What Do You Know About Auslander Algebra?

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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

A. If the projective dimension of its socle does not exceed one
Explanation
If the projective dimension of the socle (the submodule consisting of elements annihilated by all non-zero divisors) of the injective Δ-module is not greater than one, then the injective Δ-module is projective. This means that the socle has a relatively simple structure and can be generated by a small number of elements.

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• 2.

When was the Auslander algebra introduced?

• A.

1973

• B.

1974

• C.

1975

• D.

1976

B. 1974
Explanation
The Auslander algebra was introduced in 1974.

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• 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

A. Field
Explanation
In the context of considering the Auslander algebra An of k[T], k refers to a field. The Auslander algebra is defined over a field, which is a fundamental concept in abstract algebra. A field is a mathematical structure that satisfies certain properties, such as having addition, subtraction, multiplication, and division operations defined on it. In this case, k represents the field over which the Auslander algebra An is constructed.

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• 4.

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

• A.

Variable

• B.

Trigonometric value

• C.

Time

• D.

Valency

A. Variable
Explanation
The Auslander algebra An is defined over the variable T, which means that T is the correct answer. It is the variable used in the algebraic structure and represents an unknown value that can be assigned different values.

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• 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

C. Natural number
Explanation
The Auslander algebra An of k[T] is defined for a polynomial ring k[T] with a variable T raised to the power of n. In this context, n represents the degree of the polynomial, which is a natural number. Therefore, the correct answer is "Natural number."

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• 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

C. Does not exceed 1
Explanation
If the projective dimension of the socle M is at most one, it means that the socle M can be generated by at most one element. Since the socle is a submodule of M, this implies that M can also be generated by at most one element. Therefore, the projective dimension of M, which measures the minimum number of generators of M, does not exceed one.

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• 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

A. If the projective dimension of its socle does not exceed one
Explanation
If the projective dimension of the socle of the injective Δ-module does not exceed one, it means that the socle can be generated by at most one element. This implies that the socle is a projective module, as it can be freely generated by a single element. Therefore, the injective Δ-module is also projective.

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• 8.

How do we define the Auslander algebra of any representation finite algebra?

• A.

Aqua-hereditary

• B.

Quasi hereditary

• C.

Hereditary

• D.

Non-hereditary

B. Quasi hereditary
Explanation
The Auslander algebra of a representation finite algebra is defined as quasi-hereditary. This means that it has a filtration by projective modules, where each quotient module is a direct sum of indecomposable projective modules. In other words, the Auslander algebra can be decomposed into a direct sum of certain subalgebras, each of which is hereditary. This property allows for a better understanding and analysis of the Auslander algebra and its representation theory.

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• 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

A. Γ ≤ 2
Explanation
An Artin algebra Γ is called an Auslander algebra if its global dimension (gl dim) is less than or equal to 2. This means that the algebra has a relatively simple structure and its modules can be understood and studied in a more manageable way. It also implies that the algebra has certain homological properties that make it suitable for certain applications and computations.

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• 10.

Which ring is associated with the theorem?

• A.

Endomorphism ring

• B.

Exomorphism ring

• C.

Static ring

• D.

External ring

A. EndomorpHism ring
Explanation
The endomorphism ring is associated with the theorem mentioned in the question. This ring consists of all endomorphisms of a given mathematical object, such as a group or a module, and it forms a ring under composition of endomorphisms. The other options, exomorphism ring, static ring, and external ring, are not commonly used or recognized terms in mathematics.

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• 11.

Which modules of algebra A is summed up?

• A.

Decomposable modules

• B.

Incompressible modules

• C.

Normal modules

• D.

Indecomposable modules

D. Indecomposable modules
Explanation
Indecomposable modules are the ones that cannot be broken down into smaller modules. They are considered as the building blocks of larger modules and cannot be further decomposed. Therefore, the correct answer is "Indecomposable modules."

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