Auslander Algebra Trivia Quiz
The Auslander Algebra is a work of Maurice Auslander a famous American Mathematician. His work was an Artin Algebra representation. An Artin Algebra is an Auslander Algebra if gl dim Γ ≤ 2 and if 0→Γ→I→J→K→0 is a minimal injective resolution of Γ then I and J are projective Γ-modules; where Γ is the Artin Algebra. Try out this quiz.
- 1.
Which of these is a summed up module of Algebra A?
- A.
Integrated modules
- B.
Indecomposable module
- C.
Projective module
- D.
Isomorphic module
Correct Answer
B. Indecomposable moduleExplanation
An indecomposable module is a module that cannot be expressed as the direct sum of two non-zero submodules. In other words, it cannot be "broken down" into smaller modules. This makes it a summed up module of Algebra A because it represents a fundamental building block that cannot be further simplified or decomposed.Rate this question:
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- 2.
When did Auslander introduce the Algebra?
- A.
1874
- B.
1870
- C.
1974
- D.
1989
Correct Answer
C. 1974 -
- 3.
In these fomular, k[T ]=hT n i, what does k stand for?
- A.
Term
- B.
Natural value
- C.
Module
- D.
Field
Correct Answer
D. FieldExplanation
The correct answer is "Field". In the given formula, k represents the field. A field is a mathematical structure that consists of a set of elements along with operations such as addition and multiplication. In this context, k is the field used to represent the relationship between temperature (T) and the rate constant (k) in a chemical reaction.Rate this question:
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- 4.
What does T represent in the Auslander formular?
- A.
Constant
- B.
Value for Artin constant
- C.
Variable
- D.
Function
Correct Answer
C. VariableExplanation
The correct answer is "Variable." In the Auslander formula, T represents a variable. This means that its value can change and is not fixed.Rate this question:
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- 5.
What does the n in the Auslander formular stand for?
- A.
Integers
- B.
Natural Number
- C.
Rational number
- D.
Constant
Correct Answer
B. Natural NumberExplanation
The "n" in the Auslander formula stands for natural number. A natural number is a positive integer greater than zero that is used to count or label objects. In the context of the Auslander formula, it is likely used as a parameter or variable representing a specific natural number value.Rate this question:
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- 6.
Which of these symbol represents Artin Algebra in relation to Auslander Algebra?
- A.
∆
- B.
μ
- C.
Λ
- D.
Γ
Correct Answer
D. ΓExplanation
The symbol Γ represents Artin Algebra in relation to Auslander Algebra. This is because Γ is commonly used to denote the Auslander-Reiten quiver, which is a directed graph that represents the indecomposable modules of an Artin algebra. The Auslander-Reiten quiver provides important information about the structure and representation theory of the Artin algebra, making Γ the correct symbol to represent it.Rate this question:
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- 7.
What kind of ring is the sum of indecomposable module of A in the Auslander Algebra?
- A.
Module
- B.
Endomorphism
- C.
Homomorphism
- D.
Artin
Correct Answer
B. EndomorphismExplanation
The sum of indecomposable modules of A in the Auslander Algebra is an endomorphism. This means that it is a module that is both a domain and a codomain of a homomorphism, and the homomorphism is from the module to itself. In other words, it represents the set of all possible mappings from the module to itself.Rate this question:
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- 8.
Which of these satisfies the condition for an Artin Algebra to be an Auslander Algebra?
- A.
Γ ≤ 2
- B.
Γ ≥ 2
- C.
Γ < 1
- D.
Γ > 2
Correct Answer
A. Γ ≤ 2Explanation
An Artin Algebra is an Auslander Algebra if its global dimension (Γ) is less than or equal to 2. This means that the algebra has a finite projective resolution of length 2 or less.Rate this question:
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- 9.
Which of these people is connected to the Auslander Algebra?
- A.
Reinten
- B.
Becleau
- C.
Herin
- D.
Phills
Correct Answer
A. ReintenExplanation
Reinten is connected to the Auslander Algebra.Rate this question:
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- 10.
Which of these explains Auslander Algebra of any representation of finite Algebra?
- A.
Budding of Fl
- B.
Blossom of flowers
- C.
Quasi Hereditary
- D.
Passing of Gene
Correct Answer
C. Quasi HereditaryExplanation
Quasi Hereditary is an explanation for the Auslander Algebra of any representation of a finite Algebra. Quasi Hereditary refers to a property of certain categories that allows for the decomposition of modules into a direct sum of indecomposable modules. This property is closely related to the Auslander-Reiten theory, which studies the structure of modules and their relations in a given algebra. Therefore, the concept of Quasi Hereditary provides a valid explanation for the Auslander Algebra of any representation of a finite Algebra.Rate this question:
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![In these fomular, k[T ]=hT n i, what does k stand for? - ProProfs](https://media.proprofs.com/images/QM/user_images/2305925/1554960814.png "In these fomular, k[T ]=hT n i, what does k stand for?")
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Mar 18, 2023Quiz Edited by
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Apr 09, 2019Quiz Created by
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