Recent Developments in Commutative Algebra: Levico Terme, Trento 2019 (C.I.M.E. Foundation Subseries)

Preface
	Participants
Contents
List of Contributors
1 Defining Equations of Blowup Algebras
	1.1 Introduction
	1.2 Linear Type
	1.3 Perfect Ideals of Height Two
		1.3.1 The Jacobian Dual Method
		1.3.2 Graphs and Images of Rational Maps
		1.3.3 The Method of Divisors
		1.3.4 Jouanalou Duality and Rational Plane Curves
	1.4 Bounding Bidegrees of Defining Equations
		1.4.1 Linearly Presented Height Three Gorenstein Ideals with Gd
	References
2 Koszul Modules
	2.1 Syzygies of the Tangential Variety to a Rational Normal Curve
		2.1.1 The Local Description of g, Tg
		2.1.2 Characteristic Zero Interpretation of g, Tg
		2.1.3 The Betti Table of Tg in Characteristic 2
		2.1.4 The Betti Numbers of Tg for char(k)≠2
	2.2 Koszul Modules
	2.3 Weyman Modules, Hermite Reciprocity, and the Syzygies of Tg
		2.3.1 Weyman Modules
		2.3.2 Hermite Reciprocity
		2.3.3 The Coordinate Independent Description of g
		2.3.4 The Key Insight for the Proof of Theorem 2.4
	2.4 Syzygies of Curves
	Appendix
		Symmetric vs Divided Powers
		The Eagon–Northcott and Koszul Complexes
		The Buchsbaum–Rim Complex
	References
3 Gröbner Degenerations
	3.1 Gröbner Bases for Beginners
		3.1.1 Notation and Basic Definitions
		3.1.2 Gröbner Bases
		3.1.3 The Buchberger Algorithm
	3.2 Initial Ideals with Respect to Weights
		3.2.1 Weights and Monomial Orders
		3.2.2 Degeneration to the Initial Ideal
			3.2.2.1 Semicontinuity
	3.3 Fibre-Full Modules
	3.4 Squarefree Monomial Ideals and Fibre-Full Modules
	3.5 Some Discussions and Related Open Questions
		3.5.1 Cohomologically Full Rings Defined by Monomial Ideals
		3.5.2 Sagbi Bases
		3.5.3 The Category of Fibre-Full Modules
	References
4 Adams Operations in Commutative Algebra
	4.1 Classical Lambda and Adams Operations on K0
		4.1.1 Definition of K0
		4.1.2 Lambda Operations
		4.1.3 Adams Operations
		4.1.4 A Theorem of Grothendieck
	4.2 Algebraic K-Theory with Supports
		4.2.1 Grothendieck Group with Supports
		4.2.2 Complexes with Finite Length Homology
		4.2.3 Cup Product
		4.2.4 Intersection Multiplicity
		4.2.5 Functorality
		4.2.6 A Right Exact Sequence
		4.2.7 Adams Operations
		4.2.8 Frobenius
	4.3 Existence of Adams Operations, Weight Decompositions
		4.3.1 The Adams Operations of Gillet-Soulé
		4.3.2 Cyclic Adams Operations
		4.3.3 The Weight Decomposition
	4.4 Applications to Some Conjectures in Commutative Algebra
		4.4.1 Serre Vanishing Conjecture
		4.4.2 The Total Rank Conjecture
		4.4.3 Matrix Factorizations
		4.4.4 A Generalization of the Total Rank Conjecture
	References