Mathematics

We search some splitting (resp. finiteness) criteria by applying certain splitting (resp. finiteness) property of the cohomological functors. We present a connection from our approach to the realization problem of Nunke. This equipped with several applications. For instance, we recover some results of Jensen (and others) by applying simple methods. Additional applications include a computation of some numerical invariants such as the projective dimension of some injective modules and et cetera.

We introduce the notions of Koszul N -complex, C ˇ ech N -complex and telescope N -complex, explicit derived torsion and derived completion functors in the derived category D N (R) of N -complexes using the C ˇ ech N -complex and the telescope N -complex. Moreover, we give an equivalence between the category of cohomologically a -torsion N -complexes and the category of cohomologically a -adic complete N -complexes, and prove that over a commutative noetherian ring, via Koszul cohomology, via RHom cohomology (resp. ⊗ cohomology) and via local cohomology (resp. derived completion), all yield the same invariant.

Let a be an ideal of local ring (R,m) and M a finitely generated R -module and n∈N . It is shown that some results concerning cominimaxness of formal local cohomology modules.

In this note, we show that a part of [5, Remark 2.2] is not correct. Some conditions are given under which the same holds.

In this paper, we compare annihilators of Tor and Ext modules of finitely generated modules over a commutative noetherian ring. For local Cohen--Macaulay rings, one of our results refines a theorem of Dao and Takahashi.

Suppose R is a Q -Gorenstein F -finite and F -pure ring of prime characteristic p>0 . We show that if I⊆R is a compatible ideal (with all p −e -linear maps) then there exists a module finite extension R→S such that the ideal I is the sum of images of all R -linear maps S→R .

Skew completable unimodular rows of odd length are completable over polynomial extension of a local ring if dimension of local ring and length of unimodular rows are same.

We find necessary and sufficient conditions for a complete local ring containing the rationals to be the completion of a countable excellent local (Noetherian) domain. Furthermore, we find necessary and sufficient conditions for a complete local ring to be the completion of a countable noncatenary local domain, as well as necessary and sufficient conditions for it to be the completion of a countable noncatenary local unique factorization domain.

We find necessary and sufficient conditions for a complete local (Noetherian) ring to be the completion of an uncountable local (Noetherian) domain with a countable spectrum. Our results suggest that uncountable local domains with countable spectra are more common than previously thought. We also characterize completions of uncountable excellent local domains with countable spectra assuming the completion contains the rationals, completions of uncountable local unique factorization domains with countable spectra, completions of uncountable noncatenary local domains with countable spectra, and completions of uncountable noncatenary local unique factorization domains with countable spectra.

We present new conditions that characterize the componentwise linearity of Stanley-Reisner ideal of broken circuit complexes of simple matroids. We employ the notion of graded linearity (called also quasilinearity in the literature), which is more general than having linear resolution and show that componentwise linearity is equivalent to have graded linear resolution. As a result generalizing the one in \cite{RV}, we prove that the componentwise linearilty of Stanley-Reisner ideal of broken circuit complex translates to the decomposition of the matroid into specific matroids. An application to the Koszul property of Orlik-Terao ideal of hyperplane arrangements is given. We end up with some partial results on Koszul property and complete intersection of Orlik-Terao algebra of central arrangements.