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Hochsters theta invariant is defined for a pair of finitely generated modules on a hypersurface ring having only an isolated singularity. Up to a sign, it agrees with the Euler invariant of a pair of matrix factorizations. Working over the complex numbers, Buchweitz and van Straten have established an interesting connection between Hochsters theta invariant and the classical linking form on the link of the singularity. In particular, they establish the vanishing of the theta invariant if the hypersurface is even-dimensional by exploiting the fact that the (reduced) cohomology of the Milnor fiber is concentrated in odd degrees in this situation. In this paper, we give purely algebra
Let $R=Bbbk[x_1,dots,x_n]$ be a polynomial ring over a field $Bbbk$ and let $Isubset R$ be a monomial ideal preserved by the natural action of the symmetric group $mathfrak S_n$ on $R$. We give a combinatorial method to determine the $mathfrak S_n$-m
We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals associated to nu
We give an elementary proof of Grothendiecks non-vanishing Theorem: For a finitely generated non-zero module $M$ over a Noetherian local ring $A$ with maximal ideal $m$, the local cohomology module $H^{dim M}_{m}(M)$ is non-zero.
Using vanishing of graded components of local cohomology modules of the Rees algebra of the normal filtration of an ideal, we give bounds on the normal reduction number. This helps to get necessary and sufficient conditions in Cohen-Macaulay local ri
We study the first Hilbert coefficient (after the multiplicity) $e_1$ of a local ring $(A,m). $ Under various circumstances, it is also called the {bf Chern number} of the local ring $A.$ Starting from the work of D.G. Northcott in the 60s, several r