HW due Fri Feb 9Mon Feb 12 (to avoid the complex analysis midterm)
1a. Let R be a subring of S, and AS, SB be two modules. Define a surjection A@RB ->> A@SB.
1b. Can you extend this to a natural transformation between two functors between something-or-other? This question is intentionally ill-defined, and I imagine different people will experience different creative urges.
2. Let A,B be right R-modules and C,D left R-modules. Let + denote direct sum. Define isomorphisms (A + B) @R C ~ A@RC + B@RC and B @R (C + D) ~ B@RC + B@RD.
...a very dull question, but it leads to...
3. Let Mn(C) denote the ring of nxn complex matrices, and Ck x n denote the (Mk(C),Mn(C))-bimodule of kxn complex matrices. Show that matrix multiplication
Ck x n x Cn x m -> Ck x m
induces a map
Ck x n @Mn(C) Cn x m -> Ck x m
and that this map is an isomorphism.
4. An isomorphism D: R -> Rop is called an anti-automorphism of R. For example, the identity function is an anti-automorphism iff R is commutative.
a. Let MR be a right R-module. Use D to define a left module structure on it.
b. Let RN be a left R-module. Using D, we can make N* := HomR(N,R) a left R-module. Using D again, we can make (N*)* a left R-module. Find a condition on D that guarantees that this new left R-module structure on N** agrees with the old one.
c. Show the converse: if this new left R-module structure on N** agrees with the old one for all N, then D satisfies that condition.
b. Even if D is just an anti-homomorphism, not necessarily invertible, say how to use it to turn any right R-module into a left R-module.
c. Let M be a left R-module, which we turn into a right module, then into a left module again. Determine the condition on D under which this new module structure agrees with the old one (for arbitrary M).
5. Forgetting module theory for a moment, an inner product < , > on a real vector space V induces an isomorphism V -> V*, taking v |-> (w |-> < v,w >). This looks weird to us now, in that it's an isomorphism of a left module with a right module.
a. Use question 4 to state the definition of an inner product on a real vector space, being careful about left vs. right modules. (This being an algebra class, you can skip nonalgebraic conditions like "positive definite".)
b. Use question 4 to define a Hermitian inner product on a complex vector space, being careful about left vs. right modules.
6. Let A,B be two finite abelian groups (aka (Z,Z)-bimodules!). Show that A @Z B is again a finite abelian group, and say how to calculate it.
1b. Can you extend this to a natural transformation between two functors between something-or-other? This question is intentionally ill-defined, and I imagine different people will experience different creative urges.
2. Let A,B be right R-modules and C,D left R-modules. Let + denote direct sum. Define isomorphisms (A + B) @R C ~ A@RC + B@RC and B @R (C + D) ~ B@RC + B@RD.
...a very dull question, but it leads to...
3. Let Mn(C) denote the ring of nxn complex matrices, and Ck x n denote the (Mk(C),Mn(C))-bimodule of kxn complex matrices. Show that matrix multiplication
induces a map
and that this map is an isomorphism.
4. An isomorphism D: R -> Rop is called an anti-automorphism of R. For example, the identity function is an anti-automorphism iff R is commutative.
a. Let MR be a right R-module. Use D to define a left module structure on it.
b. Let RN be a left R-module. Using D, we can make N* := HomR(N,R) a left R-module. Using D again, we can make (N*)* a left R-module. Find a condition on D that guarantees that this new left R-module structure on N** agrees with the old one.
c. Show the converse: if this new left R-module structure on N** agrees with the old one for all N, then D satisfies that condition.
b. Even if D is just an anti-homomorphism, not necessarily invertible, say how to use it to turn any right R-module into a left R-module.
c. Let M be a left R-module, which we turn into a right module, then into a left module again. Determine the condition on D under which this new module structure agrees with the old one (for arbitrary M).
5. Forgetting module theory for a moment, an inner product < , > on a real vector space V induces an isomorphism V -> V*, taking v |-> (w |-> < v,w >). This looks weird to us now, in that it's an isomorphism of a left module with a right module.
a. Use question 4 to state the definition of an inner product on a real vector space, being careful about left vs. right modules. (This being an algebra class, you can skip nonalgebraic conditions like "positive definite".)
b. Use question 4 to define a Hermitian inner product on a complex vector space, being careful about left vs. right modules.
6. Let A,B be two finite abelian groups (aka (Z,Z)-bimodules!). Show that A @Z B is again a finite abelian group, and say how to calculate it.
posted by Allen Knutson @ 7:18 AM
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