complete intersection k theory and chern classes the purpose of this paper is to investigate the theory of complete intersection in noetherian commutative rings from the k theory point of view by complete intersection theory we mean questions like when whether an ideal is the image of a projective module of appropriate rank the paper has two parts in part one section 1 5 we deal with the relationship between complete intersection and k theory the part two section 6 8 is essentially devoted to construction projective modules with certain cycles as the total chern class here chern classes will take values in the associated graded ring of the grothedieck filtration and as well in the chow group in the smooth case in this paper all our rings are commutative and schemes are noetherian to avoid unnecessary complications we shall assume that all our schemes are connected for a noetherian schemex k0 x will denote the grothendieck group of locally free sheaves of finite rank over x whenever it make sense for a coherent sheaf m over x m will denote the class of m in k0 x we shall mostly be concerned with x speca where a is a noetherian commutative ring and in this case we shall also use the notation k0 a for k0 x

strong euler class of a stably free module of odd rank abstract let r be a commutative noetherian ring of dimension n 3 following a suggestion of fasel we establish a group homomorphism from van der kallen s group um n 1 r e n 1 r to the n th euler class group e n r so that 1 when n is even coincides with the homomorphism given by bhatwadekar and sridharan through euler classes 2 when n is odd is non trivial in general for an important class of rings 3 the sequence um n 1 r e n 1 r e n r e 0 n r 0 is exact where e 0 n r is the n th weak euler class group if x spec r is a smooth affine variety of dimension n over r so that the complex points of x are complete intersections and the canonical module k r is trivial then the sequence is proved to be exact on the left as well more generally let r be a commutative noetherian ring of dimension d and n be an integer such that n d 2 n 3 we also indicate how to extend our arguments to this setup to obtain a group homomorphism from um n 1 r e n 1 r to e n r

shadow hybrid monte carlo an efficient propagator in phase space of macromolecules shadow hybrid monte carlo shmc is a new method for sampling the phase space of large molecules particularly biological molecules it improves sampling of hybrid monte carlo hmc by allowing larger time steps and system sizes in the molecular dynamics md step the acceptance rate of hmc decreases exponentially with increasing system size n or time step t this is due to discretization errors introduced by the numerical integrator shmc achieves an asymptotic o n1 4 speedup over hmc by sampling from all of phase space using high order approximations to a shadow or modified hamiltonian exactly integrated by a symplectic md integrator shmc satisfies microscopic reversibility and is a rigorous sampling method shmc requires extra storage modest computational overhead and a reweighting step to obtain averages from the canonical ensemble this is validated by numerical experiments that compute observables for different molecules ranging from a small n alkane butane with four united atoms to a larger solvated protein with 14 281 atoms in these experiments shmc achieves an order magnitude speedup in sampling efficiency for medium sized proteins sampling efficiency is measured by monitoring the rate at which different conformations of the molecules dihedral angles are visited and by computing ergodic measures of some observables

rationality problem for generic tori in simple groups abstract we prove that except for several already known cases the generic torus of a simple adjoint or simply connected group is not stably rational this confirms a conjecture by le bruyn on generic norm tori

euler class group of a laurent polynomial ring local case abstract let r be a noetherian commutative ring of dimension n 2 and let a r t t 1 assume that the height of the jacobson radical of r is at least 2 let p be a projective a module of rank n dim a 1 with trivial determinant we define an abelian group called the euler class group of a denoted by e a let be an isomorphism from a to det p to the pair p we associate an element of e a called the euler class of p denoted by e p then we prove that a necessary and sufficient condition for p to have a unimodular element is the vanishing of e p in e a earlier bhatwadekar and raja sridharan have defined the euler class group of r denoted by e r and have proved similar results for projective r module of rank n later m k das defined the euler class group of the polynomial ring r t denoted by e r t and proved similar results for projective r t modules of rank n with trivial determinant

erratum to a comparison of generalized hybrid monte carlo methods with and without momentum flip j comput phys 228 2009 2256 2265 the generalized hybrid monte carlo ghmc method combines metropolis corrected constant energy simulations with a partial random refreshment step in the particle momenta the standard detailed balance condition requires that momenta are negated upon rejection of a molecular dynamics proposal step the implication is a trajectory reversal upon rejection which is undesirable when interpreting ghmc as thermostated molecular dynamics we show that a modified detailed balance condition can be used to implement ghmc without momentum flips the same modification can be applied to the generalized shadow hybrid monte carlo gshmc method numerical results indicate that ghmc gshmc implementations with momentum flip display a favorable behavior in terms of sampling efficiency i e the traditional ghmc gshmc implementations with momentum flip got the advantage of a higher acceptance rate and faster decorrelation of monte carlo samples the difference is more pronounced for ghmc we also numerically investigate the behavior of the ghmc method as a langevin type thermostat we find that the ghmc method without momentum flip interferes less with the underlying stochastic molecular dynamics in terms of autocorrelation functions and it to be preferred over the ghmc method with momentum flip the same finding applies to gshmc

