. --- Commutative algebra. --- Mathematics. --- Algebra --- Closure. --- Commutative Algebra. --- Decomposition. --- Factorization.
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Commutative algebra --- Algèbre commutative --- Algèbres commutatives
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This is the first of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains combinatorial and homological surveys. The combinatorial papers document some of the increasing focus in commutative algebra recently on the interaction between algebra and combinatorics. Specifically, one can use combinatorial techniques to investigate resolutions and other algebraic structures as with the papers of Fløystad on Boij-Söderburg theory, of Geramita, Harbourne and Migliore, and of Cooper on Hilbert functions, of Clark on minimal poset resolutions and of Mermin on simplicial resolutions. One can also utilize algebraic invariants to understand combinatorial structures like graphs, hypergraphs, and simplicial complexes such as in the paper of Morey and Villarreal on edge ideals. Homological techniques have become indispensable tools for the study of noetherian rings. These ideas have yielded amazing levels of interaction with other fields like algebraic topology (via differential graded techniques as well as the foundations of homological algebra), analysis (via the study of D-modules), and combinatorics (as described in the previous paragraph). The homological articles the editors have included in this volume relate mostly to how homological techniques help us better understand rings and singularities both noetherian and non-noetherian such as in the papers by Roberts, Yao, Hummel and Leuschke.
Algebra. --- Commutative algebra. --- Mathematics. --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Combinatirics. --- Commutative Algebra. --- Homology.
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Multiplicity (Mathematics) --- Chern classes. --- Commutative algebra. --- Classes de Chern --- Algèbre commutative --- Chern classes --- Commutative algebra
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This book includes 33 expanded abstracts of selected talks given at the two workshops "Homological Bonds Between Commutative Algebra and Representation Theory" and "Brave New Algebra: Opening Perspectives," and the conference "Opening Perspectives in Algebra, Representations, and Topology," held at the Centre de Recerca Matemàtica (CRM) in Barcelona between January and June 2015. These activities were part of the one-semester intensive research program "Interactions Between Representation Theory, Algebraic Topology and Commutative Algebra (IRTATCA)." Most of the abstracts present preliminary versions of not-yet published results and cover a large number of topics (including commutative and non commutative algebra, algebraic topology, singularity theory, triangulated categories, representation theory) overlapping with homological methods. This comprehensive book is a valuable resource for the community of researchers interested in homological algebra in a broad sense, and those curious to learn the latest developments in the area. It appeals to established researchers as well as PhD and postdoctoral students who want to learn more about the latest advances in these highly active fields of research.
Commutative algebra. --- Commutative rings. --- Commutative Rings and Algebras.
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Commutative algebra. --- Geometry, Algebraic. --- Geometry, Affine. --- Commutative rings. --- Modules (Algebra)
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