Mixing
Weak generators of the right-bounded derived category of a finite-dimensional algebra
2
$begingroup$
The setup:
- Let $A$ be a finite-dimensional $k$-algebra over some field $k$.
- Let $mathcal{B} = Hot^-(Proj A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) projective right $A$-modules which are bounded from the right. This category can be identified with the right-bounded derived category $D^-(Mod A)$ of $A$-modules.
- Let $mathcal{C} = Hot^-(proj A)$ denote the full subcategory of $mathcal{B}$ given by right-bounded complexes of finitely generated projective $A$-modules. In different terms, this category corresponds to the right-bounded derived category
$D^-(mod A)$ of finitely generated $A$-modules. - Let $P$ be a perfect object of $mathcal{B}$, that is, a bounded complex of finitely generated projective right $A$-modules.
Assume also that $P$ is a weak generator of the small category $mathcal{C}$, so for any object $X in mathcal{C}$ there is some integer $m$ and some non-zero morphism $P to X[m]$ in $mathcal{C}$.
My question:
Is $P$ already a weak generator of the big category $mathcal{B}$?
Some background:
- By a result of Jeremy Rickard, the answer is affirmative if $P$ is a partial tilting complex, that is, if $Hom_{mathcal{B}}(P,P[n])=0$ for any non-zero integer $n$.
Reference:
Proposition 5.4 in
Rickard, Jeremy, Morita theory for derived categories, J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989). ZBL0642.16034.
- By a result of Bruno J. Müller, any projective $A$-module is a (possibly infinite) direct sum of finitely generated projective $A$-modules.
Reference:
Müller, Bruno J., On semi-perfect rings, Mathematical Report No. 19, Vol. 1. Hamilton, Ont.: McMaster University, Department of Mathematics. 11 p. (1969). ZBL0226.16026.
- At least, $P$ is a weak generator of the homotopy category of complexes of projective $A$-modules which have finitely generated cohomology at each degree.
Any comments and any input will be very appreciated.
representation-theory homological-algebra
asked Jan 14 at 19:10
WayneWayne
385
$endgroup$
add a comment |
2
$begingroup$
The setup:
- Let $A$ be a finite-dimensional $k$-algebra over some field $k$.
- Let $mathcal{B} = Hot^-(Proj A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) projective right $A$-modules which are bounded from the right. This category can be identified with the right-bounded derived category $D^-(Mod A)$ of $A$-modules.
- Let $mathcal{C} = Hot^-(proj A)$ denote the full subcategory of $mathcal{B}$ given by right-bounded complexes of finitely generated projective $A$-modules. In different terms, this category corresponds to the right-bounded derived category
$D^-(mod A)$ of finitely generated $A$-modules. - Let $P$ be a perfect object of $mathcal{B}$, that is, a bounded complex of finitely generated projective right $A$-modules.
Assume also that $P$ is a weak generator of the small category $mathcal{C}$, so for any object $X in mathcal{C}$ there is some integer $m$ and some non-zero morphism $P to X[m]$ in $mathcal{C}$.
My question:
Is $P$ already a weak generator of the big category $mathcal{B}$?
Some background:
- By a result of Jeremy Rickard, the answer is affirmative if $P$ is a partial tilting complex, that is, if $Hom_{mathcal{B}}(P,P[n])=0$ for any non-zero integer $n$.
Reference:
Proposition 5.4 in
Rickard, Jeremy, Morita theory for derived categories, J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989). ZBL0642.16034.
- By a result of Bruno J. Müller, any projective $A$-module is a (possibly infinite) direct sum of finitely generated projective $A$-modules.
Reference:
Müller, Bruno J., On semi-perfect rings, Mathematical Report No. 19, Vol. 1. Hamilton, Ont.: McMaster University, Department of Mathematics. 11 p. (1969). ZBL0226.16026.
- At least, $P$ is a weak generator of the homotopy category of complexes of projective $A$-modules which have finitely generated cohomology at each degree.
Any comments and any input will be very appreciated.
representation-theory homological-algebra
asked Jan 14 at 19:10
WayneWayne
385
$endgroup$
$begingroup$
Given the rather advanced nature of this question, I think it would receive more attention on MathOverflow. If you agree, you can clickflag>in need of moderator interventionand request that a moderator migrate the question to MathOverflow.
$endgroup$
– Mike Pierce
Jan 14 at 19:41
$begingroup$
Thank you for the advice. I have posted a slightly extended version of the question on MathOverflow.
$endgroup$
– Wayne
Jan 16 at 18:50
add a comment |
2
2
2
$begingroup$
The setup:
- Let $A$ be a finite-dimensional $k$-algebra over some field $k$.
- Let $mathcal{B} = Hot^-(Proj A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) projective right $A$-modules which are bounded from the right. This category can be identified with the right-bounded derived category $D^-(Mod A)$ of $A$-modules.
- Let $mathcal{C} = Hot^-(proj A)$ denote the full subcategory of $mathcal{B}$ given by right-bounded complexes of finitely generated projective $A$-modules. In different terms, this category corresponds to the right-bounded derived category
$D^-(mod A)$ of finitely generated $A$-modules. - Let $P$ be a perfect object of $mathcal{B}$, that is, a bounded complex of finitely generated projective right $A$-modules.
Assume also that $P$ is a weak generator of the small category $mathcal{C}$, so for any object $X in mathcal{C}$ there is some integer $m$ and some non-zero morphism $P to X[m]$ in $mathcal{C}$.