vanishing of euler class groups abstract in this paper we prove some theorems about vanishing of euler class groups for example suppose b a x f 1 where a x is a polynomial ring f a x is a non zero divisor and dim b dim a 1 3 then we prove that the euler class group e b l 0 for any rank one projective b module l

reduction in the rationality problem for multiplicative invariant fields abstract for a faithful zg lattice a and a field k on which the group g acts by field automorphisms let r be the normal subgroup generated by the elements of g which act trivially on k and act as reflections on a we prove that the rationality of the multiplicative invariant field k a g over k ar g is equivalent to the rationality of k a g over k ar g where g is a particular subgroup of g such that g r g we then use this reduction result to prove that k a g is rational over k where g is the automorphism group of a crystallographic root system g acts trivially on k and a is any lattice on the space q

stably free modules in su prob 3 suslin had asked the following question let a be any affine algebra of dimension n over an algebraically closed field what is the smallest integer m such that all stably free projective modules of rank bigger than m are free all the examples in the literature of stably free non free modules have rank less than or equal to n 1 2 the aim of this note is to construct examples of such modules of large rank we construct rank p stably free non free modules over p 2 dimensional affine algebras over algebraically closed fields wherep is any prime these varieties are actually smooth and rational over c these are trivial as holomorphic vector bundles forp 2 this is classical forp 2 see ms so these are strictly algebraic examples i had described this construction in mk 1 and proved the result for p 2 we will reproduce the construction with necessary modifications in this note let p be any prime number and k any field letf x be any polynomial of degree p over k letf 0 a e k and fi xo xl f xo x1 xp f x0 x1 also let

zero cycles and projective modules let a be a reduced affine k algebra of dimension n over an algebraically closed field k let fnko a denote the sub group of ko a generated by the images of all the residue fields of all smooth maximal ideals of height n for a module of finite projective dimension m let m denote the image of m in ko a for a projective a module p of rank n we define the nth chern class of p to be cn p e l i aip where p is the dual of p suppose fnko a has no n 1 torsion then our main result th 3 8 is that p has a free direct summand of rank one if and only if cn p 0 when the characteristic of k is zero or a is normal and n 3 it is known that fnko a is torsion free le sr hence our theorem is applicable in these cases also when a is regular fnko a coincides with chn x the chow group of zero cycles of spec a and cn p coincides with the usual nth chern class as defined by grothendieck see fu when n 3 and a is regular this result was proved in mkm when n 3 and the characteristic of k is not equal to two this result in mkm was extended to the singular case by m levine in this paper we extend the results of mkm to all dimensions with the assumption stated above on torsion in fnko a we first show that cor 3 4 if i c a is a local complete intersection of height n then a i is zero in ko a if and only if i is a complete intersection corollary 3 4 together with a result of mohan kumar mk cor 1 9 here at once gives theorem 3 8 in section 1 we give some preliminaries and generalize results in mk2 these results are crucially used in the rest of the paper the basic ideas in section 1 are all taken from mk2 in section 2 we prove theorem 2 2 which strengthens a result of boratynski bo for local complete intersections with trivial co normal bundle theorem 2 2 is one of the crucial ingredients in the proofs of the results in section 3 as an amusing application of theorem 2 2 we give a new proof of a theorem of srinivas sr about torsion in zero cycles

murthy s conjecture on 0 cycles we show that the levine weibel chow group of 0 cycles mathrm ch d a of a reduced affine algebra a of dimension d ge 2 over an algebraically closed field is torsion free among several applications it implies an affirmative solution to an old conjecture of murthy in classical k theory

on algebras which are locally mathbb a 1 in codimension one let r be a noetherian normal domain call an r algebra a locally a1 in codimension one if rp r a is a polynomial ring in one variable over rp for every height one prime ideal p in r we shall describe a general structure for any faithfully flat r algebra a which is locally a1 in codimensionone and deduce results giving sufficient conditions for such an r algebra to be a locally polynomial algebra we also give a recipe for constructing r algebras which are locally a1 in codimension one when r is a normal affine spot i e a normal local domain obtained by a localisation of an affine domain we give criteria for a faithfully flat r algebra a which is locally a1 in codimensionone to be krull and a further condition for a to be noetherian the results are used to construct intricate examples of faithfully flat r algebras locally a1 in codimension one which are noetherian normal but not finitely generated

