Math 510 S26

Math 510B Spring 2026

Instructor: C. Negron
Time and location: 12:00-12:50 pm, KAP138
Office: KAP 444C
Office hours: Fridays
Books: Part I) Finite dimensional algebras, Drozd and Kirichenko.
Part II) Introduction to commutative algebra, Atiyah and Macdonald and/or Stacks project Ch 10.
Part III) Back to Finite dimensional algebras, Drozd and Kirichenko.
TA: Tianle Liu

Syllabus here.

Lectures 1_algmod.pdf, 2_findim.pdf, 3_ssmods.pdf, 4_maschkaw.pdf, 5_jrad.pdf, 6_genrel.pdf

-- HWs, midterms, etc.

Quasi-weekly HWs, four or so quizzes, one midterm, one final.

-- How to do HW :)

Step 1: Read the problems.
Step 2: Try to do the problems.
Step 3: Succeed in doing some of the problems, and write them down, but fail to do others.
Step 4: Meet with your friends to share your collective insights.
Step 5: Ask the professor if anything remains unclear.
Step 6: Complete your homework.

Never: Search for solutions online/through an LLM.

-- Topics schedule

Part I) Algebras and modules → chain conditions → Jordan-Holder theorem → semisimplicity → Jacobson radical → Artin-Wedderburn → Maschke theorem → Artinian implies Noetherian

Part II) Noetherian rings → Polynomial rings and finite type algebras → Spectra and Zariski topology → nullstellensatz → localization and primes → nilradicals → finite extensions

Part III) Bimodules → tensoring and Hom functors → ⊗-Hom adjunction → Morita theorem → Projectives, injectives, and flat modules → constructing injectives

-- Homework

Report (presumed) errors via email or ask in class. To be turned in in class. Don't turn in garbage.

When writing up the HW, summarize the question as an introduction to each problem.

HW 5 (Due March X)

1. For an Artinian ring A, and any (possibly non-finitely generated) M, prove that Jac(A)·M = M if and only if M = 0. This is to say, a stronger version of Nakayama holds in the Artinian setting.

2. For any map of rings A → B prove that the base change functor B⊗_A-: A-mod → B-mod is right exact. Explicitly, if 0 → M' → N → M → 0 is an exact sequence of A-modules prove that the resulting sequence
B⊗_AM' → B⊗_AN → B⊗_AM → 0
maps B⊗_AN surjectively onto B⊗_AM, and sends B⊗_AM' surjectively onto the kernel of the map B⊗_AN → B⊗_AM. Prove by way of example that, for a general map A → B, the functor B⊗_A- does not preserve injections.

3. For a general ring A, and a map f:M → N between finitely generated A-modules, prove that f is surjective if and only if the induced map
f̄: M/Jac(A)·M → N/Jac(A)·N
is surjective.

4. Suppose A is Artinian, and consider any finitely generated A-module M. Prove that M is semisimple if and only if Jac(A)·M = 0. [This is true for non-finitely generated M as well, but let us not concern ourselves with this.]

5. Prove that every simple representation of S₃ over k = F₃^alg is 1-dimensional, and classify all such simples. [Hint: You already know that there are two such 1-dimensional simples. So, you know something about the dimension of kS₃/Jac(kS₃).]

.6 For k = F₃^alg, describe the Jacobson radical Jac(kS₃). Determine, at least, its dimension.

HW 4 (Due Feb X)

1. (a) Let P be a projective A module, and suppose that P decomposes as a sum P ≅ M⊕M' for modules M and M'. Prove that M and M' are also projective.
(b) Prove that a module M is projective if and only if M appears as a summand of a free module ⊕_λ∈ΛA ≅ M⊕M'.

2. For any subgroup H ⊆ G, and any field k (or even commutative ring), prove that kG is projective over kH. In particular, kG is free over kH.

3. Let k be a field. For any subgroup H ⊆ G, prove that kG is non-semisimple whenever kH is non-semisimple.

4. For k of characteristic p>0, prove that kℤ/pℤ is non-semisimple. You can do this, for example, by producing a non-split extension
0 → k → M → k → 0
for the trivial module k = k_triv.

5. Let G be a finite group and k be a field of finite characteristic p. Prove that kG is non-semisimple if and only if p|order(G).

6. Provide an example of a division ring D along with a central embedding from a field k → D for which the base change k^alg⊗_kD is non-semisimple. Here one can already think, for example, of the case where D is a field and k is a subfield.

