Cotorsion pairs and model structures on Ch(R)

2011 ◽  
Vol 54 (3) ◽  
pp. 783-797 ◽  
Author(s):  
Gang Yang ◽  
Zhongkui Liu

AbstractWe show that if the given cotorsion pair $(\mathcal{A},\mathcal{B})$ in the category of modules is complete and hereditary, then both of the induced cotorsion pairs in the category of complexes are complete. We also give a cofibrantly generated model structure that can be regarded as a generalization of the projective model structure.

Author(s):  
Aimin Xu

Let [Formula: see text] be either the category of [Formula: see text]-modules or the category of chain complexes of [Formula: see text]-modules and [Formula: see text] a cofibrantly generated hereditary abelian model structure on [Formula: see text]. First, we get a new cofibrantly generated model structure on [Formula: see text] related to [Formula: see text] for any positive integer [Formula: see text], and hence, one can get new algebraic triangulated categories. Second, it is shown that any [Formula: see text]-strongly Gorenstein projective module gives rise to a projective cotorsion pair cogenerated by a set. Finally, let [Formula: see text] be an [Formula: see text]-module with finite flat dimension and [Formula: see text] a positive integer, if [Formula: see text] is an exact sequence of [Formula: see text]-modules with every [Formula: see text] Gorenstein injective, then [Formula: see text] is injective.


Author(s):  
Wenjing Chen ◽  
Zhongkui Liu

In this paper, we construct some model structures corresponding Gorenstein [Formula: see text]-modules and relative Gorenstein flat modules associated to duality pairs, Frobenius pairs and cotorsion pairs. By investigating homological properties of Gorenstein [Formula: see text]-modules and some known complete hereditary cotorsion pairs, we describe several types of complexes and obtain some characterizations of Iwanaga–Gorenstein rings. Based on some facts given in this paper, we find new duality pairs and show that [Formula: see text] is covering as well as enveloping and [Formula: see text] is preenveloping under certain conditions, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-injective modules and [Formula: see text] denotes the class of Gorenstein [Formula: see text]-flat modules. We give some recollements via projective cotorsion pair [Formula: see text] cogenerated by a set, where [Formula: see text] denotes the class of Gorenstein [Formula: see text]-projective modules. Also, many recollements are immediately displayed through setting specific complete duality pairs.


2017 ◽  
Vol 24 (04) ◽  
pp. 577-602 ◽  
Author(s):  
Xiaoyan Yang ◽  
Tianya Cao

Given a cotorsion pair ([Formula: see text], [Formula: see text]) in an abelian category [Formula: see text] , we define cotorsion pairs ([Formula: see text], dg[Formula: see text]) and (dg[Formula: see text], [Formula: see text]) in the category [Formula: see text]N([Formula: see text]) of N-complexes on [Formula: see text]. We prove that if the cotorsion pair ([Formula: see text], [Formula: see text]) is complete and hereditary in a bicomplete abelian category, then both of the induced cotorsion pairs are complete, compatible and hereditary. We also create complete cotorsion pairs (dw[Formula: see text], (dw[Formula: see text])⊥), (ex[Formula: see text], (ex[Formula: see text])⊥) and (⊥(dw[Formula: see text]), dw[Formula: see text]), (⊥(ex[Formula: see text]); ex[Formula: see text]) in a termwise manner by starting with a cotorsion pair ([Formula: see text], [Formula: see text]) that is cogenerated by a set. As applications of these results, we obtain more abelian model structures from the cotorsion pairs.


2017 ◽  
Vol 59 (3) ◽  
pp. 685-703 ◽  
Author(s):  
AIMIN XU

AbstractGiven a complete hereditary cotorsion pair$(\mathcal{X}, \mathcal{Y})$, we introduce the concept of$(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$-Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integern, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤nover a left Noetherian ringR. Similarly, ifRis a left coherent ring in which all flat leftR-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤n. These structures have their analogous in the category of chain complexes.


2013 ◽  
Vol 13 (03) ◽  
pp. 1350106
Author(s):  
REN WEI ◽  
ZHONGKUI LIU

In this paper, we first give an alternative characterization of the derived functor Ext via the Quillen model structure on the category of complexes induced by a given cotorsion pair [Formula: see text] in the category of modules, then based on this, we consider homological dimensions of complexes related to [Formula: see text]. As applications, we extend Gorenstein projective dimension of homologically bounded below complexes (in the sense of Christensen and coauthors) to unbounded complexes whenever R is Gorenstein. Moreover, we extend Stenström's FP-injective dimension from modules to complexes, define FP-projective dimension for complexes, and characterize Noetherian and von Neumann regular rings by these dimensions.


