Free resolutions for differential modules over differential algebras

A free resolution (R, d + h) → (M, d) for a DG-module (M, d) over a DG-algebra (A, d) is constructed in the sense of a perturbation of the differential in a free bigraded resolution (R, d) → M of the underlying graded module M over an underlying graded algebra A. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.     

Permutation resolutions for Specht modules

For every composition λ of a positive integer r, we construct a finite chain complex whose terms are direct sums of permutation modules M ^μ for the symmetric group \(\mathfrak{S}_{r}\) with Young subgroup stabilizers \(\mathfrak{S}_{\mu}\). The construction is combinatorial and can be carried out over every commutative base ring k. We conjecture that for every partition λ the chain complex has homology concentrated in one degree (at the end of the complex) and that it is isomorphic to the dual of the Specht module S ^λ . We prove the exactness in special cases.     

Free resolutions for multiple point spaces

This paper relates the multiple point spaces in the source and target of a corank 1 map-germ ${(\mathbb {C}^n, 0)\to(\mathbb {C}^{n+1}, 0)}$ . Let f be such a map-germ, and, for 1 ≤ k ≤ multiplicity( f ), let D ^k ( f ) be its k’th multiple point scheme – the closure of the set of ordered k-tuples of pairwise distinct points sharing the same image. There are natural projections D ^k+1( f ) → D ^k ( f ), determined by forgetting one member of the (k + 1)-tuple. We prove that the matrix of a presentation of ${\mathcal {O}_{D^{k+1}(f)}}$ over ${\mathcal {O}_{D^k(f)}}$ appears as a certain submatrix of the matrix of a suitable presentation of ${\mathcal {O}_{\mathbb {C}^n,0}}$ over ${\mathcal {O}_{\mathbb {C}^{n+1},0}}$ . This does not happen for germs of corank > 1.     

Mukai flops and deformations of symplectic resolutions

We prove that two projective symplectic resolutions of are connected by Mukai flops in codimension 2 for a finite sub-group G <Sp(2n). It is also shown that two projective symplectic resolutions of are deformation equivalent.     

Representations of matroids and free resolutions for multigraded modules

Let k be a field, let R=k[x₁,…,xm] be a polynomial ring with the standard Zm-grading (multigrading), let L be a Noetherian multigraded R-module, and let be a finite free multigraded presentation of L over R. Given a choice S of a multihomogeneous basis of E, we construct an explicit canonical finite free multigraded resolution T_•(Φ,S) of the R-module L. In the case of monomial ideals our construction recovers the Taylor resolution. A main ingredient of our work is a new linear algebra construction of independent interest, which produces from a representation ? over k of a matroid M a canonical finite complex of finite dimensional k-vector spaces T_•(?) that is a resolution of Ker?. We also show that the length of T_•(?) and the dimensions of its components are combinatorial invariants of the matroid M, and are independent of the representation map ?.     

Characteristic-free resolutions of Weyl and Specht modules

We construct explicit resolutions of Weyl modules by divided powers and of co-Specht modules by permutational modules. We also prove a conjecture by Boltje and Hartmann (2010) [7] on resolutions of co-Specht modules.     

A family of cohen-macaulay modules over singularities of type xt + y3

Let K be a field with char K ≠ 3 and it two positive integers such that 1 ≤i <t/2,t ≠ 3i. The classification problem for maximal Cohen-Macaulay modules over K[[X,Y]]/(X^t+Y³ ) is complicated if t≥ 6, because there exist parameter families of non-isomorphic maximal Cohen-Macaulay modules [Sc], or [GK], [Yo, Ch.9] and [DG]). Here we describe parameter families of such modules N, such that N/YN is a direct sum of copies of K[[X]]/(X ⁱ)K[[X]]/(X^t-i ).     

An Alexandrov topology for maximal Cohen-Macaulay modules

Using the theory of cohomology annihilators, we define a family of topologies on the set of isomorphism classes of maximal Cohen-Macaulay modules over a Gorenstein ring. We study compactness of these topologies.     

Free ordered modules

We establish the necessary and sufficient condition on a partially ordered set S such that a free ordered R-module (R is a linearly ordered ring without divisors of zero) over the set S is o-isomorphic with a free ordered R-module over a trivially ordered set.Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 477–487, October, 1972.