scholarly journals The Koszul Complex is the Cotangent Complex

2011 ◽  
Vol 2012 (3) ◽  
pp. 607-650 ◽  
Author(s):  
Joan Millès
Keyword(s):  
Koszul Complex ◽  
Cotangent Complex

Representations of the fundamental group and the torsor of deformations. Global aspects

2017 ◽  
Author(s):  
Ahmed Abbes ◽  
Michel Gros
Keyword(s):  
Fundamental Group ◽  
Koszul Complex ◽  
Finiteness Conditions ◽  
Inverse Systems ◽  
Logarithmic Geometry

This chapter continues the construction and study of the p-adic Simpson correspondence and presents the global aspects of the theory of representations of the fundamental group and the torsor of deformations. After fixing the notation and general conventions, the chapter develops preliminaries and then introduces the results and complements on the notion of locally irreducible schemes. It also fixes the logarithmic geometry setting of the constructions and considers a number of results on the Koszul complex. Finally, it develops the formalism of additive categories up to isogeny and describes the inverse systems of a Faltings ringed topos, with a particular focus on the notion of adic modules and the finiteness conditions adapted to this setting. The chapter rounds up the discussion with sections on Higgs–Tate algebras and Dolbeault modules.

scholarly journals A Koszul complex over skew polynomial rings

2019 ◽  
Vol 47 (7) ◽  
pp. 2904-2919
Author(s):  
Josep Àlvarez Montaner ◽  
Alberto F. Boix ◽  
Santiago Zarzuela
Keyword(s):  
Polynomial Rings ◽  
Koszul Complex ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

Graded Complexes Over Power Series Rings

1986 ◽  
Vol 38 (1) ◽  
pp. 158-178 ◽  
Author(s):  
Paul Roberts
Keyword(s):  
Power Series ◽  
Local Ring ◽  
Simple Structure ◽  
Free Resolution ◽  
Koszul Complex ◽  
Local Rings ◽  
Power Series Ring ◽  
Series Ring ◽  
Noetherian Local Ring ◽  
Power Series Rings

A common method in studying a commutative Noetherian local ring A is to find a regular subring R contained in A so that A becomes a finitely generated R-module, and in this way one can obtain some information about the original ring by applying what is known about regular local rings. By the structure theorems of Cohen, if A is complete and contains a field, there will always exist such a subring R, and R will be a power series ring k[[X1, …, Xn]] = k[[X]] over a field k. In this paper we show that if R is chosen properly, the ring A (or, more generally, an A-module M), will have a comparatively simple structure as an R-module. More precisely, A (or M) will have a free resolution which resembles the Koszul complex on the variables (X1, …, Xn) = (X); such a complex will be called an (X)-graded complex and will be given a precise definition below.

scholarly journals Golod–Shafarevich-Type Theorems and Potential Algebras

2018 ◽  
Vol 2019 (15) ◽  
pp. 4822-4844 ◽  
Author(s):  
Natalia Iyudu ◽  
Agata Smoktunowicz
Keyword(s):  
Linear Growth ◽  
Complete Classification ◽  
Koszul Complex ◽  
Model Program ◽  
Minimal Model Program ◽  
Infinite Dimensional ◽  
Homogeneous Potential ◽  
Finite Dimensional ◽  
The Difference

Abstract Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Gröbner basis theory and generalized Golod–Shafarevich-type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras. Using Gröbner bases techniques and arguing in terms of associated truncated algebra we prove that they cannot have dimension smaller than 8. This answers a question of Wemyss [21], related to the geometric argument of Toda [17]. We derive from the improved version of the Golod–Shafarevich theorem, that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove that potential algebra for any homogeneous potential of degree $n\geqslant 3$ is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class $\mathcal {P}_{n}$ of potential algebras with homogeneous potential of degree $n+1\geqslant 4$, the minimal Hilbert series is $H_{n}=\frac {1}{1-2t+2t^{n}-t^{n+1}}$, so they are all infinite dimensional. Moreover, growth could be polynomial (but nonlinear) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar–Vafa invariants.

scholarly journals DETECTING KOSZULNESS AND RELATED HOMOLOGICAL PROPERTIES FROM THE ALGEBRA STRUCTURE OF KOSZUL HOMOLOGY

2018 ◽  
Vol 238 ◽  
pp. 47-85 ◽  
Author(s):  
AMANDA CROLL ◽  
ROGER DELLACA ◽  
ANJAN GUPTA ◽  
JUSTIN HOFFMEIER ◽  
VIVEK MUKUNDAN ◽  
...  
Keyword(s):  
Poincaré Series ◽  
Koszul Complex ◽  
Local Rings ◽  
Common Denominator ◽  
Algebra Structure ◽  
Koszul Algebra ◽  
Multiplicative Structure ◽  
Poincare Series ◽  
Koszul Homology ◽  
Finitely Generated Modules

Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^{R}$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicative structure of $\operatorname{H}^{R}$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Poincaré series. As an application, we show that the Poincaré series of all finitely generated modules over a stretched Cohen–Macaulay local ring are rational, sharing a common denominator.

scholarly journals Ideals with sliding depth

1985 ◽  
Vol 99 ◽  
pp. 159-172 ◽  
Author(s):  
J. Herzog ◽  
W.V. Vasconcelos ◽  
R. Villarreal
Keyword(s):  
Koszul Complex ◽  
Complete Intersections

We study here a class of ideals of a Cohen-Macaulay ring {R, m} somewhat intermediate between complete intersections and general Cohen-Macaulay ideals. Its definition, while a bit technical, rapidly leads to the development of its elementary properties. Let I = (x1 … xn) = (x) be an ideal of R and denote by H*(x) the homology of the ordinary Koszul complex K*(x) built on the sequence x. It often occurs that the depth of the module Hi i > 0, increases with i (as usual, we set depth (0) = ∞).

scholarly journals A note on the acyclicity of the Koszul complex of a module

2007 ◽  
Vol 45 (2) ◽  
pp. 273-278 ◽  
Author(s):  
José M. Giral ◽  
Francesc Planas-Vilanova
Keyword(s):  
Koszul Complex
Sign in / Sign up

Export Citation Format

Share Document