My question:
Is $P$ already a weak generator of the big category $mathcal{B}$?
Some background:
- By a result of Jeremy Rickard, the answer is affirmative if $P$ is a partial tilting complex, that is, if $Hom_{mathcal{B}}(P,P[n])=0$ for any non-zero integer $n$.
Reference:
Proposition 5.4 in
Rickard, Jeremy, Morita theory for derived categories, J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989). ZBL0642.16034.
- By a result of Bruno J. Müller, any projective $A$-module is a (possibly infinite) direct sum of finitely generated projective $A$-modules.
Reference:
Müller, Bruno J., On semi-perfect rings, Mathematical Report No. 19, Vol. 1. Hamilton, Ont.: McMaster University, Department of Mathematics. 11 p. (1969). ZBL0226.16026.
- At least, $P$ is a weak generator of the homotopy category of complexes of projective $A$-modules which have finitely generated cohomology at each degree.
Any comments and any input will be very appreciated.
representation-theory homological-algebra
asked Jan 14 at 19:10
WayneWayne
385
$endgroup$
The setup:
- Let $A$ be a finite-dimensional $k$-algebra over some field $k$.
- Let $mathcal{B} = Hot^-(Proj A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) projective right $A$-modules which are bounded from the right. This category can be identified with the right-bounded derived category $D^-(Mod A)$ of $A$-modules.
- Let $mathcal{C} = Hot^-(proj A)$ denote the full subcategory of $mathcal{B}$ given by right-bounded complexes of finitely generated projective $A$-modules. In different terms, this category corresponds to the right-bounded derived category
$D^-(mod A)$ of finitely generated $A$-modules. - Let $P$ be a perfect object of $mathcal{B}$, that is, a bounded complex of finitely generated projective right $A$-modules.
Assume also that $P$ is a weak generator of the small category $mathcal{C}$, so for any object $X in mathcal{C}$ there is some integer $m$ and some non-zero morphism $P to X[m]$ in $mathcal{C}$.
My question:
Is $P$ already a weak generator of the big category $mathcal{B}$?
Some background:
- By a result of Jeremy Rickard, the answer is affirmative if $P$ is a partial tilting complex, that is, if $Hom_{mathcal{B}}(P,P[n])=0$ for any non-zero integer $n$.
Reference:
Proposition 5.4 in
Rickard, Jeremy, Morita theory for derived categories, J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989). ZBL0642.16034.
- By a result of Bruno J. Müller, any projective $A$-module is a (possibly infinite) direct sum of finitely generated projective $A$-modules.
Reference:
Müller, Bruno J., On semi-perfect rings, Mathematical Report No. 19, Vol. 1. Hamilton, Ont.: McMaster University, Department of Mathematics. 11 p. (1969). ZBL0226.16026.
- At least, $P$ is a weak generator of the homotopy category of complexes of projective $A$-modules which have finitely generated cohomology at each degree.
Any comments and any input will be very appreciated.
representation-theory homological-algebra
representation-theory homological-algebra
asked Jan 14 at 19:10
WayneWayne
385
asked Jan 14 at 19:10
WayneWayne
385
asked Jan 14 at 19:10
WayneWayne
385
asked Jan 14 at 19:10
WayneWayne
385
asked Jan 14 at 19:10
WayneWayne
385
385
$begingroup$
Given the rather advanced nature of this question, I think it would receive more attention on MathOverflow. If you agree, you can clickflag>in need of moderator interventionand request that a moderator migrate the question to MathOverflow.
$endgroup$
– Mike Pierce
Jan 14 at 19:41
$begingroup$
Thank you for the advice. I have posted a slightly extended version of the question on MathOverflow.
$endgroup$
– Wayne
Jan 16 at 18:50
add a comment |
$begingroup$
Given the rather advanced nature of this question, I think it would receive more attention on MathOverflow. If you agree, you can clickflag>in need of moderator interventionand request that a moderator migrate the question to MathOverflow.
$endgroup$
– Mike Pierce
Jan 14 at 19:41
$begingroup$
Thank you for the advice. I have posted a slightly extended version of the question on MathOverflow.
$endgroup$
– Wayne
Jan 16 at 18:50
$begingroup$
Given the rather advanced nature of this question, I think it would receive more attention on MathOverflow. If you agree, you can click
flag>in need of moderator intervention and request that a moderator migrate the question to MathOverflow.$endgroup$
– Mike Pierce
Jan 14 at 19:41
$begingroup$
Given the rather advanced nature of this question, I think it would receive more attention on MathOverflow. If you agree, you can click
flag>in need of moderator intervention and request that a moderator migrate the question to MathOverflow.$endgroup$
– Mike Pierce
Jan 14 at 19:41
$begingroup$
Thank you for the advice. I have posted a slightly extended version of the question on MathOverflow.
$endgroup$
– Wayne
Jan 16 at 18:50
$begingroup$
Thank you for the advice. I have posted a slightly extended version of the question on MathOverflow.
$endgroup$
– Wayne
Jan 16 at 18:50
add a comment |
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$begingroup$
Given the rather advanced nature of this question, I think it would receive more attention on MathOverflow. If you agree, you can click
flag>in need of moderator interventionand request that a moderator migrate the question to MathOverflow.$endgroup$
– Mike Pierce
Jan 14 at 19:41
$begingroup$
Thank you for the advice. I have posted a slightly extended version of the question on MathOverflow.
$endgroup$
– Wayne
Jan 16 at 18:50