1999 ◽  
Vol 39 (4) ◽  
pp. 55-60 ◽  
Author(s):  
J. Alex ◽  
R. Tschepetzki ◽  
U. Jumar ◽  
F. Obenaus ◽  
K.-H. Rosenwinkel

Activated sludge models are widely used for planning and optimisation of wastewater treatment plants and on line applications are under development to support the operation of complex treatment plants. A proper model is crucial for all of these applications. The task of parameter calibration is focused in several papers and applications. An essential precondition for this task is an appropriately defined model structure, which is often given much less attention. Different model structures for a large scale treatment plant with circulation flow are discussed in this paper. A more systematic method to derive a suitable model structure is applied to this case. Results of a numerical hydraulic model are used for this purpose. The importance of these efforts are proven by a high sensitivity of the simulation results with respect to the selection of the model structure and the hydraulic conditions. Finally it is shown, that model calibration was possible only by adjusting to the hydraulic behaviour and without any changes of biological parameters.


2014 ◽  
Vol 57 (2) ◽  
pp. 263-284 ◽  
Author(s):  
XIAOYAN YANG

AbstractWe define model structures on a triangulated category with respect to some proper classes of triangles and give a general study of triangulated model structures. We look at the relationship between these model structures and cotorsion pairs with respect to a proper class of triangles on the triangulated category. In particular, we get Hovey's one-to-one correspondence between triangulated model structures and complete cotorsion pairs with respect to a proper class of triangles. Some applications are given.


2013 ◽  
Vol 17 (10) ◽  
pp. 4227-4239 ◽  
Author(s):  
W. R. van Esse ◽  
C. Perrin ◽  
M. J. Booij ◽  
D. C. M. Augustijn ◽  
F. Fenicia ◽  
...  

Abstract. Models with a fixed structure are widely used in hydrological studies and operational applications. For various reasons, these models do not always perform well. As an alternative, flexible modelling approaches allow the identification and refinement of the model structure as part of the modelling process. In this study, twelve different conceptual model structures from the SUPERFLEX framework are compared with the fixed model structure GR4H, using a large set of 237 French catchments and discharge-based performance metrics. The results show that, in general, the flexible approach performs better than the fixed approach. However, the flexible approach has a higher chance of inconsistent results when calibrated on two different periods. When analysing the subset of 116 catchments where the two approaches produce consistent performance over multiple time periods, their average performance relative to each other is almost equivalent. From the point of view of developing a well-performing fixed model structure, the findings favour models with parallel reservoirs and a power function to describe the reservoir outflow. In general, conceptual hydrological models perform better on larger and/or wetter catchments than on smaller and/or drier catchments. The model structures performed poorly when there were large climatic differences between the calibration and validation periods, in catchments with flashy flows, and in catchments with unexplained variations in low flow measurements.


2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Hao Liu ◽  
Keqiang Yue ◽  
Siyi Cheng ◽  
Chengming Pan ◽  
Jie Sun ◽  
...  

Diabetic retinopathy (DR) is one of the most common complications of diabetes and the main cause of blindness. The progression of the disease can be prevented by early diagnosis of DR. Due to differences in the distribution of medical conditions and low labor efficiency, the best time for diagnosis and treatment was missed, which results in impaired vision. Using neural network models to classify and diagnose DR can improve efficiency and reduce costs. In this work, an improved loss function and three hybrid model structures Hybrid-a, Hybrid-f, and Hybrid-c were proposed to improve the performance of DR classification models. EfficientNetB4, EfficientNetB5, NASNetLarge, Xception, and InceptionResNetV2 CNNs were chosen as the basic models. These basic models were trained using enhance cross-entropy loss and cross-entropy loss, respectively. The output of the basic models was used to train the hybrid model structures. Experiments showed that enhance cross-entropy loss can effectively accelerate the training process of the basic models and improve the performance of the models under various evaluation metrics. The proposed hybrid model structures can also improve DR classification performance. Compared with the best-performing results in the basic models, the accuracy of DR classification was improved from 85.44% to 86.34%, the sensitivity was improved from 98.48% to 98.77%, the specificity was improved from 71.82% to 74.76%, the precision was improved from 90.27% to 91.37%, and the F1 score was improved from 93.62% to 93.9% by using hybrid model structures.


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