On the realization space of the cube Adiprasito, Karim; Kalmanovich, Daniel; Nevo, Eran
Journal of the European Mathematical Society : JEMS,
01/2024, Volume:
26, Issue:
1
Journal Article
We prove that in the polynomial ring Q=kx,y,z,w, with k an algebraically closed field of characteristic zero, all Gorenstein homogeneous ideals I satisfying the inclusions (x,y,z,w)4⊆I⊆(x,y,z,w)2 can ...be obtained by doubling from a grade three perfect ideal J⊂I such that Q/J is a locally Gorenstein ring. Moreover, a graded minimal free resolution of the Q-module Q/I can be completely described in terms of a graded minimal free resolution of the Q-module Q/J and a homogeneous embedding of a shift of the canonical module ωQ/J into Q/J.
In their paper 1, H. Ananthnarayan, L. Avramov, and W.F. Moore introduced a connected sum construction for local Gorenstein rings A,B over a local Gorenstein ring T, which, in the graded Artinian ...case, can be viewed as an algebraic analogue of the topological construction of the same name. We give two alternative descriptions of this algebraic connected sum: the first uses algebraic analogues of Thom classes of vector bundles and Gysin homomorphisms, the second is in terms of Macaulay dual generators. We also investigate the extent to which the connected sum of A,B over an Artinian Gorenstein algebra T preserves the weak or strong Lefschetz property, thus providing new classes of rings which satisfy these properties.
We show that under mild conditions, the connected sum
$M\# N$
of simply connected, closed, orientable n-dimensional Poincaré Duality complexes M and N is hyperbolic and has no homotopy exponent at ...all but finitely many primes, verifying a weak version of Moore’s conjecture. This is derived from an elementary framework involving
$CW$
-complexes satisfying certain conditions.
In 2012, Ananthnarayan, Avramov and Moore give a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. Given a Gorenstein ring, one would like to know ...whether it decomposes as a connected sum and if so, what are its components. We answer these questions in the Artinian case and investigate conditions on the ring which force it to be indecomposable as a connected sum. We further give a characterization for Gorenstein Artin local rings to be decomposable as connected sums, and as a consequence, obtain results about its Poincaré series and minimal number of generators of its defining ideal. Finally, we show that the indecomposable components appearing in the connected sum decomposition are unique up to isomorphism.
We record an answer to the question “In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?”. In the process of doing so, we are ...naturally led to ask “For which values of ℓ is the connected sum of ℓ closed almost complex manifolds necessarily an almost complex manifold?”. We answer this question, along with its non-compact analogue, using obstruction theory and Yang's results on the existence of almost complex structures on (n−1)-connected 2n-manifolds. Finally, we partially extend Datta and Subramanian's result on the nonexistence of almost complex structures on products of two even spheres to rational homology spheres by using the index of the twisted spinc Dirac operator.
Let
W
be a closed area enlargeable manifold in the sense of Gromov-Lawson and
M
be a noncompact spin manifold, the authors show that the connected sum
M#W
admits no complete metric of positive scalar ...curvature. When
W = T
n
, this provides a positive answer to the generalized Geroch conjecture in the spin setting.
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