scholarly journals Quadratic Gorenstein Rings and the Koszul Property II

2021 ◽  
Author(s):  
Matthew Mastroeni ◽  
Hal Schenck ◽  
Mike Stillman
Keyword(s):  
Gorenstein Ring ◽  
Gorenstein Rings ◽  
Koszul Property ◽  
Answering Questions

Abstract Conca–Rossi–Valla [6] ask if every quadratic Gorenstein ring $R$ of regularity three is Koszul. In [15], we use idealization to answer their question, proving that in nine or more variables there exist quadratic Gorenstein rings of regularity three, which are not Koszul. In this paper, we study the analog of the Conca–Rossi–Valla question when the regularity of $R$ is four or more. Let $R$ be a quadratic Gorenstein ring having ${\operatorname {codim}} \ R = c$ and ${\operatorname {reg}} \ R = r \ge 4$. We prove that if $c = r+1$ then $R$ is always Koszul, and for every $c \geq r+2$, we construct quadratic Gorenstein rings that are not Koszul, answering questions of Matsuda [16] and Migliore–Nagel [19].

A Gorenstein Ring with Larger Dilworth Number than Sperner Number

2000 ◽  
Vol 43 (1) ◽  
pp. 100-104 ◽  
Author(s):  
James S. Okon ◽  
J. Paul Vicknair
Keyword(s):  
Local Ring ◽  
Gorenstein Ring ◽  
Embedding Dimension ◽  
Gorenstein Rings

AbstractA counterexample is given to a conjecture of Ikeda by finding a class of Gorenstein rings of embedding dimension 3 with larger Dilworth number than Sperner number. The Dilworth number of is computed when A is an unramified principal Artin local ring.

scholarly journals Almost Gorenstein rings arising from fiber products

2020 ◽  
pp. 1-18
Author(s):  
Naoki Endo ◽  
Shiro Goto ◽  
Ryotaro Isobe
Keyword(s):  
Local Ring ◽  
The Other ◽  
Gorenstein Ring ◽  
Local Rings ◽  
Regular Local Ring ◽  
Gorenstein Rings ◽  
Fiber Product

Abstract The purpose of this paper is, as part of the stratification of Cohen–Macaulay rings, to investigate the question of when the fiber products are almost Gorenstein rings. We show that the fiber product $R \times _T S$ of Cohen–Macaulay local rings R, S of the same dimension $d>0$ over a regular local ring T with $\dim T=d-1$ is an almost Gorenstein ring if and only if so are R and S. In addition, the other generalizations of Gorenstein properties are also explored.

scholarly journals A Note on Gorenstein Rings of Embedding Codimension Three

1973 ◽  
Vol 50 ◽  
pp. 227-232 ◽  
Author(s):  
Junzo Watanabe
Keyword(s):  
Local Ring ◽  
Arbitrary Dimension ◽  
Three Dimensional ◽  
Regular Sequence ◽  
Complete Intersections ◽  
Gorenstein Ring ◽  
Regular Local Ring ◽  
Gorenstein Rings ◽  
Number Of Generators ◽  
Five Elements

Let A = R/, where R is a regular local ring of arbitrary dimension and is an ideal of R. If A is a Gorenstein ring and if height = 2, it is easily proved that A is a complete intersection, i.e., is generated by two elements (Serre [5], Proposition 3). Hence Gorenstein rings which are not complete intersections are of embedding codimension at least three. An example of these rings is found in Bass’ paper [1] (p. 29). This is obtained as a quotient of a three dimensional regular local ring by an ideal which is generated by five elements, i.e., generated by a regular sequence plus two more elements. In this paper, suggested by this example, we prove that if A is a Gorenstein ring and if height = 3, then is minimally generated by an odd number of elements. If A has a greater codimension, presumably there is no such restriction on the minimal number of generators for , as will be conceived from the proof.

scholarly journals Acyclic Complexes and Gorenstein Rings

2020 ◽  
Vol 27 (03) ◽  
pp. 575-586
Author(s):  
Sergio Estrada ◽  
Alina Iacob ◽  
Holly Zolt
Keyword(s):  
Analogous Result ◽  
Gorenstein Ring ◽  
Gorenstein Rings ◽  
Injective Modules ◽  
Flat Modules ◽  
Coherent Ring ◽  
Acyclic Complexes ◽  
Gorenstein Flat ◽  
Gorenstein Injective

For a given class of modules [Formula: see text], let [Formula: see text] be the class of exact complexes having all cycles in [Formula: see text], and dw([Formula: see text]) the class of complexes with all components in [Formula: see text]. Denote by [Formula: see text][Formula: see text] the class of Gorenstein injective R-modules. We prove that the following are equivalent over any ring R: every exact complex of injective modules is totally acyclic; every exact complex of Gorenstein injective modules is in [Formula: see text]; every complex in dw([Formula: see text][Formula: see text]) is dg-Gorenstein injective. The analogous result for complexes of flat and Gorenstein flat modules also holds over arbitrary rings. If the ring is n-perfect for some integer n ≥ 0, the three equivalent statements for flat and Gorenstein flat modules are equivalent with their counterparts for projective and projectively coresolved Gorenstein flat modules. We also prove the following characterization of Gorenstein rings. Let R be a commutative coherent ring; then the following are equivalent: (1) every exact complex of FP-injective modules has all its cycles Ding injective modules; (2) every exact complex of flat modules is F-totally acyclic, and every R-module M such that M+ is Gorenstein flat is Ding injective; (3) every exact complex of injectives has all its cycles Ding injective modules and every R-module M such that M+ is Gorenstein flat is Ding injective. If R has finite Krull dimension, statements (1)–(3) are equivalent to (4) R is a Gorenstein ring (in the sense of Iwanaga).

