Let A be the ring obtained by localizing the polynomial ring κ[X, Y, Z, W ] over a field κ at the maximal ideal (X, Y, Z, W) and modulo the ideal (XW − Y Z). Let p be the ideal of A generated by X and Y. We study the module structure of a minimal injective resolution of A/p in details using local cohomology. Applications include the description of Ext i A (M, A/p), where M is a module construct...

Let A be the ring obtained by localizing the polynomial ring κ[X, Y, Z, W ] over a field κ at the maximal ideal (X, Y, Z, W) and modulo the ideal (XW − Y Z). Let p be the ideal of A generated by X and Y. We study the module structure of a minimal injective resolution of A/p in details using local cohomology. Applications include the description of Ext i A (M, A/p), where M is a module construct...

Let A be the ring obtained by localizing the polynomial ring κ[X,Y, Z,W ] over a field κ at the maximal ideal (X, Y, Z,W ) and modulo the ideal (XW − Y Z). Let p be the ideal of A generated by X and Y . We study the module structure of a minimal injective resolution of A/p in details using local cohomology. Applications include the description of Ext A (M,A/p), where M is a module constructed b...

We present an Avoidance Principle for certain graded rings. As an application we fill a gap in the proof of a result by Brodmann, Rohrer and Sazeedeh about the antipolynomiality of the Hilbert-Samuel multiplicity of the graded components of the local cohomology modules of a finitely generated module over a Noetherian homogeneous ring with two-dimensional local base ring. Let R = ⊕ n∈N0 Rn be a ...

We study a generalization of the Canonical Element Conjecture. In particular we show that given a nonregular local ring (A, m) and an i > 0, there exist finitely generated A-modules M such that the canonical map from ExtA(M/mM,Syzi(M/mM)) to Hi m (M,Syzi(M/mM)) is nonzero. Moreover, we show that even when M has infinite projective dimension and i > dim(A), studying these maps sheds light on the...

In the theory of deformation of Okamoto-Painlevé pair (S, Y ), a local cohomology group H D (ΘS(− logD)) plays an important role. In this paper, we estimate the local cohomology group of pair (S, Y ) for several types, and obtain the following results. For a pair (S, Y ) corresponding to the space of initial conditions of the Painlevé equations, we show that the local cohomology group H D (ΘS(−...

Let A be a (G, χ)-Hopf algebra with bijection antipode and let M be a G-graded A-bimodule. We prove that there exists an isomorphism HH∗gr(A,M) ∼= Ext ∗ A-gr(K, (M)), where K is viewed as the trivial graded A-module via the counit of A, M is the adjoint A-module associated to the graded A-bimodule M and HHgr denotes the G-graded Hochschild cohomology. As an application, we deduce that the cohom...

We prove that over a commutative noetherian ring the three approaches to introducing depth for complexes: via Koszul homology, via Ext modules, and via local cohomology, all yield the same invariant. Using this result, we establish a far reaching generalization of the classical AuslanderBuchsbaum formula for the depth of finitely generated modules of finite projective dimension. We extend also ...

The face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is nonsingular, i.e., has the homology of a wedge of spheres of the expected dimension. This is derived from an enumerative result for local cohomology of face rings modulo generic linear forms, as compared with local cohomolog...

For a Specht module S for the symmetric group Σd, the cohomology H (Σd,S ) is known only in degree i = 0. We give a combinatorial criterion equivalent to the nonvanishing of the degree i = 1 cohomology, valid in odd characteristic. Our condition generalizes James’ solution in degree zero. We apply this combinatorial description to give some computations of Specht module cohomology, together wit...