What Do You Know About Auslander Algebra?

11 Questions | Total Attempts: 105

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What Do You Know About Auslander Algebra? - Quiz

In the mid-1970s, famous American homological algebra and commutative algebra mathematician Maurice Auslander (August 3, 1926 - November 18, 1994) introduced the Auslander algebra. Ever since, the concept has been widely used in the algebra branch of mathematics. Basically, the Auslander algebra of an algebra A is the endomorphism ring of the total of the indecomposable modules of A.


Questions and Answers
  • 1. 
    How do we know if the injective Δ-module is projective? 
    • A. 

      If the projective dimension of its socle does not exceed one

    • B. 

      If the projective dimension of its socle is more than one

    • C. 

      If the projective dimension of its socle is between 1 and -1

    • D. 

      If the projective dimension of its socle equals to one t 

  • 2. 
     When was the Auslander algebra introduced?
    • A. 

      1973

    • B. 

      1974

    • C. 

      1975

    • D. 

      1976

  • 3. 
    When considering Auslander algebra An of k[T ]=hT n i, what is k? 
    • A. 

      Field

    • B. 

      Variable

    • C. 

      Natural number

    • D. 

      Kepler value

  • 4. 
    When considering Auslander algebra An of k[T ]=hT n i, what is T?
    • A. 

      Variable

    • B. 

      Trigonometric value

    • C. 

      Time

    • D. 

      Valency

  • 5. 
    When considering Auslander algebra An of k[T ]=hT n i, what is n? 
    • A. 

      Real number

    • B. 

      Whole number

    • C. 

      Natural number

    • D. 

      Constant number

  • 6. 
    For a Δ-module M, what is the projective dimension of M when the projective dimension of its socle M is at most one?
    • A. 

      Exceeds 1

    • B. 

      Less than -1

    • C. 

      Does not exceed 1

    • D. 

      More than -1

  • 7. 
    How do we know if the injective Δ-module is projective? 
    • A. 

      If the projective dimension of its socle does not exceed one

    • B. 

      If the projective dimension of its socle is more than one

    • C. 

      If the projective dimension of its socle is between 1 and -1

    • D. 

      If the projective dimension of its socle equals to one t 

  • 8. 
    How do we define the Auslander algebra of any representation finite algebra?
    • A. 

      Aqua-hereditary

    • B. 

      Quasi hereditary

    • C. 

      Hereditary

    • D. 

      Non-hereditary

  • 9. 
    An Artin algebra Γ is called an Auslander algebra if gl dim... 
    • A. 

      Γ ≤ 2

    • B. 

      Γ = 2

    • C. 

      Γ < 2

    • D. 

      Γ > 2

  • 10. 
    Which ring is associated with the theorem?
    • A. 

      Endomorphism ring

    • B. 

      Exomorphism ring

    • C. 

      Static ring

    • D. 

      External ring

  • 11. 
    Which modules of algebra A is summed up? 
    • A. 

      Decomposable modules

    • B. 

      Incompressible modules

    • C. 

      Normal modules

    • D. 

      Indecomposable modules

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