scholarly journals Quadratic Gorenstein rings and the Koszul property I

2020 ◽  
Vol 374 (2) ◽  
pp. 1077-1093 ◽  
Author(s):  
Matthew Mastroeni ◽  
Hal Schenck ◽  
Mike Stillman
Keyword(s):  
Gorenstein Rings ◽  
Koszul Property

Relative Gorenstein rings and duality pairs

2019 ◽  
Vol 19 (08) ◽  
pp. 2050147
Author(s):  
Junpeng Wang ◽  
Zhenxing Di
Keyword(s):  
Model Category ◽  
Gorenstein Ring ◽  
Gorenstein Rings ◽  
Duality Pairs ◽  
Coherent Rings

Let [Formula: see text] be a ring (not necessarily commutative) and [Formula: see text] a bi-complete duality pair. We investigate the notions of (flat-typed) [Formula: see text]-Gorenstein rings, which unify Iwanaga–Gorenstein rings, Ding–Chen rings, AC-Gorenstein rings and Gorenstein [Formula: see text]-coherent rings. Using an abelian model category approach, we show that [Formula: see text] and [Formula: see text], the homotopy categories of all exact complexes of projective and injective [Formula: see text]-modules, are triangulated equivalent whenever [Formula: see text] is a flat-typed [Formula: see text]-Gorenstein ring.

O tym, co minęło, lecz nie zostało zapomniane: Badania archeologiczne na terenie byłego obozu zagłady w Treblince

2012 ◽  
pp. 83-118
Author(s):  
Caroline Sturdy Colls
Keyword(s):  
Temporal Analysis ◽  
Archaeological Survey ◽  
Mass Graves ◽  
Non Invasive ◽  
Physical Evidence ◽  
Spatial And Temporal Analysis ◽  
The Holocaust ◽  
Public Consciousness ◽  
Level Of Knowledge ◽  
Answering Questions

Public impression of the Holocaust is unquestionably centred on knowledge about, and the image of, Auschwitz-Birkenau – the gas chambers, the crematoria, the systematic and industrialized killing of victims. Conversely, knowledge of the former extermination camp at Treblinka, which stands in stark contrast in terms of the visible evidence that survives pertaining to it, is less embedded in general public consciousness. As this paper argues, the contrasting level of knowledge about Auschwitz- Birkenau and Treblinka is centred upon the belief that physical evidence of the camps only survives when it is visible and above-ground. The perception of Treblinka as having been “destroyed” by the Nazis, and the belief that the bodies of all of the victims were cremated without trace, has resulted in a lack of investigation aimed at answering questions about the extent and nature of the camp, and the locations of mass graves and cremation pits. This paper discusses the evidence that demonstrates that traces of the camp do survive. It outlines how archival research and non-invasive archaeological survey has been used to re-evaluate the physical evidence pertaining to Treblinka in a way that respects Jewish Halacha Law. As well as facilitating spatial and temporal analysis of the former extermination camp, this survey has also revealed information about the cultural memory.

MODIFIKASI MODEL PEMBELAJARAN PROBLEM POSING DENGAN STRATEGI PEMBELAJARAN TUGAS DAN PAKSA

2019 ◽  
Vol 2 ◽  
Author(s):  
Lita Amalia ◽  
Alda Dwiyana Putri ◽  
Alfajri Mairizki Nurfansyah
Keyword(s):  
Student Learning ◽  
Learning Outcomes ◽  
Learning Process ◽  
Learning Strategy ◽  
Student Learning Outcomes ◽  
Learning Model ◽  
Problem Posing ◽  
Reasoning Skills ◽  
The Impact ◽  
Answering Questions

The purpose of this paper is to describe the Problem Posing learning model with Task and Forced Strategy. As for the background of this writing is because of difficulties in understanding the material and also lack of enthusiasm of students in learning the material so that the impact on student learning outcomes is still low. The low student learning outcomes are, of course, many factors, one of which is the problem of applying a learning model that is still teacher-centered, so students tend to be passive. For this reason, the teacher can use the Problem Posing learning model that is modified by the task and force strategy (Task and Forced). Problem Posing learning model is a learning model that requires students to develop their systematic reasoning skills in making questions and answering questions. While the task and force strategy (Task and Forced) is a learning strategy that has little effect on students to complete the task until it is completed and on time to avoid the punishment given by the teacher as a consequence. So that students will be motivated in listening, understanding the material delivered and doing assignments on time. By combining this model and strategy can be a solution so that the learning process becomes quality.

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