What Do You Know About Auslander Algebra?
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.
- 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
![When considering Auslander algebra An of k[T ]=hT n i, what is k? - ProProfs](https://media.proprofs.com/images/QM/user_images/2284644/1529818846.png "When considering Auslander algebra An of k[T ]=hT n i, what is k?")
![When considering Auslander algebra An of k[T ]=hT n i, what is T? - ProProfs](https://media.proprofs.com/images/QM/user_images/2284644/1529818805.png "When considering Auslander algebra An of k[T ]=hT n i, what is T?")
![When considering Auslander algebra An of k[T ]=hT n i, what is n? - ProProfs](https://media.proprofs.com/images/QM/user_images/2284644/1529818760.jpeg "When considering Auslander algebra An of k[T ]=hT n i, what is n?")
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