HW 3 (Due Feb 13)

1. For the regular A module, prove that the right action of A on itself provides a ring isomorphism A^op → End_A(A).

2. (a) Prove that ℤ/nℤ has precisely n non-isomorphic 1-dimensional (simple) representations L_i over C.
(b) Prove that each simple 1-dimensonal representation L_i admits an injective Cℤ/nℤ-module map L_i → Cℤ/nℤ, and that this map is unique up to scaling.
(c) Provide an isomorphism of Cℤ/nℤ-modules
⊕_i=1ⁿ L_i → Cℤ/nℤ.
(d) Prove that every finite-dimensional ℤ/nℤ-representation M over C decomposes as
M ≅ ⊕_i=1ⁿ m_i·L_i
where each m_i=[L_i:M].

3. Suppose that a group G acts on a k-algebra A by algebra automorphisms, *: G×A → A. Endow the vector space A⋊G = A⊗_kkG with the uniquue bilinear map ·: A⋊G × A⋊G → A⋊G which appears on monomials as
(a⊗g)·(a'⊗g') =a(g*a')⊗gg'.
Prove that A⋊G is a ring.

4. Let q ∈ C^× be an n-th root of unity, and consider the ℤ/nℤ-action on C[x] by the algebra automorphisms ā·p(x) = p(q^a·x). Take A_q = C[x]⋊ℤ/nℤ.
(a) Prove that M_r = A_q/A_q·x^r is a free module over Cℤ/nℤ of rank r. Give a basis for M_r over Cℤ/nℤ.
(b) Calculate Rad(M_r), M_r/Rad(M_r), and soc(M_r).

5. For any finite length semisimple module M over a ring A, prove that there are division rings D₁,..., D_t and integers n₁,..., n_t for which
End_A(M) ≅ ∏_i=1^t M_{n_i}(D_i)
as rings.

HW 2 (Due Wed Feb 4)

1. Let k be a field of characteristic other than 2 or 3. Prove that the quotient module L(2) = k³/image(k_triv) of the permutation module over kS₃ along the inclusion k_triv → k³, 1 ↦ e₁+e₂+e₃, is a simple kS₃-module. [Remark: As pointed out in class, this representation is actually semisimple in characteristic 2 as well. However, let us not worry about this case.]

2. For k as in question 1, prove that the action map kS₃ → End_k(L(2)) is surjective. In particular, observe that the 2 x 2 matrix ring appears as a quotient of kS₃.

3. (a) Prove that any PID is Noetherian.
(b) Prove that ℤ and k[x], for k a field, are Noetherian but not Artinian.

4. For distinct irreducible poynomials p₁,..., p_r in Q[x], and any integers m_i>0, take
M = Q[x]/(p₁^m₁ ⋯ p_r^m_r).
For α in the algebraic closure Q^alg, prove that [Q(α):M] > 0 if and only if p_i(α) = 0 at some i, and in this case [Q(α):M] = m_i.
[Remark: Here each Q(α) is considered as a Q[x]-module via the action f(x)·ζ=f(α)ζ.]

5. (a) For any finite-dimensional C[x]-module M, prove that length(M) = dim_C(M).
(b) Prove that there are finite-dimensional F₅S₃-modules for which length(M) < dim_F₅(M).

HW 1 (Due Mon Jan 26) [Turn in to Tianle Liu's mailbox]

1. Let k be a commutative ring and A be a k-algebra. For any additive group M, prove that an A-module structure on M is equivalent to the choice of a k-module structure on M and a k-algebra map φ: A → End_k(M).

2. For any field k, prove that the only ideals in M_n(k) are 0 and M_n(k) itself. [Hint: Consider row and column reduction.]

3. Let k be a field.
(a) Prove that the standard module 𝕍 = kⁿ over M_n(k) is simple.
(b) Prove that End_{M_n(k)}(𝕍) = k·id_𝕍. [Hint: Think about eigenvectors for the actions of the diagonals.]

4. Classify all simple modules over Q[x].

5. For any A-module M define the annihilator Ann_A(M) ⊆ A as the subset consisting of all a in A for which a·M=0. Show that Ann_A(M) is an ideal in A.

6. For any ideal I in an algebra A, with corresponding quotient map π: A → A/I, prove that the restriction functor
res_π: A/I-mod → A-mod
is fully faithful. Prove, furthermore, that this functor is an equivalence onto the full subcategory of A-modules which are annihilated by I.

7. (a) For any ideal I ⊆ A and A-module M, prove that the span I·M of all elements of the form x·m with x in I and m in M is a submodule in M.
(b) Prove that the quotient map π_M: M → M/I·M induces an isomorphism of A/I-modules (A/I)⊗_AM ≅ M/I·M. [Hint: Prove that π_M has the requisite universal property.]