Results on finite-dimensional algebras, selection, Verdier quotients, bimodule complexes and trivial extensions, with their assumptions

The course proves the statements below; each link leads to the statement with its assumptions and its proof. The lessons are self-checked by the writing AI; independent mathematical review is not complete.

• Nakayama's lemma — Lemma 2.3. A finitely generated module M over a finite-dimensional algebra A with (rad A)M = M is zero.
• The semisimple quotient — Proposition 2.5. A/rad A is semisimple, M/(rad A)M is semisimple for finitely generated M, and every simple module is (A/rad A)e for an idempotent e.
• Lifting idempotents — Lemma 3.1. Modulo a nilpotent ideal, idempotents lift by an explicit binomial formula.
• Projective covers — Proposition 3.2. Every finitely generated module over a finite-dimensional algebra has a projective cover with kernel in the radical.
• Projective dimension from the top — Theorem 4.3. For finitely generated N, pd N = sup{i : Tor_i(A/rad A, N) != 0}, computed by a minimal resolution.
• A finite presentation — Lemma 2.1. The group with generators T_l, U_i(r), V_i(r), W_ij(r) and relations (G1)-(G5) is finitely presented on fifteen generators.
• Central involutions — Lemma 3.1. z_N = [U_i(a), V_i(b)] with a + b = N does not depend on i, a, b, is central, and z_N^2 = 1.
• Matrix quotients — Lemma 4.1. The group maps onto finite subgroups of GL_5(F_2[t]/(t^m - 1)) in which z_0, ..., z_(m-1) are independent central involutions.
• Selection with unbounded extinction (OpenAI) — Proposition 5.1. For R = CG, e = (1 + z_0)/2 and the shift alpha there are finite-dimensional modules Y_m with H^j(Y_m) != 0 for j < m and H^m(Y_m) = 0, where H(Y) is eY twisted by alpha.
• Quadratic presentations — Lemma 1.1. Every finitely presented algebra has a presentation whose relations have degree at most two.
• Directed algebras — Proposition 2.2. A directed algebra with n idempotents has global dimension at most n - 1 on both sides; the three-vertex algebra B has global dimension at most two.
• The quotient action — Lemma 2.2. In the Verdier quotient by the cones of s and s', x_h -> s^(-1)a_h defines a unital homomorphism from R to the endomorphisms of e_0 B.
• Evaluation — Proposition 3.2. Tensoring with M(Y) factors through the Verdier quotient and recovers the R-action on Y.
• The odd double — Lemma 4.3. For an idempotent of an object of a triangulated category with image U in the idempotent completion, U + U[1] and U + U[3] are isomorphic to objects of the category.
• A diagram for the selection — Proposition 5.4. A fixed B-diagram in the Verdier quotient evaluates at every Y to eY + (eY)[3], with the arrows acting through alpha.
• Diagram lifting — Lemma 2.1. Every B-diagram in the Verdier quotient is isomorphic to the image of a B-diagram in the homotopy category.
• Fixed lifted diagram — Proposition 3.1. Complexes D_i, chain maps f_a and homotopies h_rho, chosen before Y, whose evaluation at every Y is the selection diagram.
• The three-column complex — Lemma 1.1. The arrows, the maps f_a and the homotopies h_rho form one bounded complex P of B-bimodules, projective on the right.
• Vertices — Lemma 2.1. Each D_i is homotopy equivalent to e_i P.
• Realization of selection (OpenAI) — Theorem 3.1. P (x)^L_B M(Y) = M(H(Y)) + M(H(Y))[3] in D(B) for every finite-dimensional R-module Y.
• Iterates — Corollary 3.2. The j-th iterate of P (x)^L_B - on M(Y) is a sum of binomially many shifts of M(H^j(Y)).
• Ordinary simulation (OpenAI) — Proposition 3.1. D = B x (B (x) C) has finite global dimension, and two steps of F = X (x)^L_D - on the first factor are P[b_*] (x)^L_B -.
• Even iterates — Corollary 3.2. F^(2j) M(Y) is a sum of shifts of M(H^j(Y)).
• Bar decomposition (OpenAI) — Proposition 2.2. For a D-module N inflated to the trivial extension A = D x X, D (x)^L_A N is the sum over r of (F^r N)[r].
• Detecting projective dimension (OpenAI) — Corollary 3.2. If D has finite global dimension on both sides: F^t N = 0 gives pd_A N finite, and F^r N != 0 gives pd_A N >= r.
• Infinite little finitistic dimension (OpenAI) — Theorem 4.1. A finite-dimensional complex algebra A with modules N_m satisfying 2m - 2 <= pd N_m < infinity; findim A = Findim A = infinity.
• Left-right asymmetry (OpenAI, under named results) — Corollary 5.1. Under the named statements (CUM), (RIC) and (MOR): an algebra with both finitistic dimensions infinite on the left and zero on the right.

Reading order

Read the lessons in order; each uses the constructions of the previous ones. The first lesson is also a self-contained introduction to minimal resolutions over finite-dimensional algebras.