hereditary rings with countably generated cotorsion envelope abstract let r be a left hereditary ring we show that if the left cotorsion envelope c r r of r is countably generated then r is a semilocal ring in particular we deduce that c r r is finitely generated if and only if r is a semiperfect cotorsion ring our proof is based on set theoretical counting arguments we also discuss some possible extensions of this result

cartesian modules over representations of small categories we introduce the new concept of cartesian module over a pseudofunctor r from a small category to the category of small preadditive categories already the case when r is a strict functor taking values in the category of commutative rings is sucient

the schroder bernstein problem for modules abstract in this paper we study the schroder bernstein problem for modules we obtain a positive solution for the schroder bernstein problem for modules invariant under endomorphisms of their general envelopes under some mild conditions that are always satisfied for example in the case of injective pure injective or cotorsion envelopes in the particular cases of injective envelopes and pure injective envelopes we are able to extend it further and we show that the schroder bernstein problem has a positive solution even for modules that are invariant only under automorphisms of their injective envelopes or pure injective envelopes

the bogomolov miyaoka yau inequality for logarithmic surfaces in positive characteristic we generalize bogomolov s inequality for higgs sheaves and the bogomolov miyaoka yau inequality in positive characteristic to the logarithmic case we also generalize shepherd barron s results on bogomolov s inequality on surfaces of special type from rank 2 to the higher rank case we use these results to show some examples of smooth nonconnected curves on smooth rational surfaces that cannot be lifted modulo p2 these examples contradict some claims by xie

bogomolov s inequality for higgs sheaves in positive characteristic we prove bogomolov s inequality for higgs sheaves on varieties in positive characteristic p that can be lifted modulo p 2 this implies the miyaoka yau inequality on surfaces of non negative kodaira dimension liftable modulo p 2 this result is a strong version of shepherd barron s conjecture our inequality also gives the first algebraic proof of bogomolov s inequality for higgs sheaves in characteristic zero solving the problem posed by narasimhan

semistable sheaves in positive characteristic we prove maruyama s conjecture on the boundedness of slope semistable sheaves on a projective variety defined over a noetherian ring our approach also gives a new proof of the boundedness for varieties defined over a characteristic zero field this result implies that in mixed characteristic the moduli spaces of gieseker semistable sheaves are projective schemes of finite type the proof uses a new inequality bounding slopes of the restriction of a sheaf to a hypersurface in terms of its slope and the discriminant this inequality also leads to effective restriction theorems in all characteristics improving earlier results in characteristic zero

uniruled surfaces of general type we give a systematic construction of uniruled surfaces in positive characteristic using this construction we find surfaces of general type with non trivial global vector fields surfaces with arbitrarily non reduced picard schemes as well as surfaces with inseparable canonical maps in particular we show that some previously known pathologies are not sporadic but exist in abundance

sigma cotorsion rings abstract it is proved that a ring r is right perfect if and only if it is cotorsion as a right module over itself several other conditions are shown to be equivalent for example that every pure submodule of a free right r module is strongly pure essential in a direct summand or that the countable direct sum c r 0 of the cotorsion envelope of r r is cotorsion if c r is a flat cotorsion module then c r admits a decomposition into a direct sum of indecomposable modules with a local endomorphism ring the jacobson radical j s of the endomorphism ring s end r c is characterized as the maximum ideal that acts locally t nilpotently on c r if r is semilocal and c c r then the radical consists of those endomorphisms f c c whose image is contained in cj

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

model category structures arising from drinfeld vector bundles we present a general construction of model category structures on the category ch qco x of unbounded chain complexes of quasi coherent sheaves on a semi separated scheme x the construction is based on making compatible the filtrations of individual modules of sections at open affine subsets of x it does not require closure under direct limits as previous methods we apply it to describe the derived category mathbb d qco x via various model structures on ch qco x as particular instances we recover recent results on the flat model structure for quasi coherent sheaves our approach also includes the case of infinite dimensional vector bundles and of restricted flat mittag leffler quasi coherent sheaves as introduced by drinfeld finally we prove that the unrestricted case does not induce a model category structure as above in general

supersymmetric extension of the scalar born infeld equation in this paper a supersymmetric extension of the scalar born infeld equation is constructed through a superspace formalism which involves the addition of one independent fermionic grassmann variable to the existing bosonic spacetime coordinates the bosonic scalar field is replaced by a fermionic superfield which is composed of two component fields one bosonic and one fermionic the resulting equation is invariant under a space supersymmetric transformation and in its most general form involves four arbitrary parameters for a certain specific case the lie superalgebra of symmetries was identified and the one dimensional subalgebras were systematically classified into splitting and non splitting conjugate classes a number of group invariant solutions were obtained including polynomial solutions solutions by radicals and solitary waves including bumps kinks and doubly periodic solutions