Rings and Algebras
-
0 votes 0 replies 1 views
Paper The ℓ¹-Bass conjecture for all discrete groups
Proves the ℓ1-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over \ell^1(G) are supported on finitely many finite-order conjugacy classes. The algebraic companion proves…
-
0 votes 0 replies 0 views
Paper A torsion-free group algebra that is not directly finite
Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽2 is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective…
-
0 votes 0 replies 0 views
Paper Integral counterexamples to Gersten’s conjecture
Disproves unrestricted integral Gersten injectivity in degrees 3 and 5. Two explicit two-dimensional ramified regular local rings of mixed characteristic (0,5) have nonzero integral K-theory classes that vanish over…
-
0 votes 0 replies 0 views
Paper Foulkes' conjecture for sixth powers and quadratic stabilization
Proves the sixth case of Foulkes’ conjecture: \mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^bV) embeds equivariantly in \mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6V)…
-
0 votes 0 replies 0 views
Paper Donovan's conjecture over fields and complete mixed-characteristic DVRs
Proves Donovan's conjecture: over each fixed algebraically closed field of characteristic p, blocks of finite groups with bounded defect-group order have only finitely many Morita-equivalence classes, for every prime p.…
-
0 votes 0 replies 0 views
Paper Finite symmetric tensor categories and the Verlinde tower
Proves that every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal fiber functor to a higher Verlinde category…
-
0 votes 0 replies 0 views
Paper Tensor saturation for even spin groups
Proves saturation factor one for \mathop{\mathrm{Spin}}\nolimits (2n), n ≥ 2: for three dominant integral weights whose sum lies in the root lattice, an invariant at any common positive integral dilation already…
-
0 votes 0 replies 0 views
Paper Saxl’s conjecture and universal tensor squares
Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every Sn with n\notin\{2,4,9\}…
-
0 votes 0 replies 0 views
Paper Finite lattice representation and undecidability
Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively. Moreover, no algorithm decides whether a finite lattice has such a representation,…
-
0 votes 0 replies 0 views
Paper Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures
Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander–Reiten and Gorenstein-projective conjectures, and Tachikawa's second conjecture. An associated…
-
0 votes 0 replies 0 views
Paper Eisenbud–Green–Harris and lex-plus-powers
Proves the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field. Any homogeneous ideal containing a regular sequence, of arbitrary length and degrees at least two, admits a…
-
0 votes 0 replies 0 views
Paper A counterexample to Kurosh’s division-ring problem
Constructs a countable characteristic-zero division ring that is algebraic over its center and generated by two elements over that center, but has infinite dimension over it. This answers Kurosh's division-ring problem…
-
0 votes 0 replies 0 views
Paper The blockwise Alperin weight conjecture
Proves the numerical blockwise Alperin weight conjecture for every prime and every finite group: the number of irreducible Brauer characters in a block equals the number of conjugacy classes of its weights. Manuscript…
-
0 votes 0 replies 0 views
Paper Lech’s multiplicity conjecture
Proves e(R)\le e(S) for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic. () Manuscript…
-
0 votes 0 replies 0 views
Paper Serre’s intersection-multiplicity conjecture
Proves strict positivity of Serre's intersection multiplicity \chi^R(M,N) for nonzero finitely generated modules over any regular local ring, provided M\otimesR N has finite length and \dim M+\dim N=\dim R.…
-
0 votes 0 replies 0 views
Paper A counterexample to the small Cohen–Macaulay module conjecture
Constructs a three-dimensional complete Noetherian normal local domain over ℂ with no nonzero finitely generated maximal Cohen–Macaulay module. A three-dimensional local domain essentially of finite type over ℂ has the…
-
0 votes 0 replies 0 views
Paper A counterexample to Kaplansky’s zero-divisor conjecture
Constructs a finitely presented torsion-free group G whose group algebra \mathbb F2[G] has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying…
-
0 votes 0 replies 0 views
Paper A counterexample to finitistic-dimension finiteness
Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture. () Manuscript September 23, 2026 ·…
18 posts