Algebra

We prove Serre's positivity conjecture, including ramified mixed characteristic. If two nonzero finitely generated modules over a regular local ring have tensor product of finite length and complementary dimensions, then their intersection multiplicity is strictly positive.

We prove Lech's multiplicity conjecture: Hilbert–Samuel multiplicity cannot decrease under a flat local homomorphism of nonzero Noetherian local rings. The result holds in arbitrary dimension, with no restrictions on the residue fields or characteristics.

We construct a three-dimensional complete Noetherian normal local domain containing ℂ, with residue field ℂ, that has no nonzero finitely generated maximal Cohen–Macaulay module. This disproves the domain form of the small Cohen–Macaulay module conjecture.

We disprove Kaplansky's zero-divisor conjecture by constructing a finitely presented torsion-free group G for which \(\mathbb F_2[G]\) has nonzero zero divisors. The group admits a finite two-dimensional classifying space.

We construct a finitely presented torsion-free counterexample to Kaplansky's direct finiteness conjecture over the field of two elements. The group admits a finite two-dimensional classifying complex.

We construct a counterexample to Kaplansky's direct-finiteness conjecture in odd characteristic. For one specified odd prime p, we obtain a field K of order p⁴, a finitely generated group G containing torsion, and finite sums \(a,b\in K[G]\) with \(ab=1\) but \(ba\ne1\). The same elements define a cellular automaton on K^G that is injective but not surjective.

We disprove the unrestricted group-ring Determinant Conjecture. We construct a finitely generated group G and a square matrix over \(\mathbb Z[G]\) that is invertible over \(\mathbb Q[G]\) and has Fuglede–Kadison determinant strictly between zero and one. The logarithmic integral defining the determinant is finite.

We disprove Kaplansky's direct-finiteness conjecture by constructing a finite field K of characteristic two, a finitely presented group G, and finite sums \(a,b\in K[G]\) with \(ab=1\) but \(ba\ne1\). The group G is nonsofic. The same elements define a cellular automaton on K^G that is injective but not surjective, disproving Gottschalk's surjunctivity conjecture.

We construct a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.

We disprove the Auslander–Reiten conjecture for Artin algebras. We construct a finite-dimensional algebra Λ over \(k=\mathbb F_2(q,H_1,H_2)\) and a finite-dimensional nonprojective left module Z such that \(\mathop{\mathrm{Ext}}\nolimits ^i_\Lambda(Z,Z)=\mathop{\mathrm{Ext}}\nolimits ^i_\Lambda(Z,\Lambda)=0\) for every i > 0. The module is Gorenstein-projective, so the same example also disproves the Gorenstein-projective conjecture. Both counterexamples persist after every extension of k.

We disprove Tachikawa's second conjecture by constructing a finite-dimensional symmetric algebra over \(k=\mathbb F_2(q,H_1,H_2)\) with a finite-dimensional nonprojective module whose self-extension groups vanish in every positive degree. The associated endomorphism algebra also gives counterexamples to the classical, generalized, and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. These conclusions persist after every extension of k.

We prove the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field for homogeneous regular sequences of any positive length, with degrees at least two. For every homogeneous ideal containing such a sequence, the corresponding lex-plus-powers ideal has the same Hilbert function and at least as large a graded Betti number in every homological and internal degree.

We prove the Eisenbud–Green–Harris conjecture over every characteristic-zero field for homogeneous regular sequences of any positive length, with degrees at least two. Thus every homogeneous ideal containing such a sequence has the Hilbert function of a monomial ideal containing the corresponding pure powers.

We construct a countable division ring of characteristic zero that is algebraic over its center, generated by two elements as an algebra over that center, and infinite-dimensional over it. This gives a negative answer to the Kurosh problem for division rings.

We prove the numerical blockwise Alperin weight conjecture for every finite group and every prime p. For each p-block B, the number of irreducible Brauer characters in B equals the number of conjugacy classes of B-weights.

We prove integral Donovan finiteness: for every prime p and positive integer M, the blocks of all finite groups with defect groups of order at most M have only finitely many Morita equivalence classes over \(W(\overline{\mathbb F}_p)\). The defect groups need not be abelian, and the result includes p = 2. The same bounded-defect finiteness holds over each fixed complete discrete valuation ring of characteristic zero with algebraically closed residue field of characteristic p, including ramified rings.

We prove Donovan's conjecture over every algebraically closed field K of characteristic p > 0. For each fixed K and bound on defect-group order, blocks of finite groups represent only finitely many K-linear Morita equivalence classes. The defect groups need not be abelian, and the result includes p = 2.

We prove the saturation conjecture for \(\mathop{\mathrm{Spin}}\nolimits (2n)\), n ≥ 2. If three dominant integral weights sum to an element of the root lattice, then the existence of a nonzero tensor invariant after a positive integral dilation implies the existence of one at the original weights.

For every positive integer n other than 2, 4, and 9, we prove that some irreducible complex representation of S_n has a tensor square containing every irreducible representation. This resolves the tensor square conjecture for symmetric groups affirmatively.

For every staircase partition, we prove that the tensor square of the corresponding irreducible complex representation of the symmetric group contains every irreducible representation of that group. This proves Saxl's conjecture.

We give an explicit colored-graph characterization of the finite nonempty lattices that occur as full congruence lattices of finite algebras, and prove that deciding this representation property is undecidable. In particular, the finite lattice representation problem has a negative answer. We also prove that recognition of full subgroup intervals in finite groups is undecidable.

We give a negative solution to the finite lattice representation problem. We prove that there is a finite nonempty lattice that is not the full congruence lattice of any finite nonempty algebra of any finite signature.

We prove the ℓ¹-Bass conjecture for every discrete group. The Hattori–Stallings trace of every idempotent matrix over the complex ℓ¹ group algebra is supported on finitely many conjugacy classes of finite-order elements.

We prove the complex group-ring Bass trace conjecture for every discrete group: the Hattori–Stallings trace of a finitely generated projective module over its complex group ring is supported on conjugacy classes of finite-order elements. As a consequence, for every torsion-free group G and every commutative unital domain R of characteristic zero, the only idempotents in \(RG\) are 0 and 1. This proves Kaplansky's idempotent conjecture in characteristic zero.

We prove the finite case of the Benson–Etingof–Ostrik conjecture: every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal functor to a finite higher Verlinde category \(\mathop{\mathrm{Ver}}\nolimits _{p^n}(k)\). The result includes characteristic two, and the level n may depend on the category.

We construct a two-dimensional ramified regular local ring A in mixed characteristic \((0,5)\) for which the integral map \(K_3(A)\to K_3(\mathop{\mathrm{Frac}}\nolimits A)\) has nonzero kernel. This gives a negative answer to the unrestricted integral Gersten conjecture.

We construct a two-dimensional ramified regular local ring A of mixed characteristic \((0,5)\) for which \(K_5(A)\to K_5(\mathop{\mathrm{Frac}}\nolimits A)\) has a nonzero kernel. This disproves Gersten's conjecture in its unrestricted integral form.

For every finite-dimensional complex vector space V, we prove that the canonical Foulkes–Howe map \(\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^a V)\longrightarrow\mathop{\mathrm{Sym}}\nolimits ^a(\mathop{\mathrm{Sym}}\nolimits ^b V)\) is surjective whenever a ≥ 2 and \(b\ge a(a-1)\). This gives a quadratic stabilization bound independent of \(\dim V\).

We prove the sixth-symmetric-power case of Foulkes' conjecture. For every integer b ≥ 6 and every finite-dimensional complex vector space V, there is a \(\mathop{\mathrm{GL}}\nolimits (V)\)-equivariant injection \(\mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^b V)\hookrightarrow\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6 V)\).