long run accuracy of variational integrators in the stochastic context this paper presents a lie trotter splitting for inertial langevin equations geometric langevin algorithm and analyzes its long time statistical properties the splitting is defined as a composition of a variational integrator with an ornstein uhlenbeck flow assuming that the exact solution and the splitting are geometrically ergodic the paper proves the discrete invariant measure of the splitting approximates the invariant measure of inertial langevin equations to within the accuracy of the variational integrator in representing the hamiltonian in particular if the variational integrator admits no energy error then the method samples the invariant measure of inertial langevin equations without error numerical validation is provided using explicit variational integrators with first second and fourth order accuracy

a patch that imparts unconditional stability to explicit integrators for langevin like equations this paper extends the results in 8 to stochastic differential equations sdes arising in molecular dynamics it implements a patch to explicit integrators that consists of a metropolis hastings step the patched integrator preserves the sde s equilibrium distribution and is accurate on finite time intervals as a corollary this paper proves the integrator s accuracy in estimating finite time dynamics along an infinitely long solution a first in molecular dynamics the paper also covers multiple time steps holonomic constraints and scalability finally the paper provides numerical tests supporting the theory

on invariance of the euler class groups under a subintegral base change abstract let r be a commutative noetherian ring and r s be a subintegral extension it is proved that the top euler class group of r is isomorphic to the top euler class group of s an example is given to show that similar conclusion does not hold for integral but not subintegral extension of rings

symmetric functions and the phase problem in crystallography the calculation of crystal structure from x ray diffraction data requires that the phases of the structure factors fourier coefficients determined by scattering be deduced from the absolute values of those structure factors motivated by a question of herbert hauptman we consider the problem of determining phases by direct algebraic means in the case of crystal structures with n equal atoms in the unit cell with n small we rephrase the problem as a question about multiplicative invariants for a particular finite group action we show that the absolute values form a generating set for the field of invariants of this action and consider the problem of making this theorem constructive and practical the most promising approach for deriving explicit formulas uses sagbi bases

projective modules over smooth real affine varieties let x spec a be a smooth affine variety of dimension n 2 over a field k and p be a projective a module of rank n a result of murthy mu theorem 3 8 says that if k is an algebraically closed field and if the top chern class cn p in ch0 x is zero then p splits off a free summand of rank 1 i e p a q we note that over any base field the vanishing of the top chern class is a necessary condition for p to split off a free summand of rank 1 however as is shown by the example of the tangent bundle of an even dimensional real sphere this condition is not sufficient in view of these examples it is of interest to explore whether one can classify examples of projective modules over smooth real varieties which have the property that the top chern class of the projective module vanishes but the projective module does not split off a free summand of rank one more precisely we ask the following

twisted multiplicative field invariants noether s problem and galois extensions noether s problem concerns fields f xg 1 g e g g f g where f is a field g is a finite group g acts by permuting the indeterminants x in the obvious way and f g refers to the field of invariant elements in showing the f g s were nonrational for certain g the author was led to consider multiplicative invariant fields ss which are defined as follows let g be a finite group and q a g lattice that is q is a finitely generated z g module which is free as an abelian group form the group algebra f g and its field of fractions f q g acts naturally on f q and the multiplicative invariant field is f q in ss it was shown that for certain q f q g was essentially equivalent to actually stably isomorphic to f g for g a split extension of g with abelian kernel this leads one to ask whether f g can be expressed as a kind of invariant field for g when g is a nonsplit extension of g the answer is yes and is the core of this paper in fact f g will be shown to be a so called a twisted multiplicative invariant field of g the goal of this paper is to present this fact and then draw a series of conclusions using it such a twisted invariant fields have appeared explicitly in previous works in ss these fields arise as another way of describing the invariant fields of reductive algebraic groups note that in s6 the finite group is the weyl group and the lattice is derived from the root lattice in s7 the unramified brauer groups of a twisted invariant fields are calculated

multiple time stepping generalized hybrid monte carlo methods performance of the generalized shadow hybrid monte carlo gshmc method 1 which proved to be superior in sampling efficiency over its predecessors 2 4 molecular dynamics and hybrid monte carlo can be further improved by combining it with multi time stepping mts and mollification of slow forces we demonstrate that the comparatively simple modifications of the method not only lead to better performance of gshmc itself but also allow for beating the best performed methods which use the similar force splitting schemes in addition we show that the same ideas can be successfully applied to the conventional generalized hybrid monte carlo method ghmc the resulting methods mts ghmc and mts gshmc provide accurate reproduction of thermodynamic and dynamical properties exact temperature control during simulation and computational robustness and efficiency mts ghmc uses a generalized momentum update to achieve weak stochastic stabilization to the molecular dynamics md integrator mts gshmc adds the use of a shadow modified hamiltonian to filter the md trajectories in the hmc scheme we introduce a new shadow hamiltonian formulation adapted to force splitting methods the use of such hamiltonians improves the acceptance rate of trajectories and has a strong impact on the sampling efficiency of the method both methods were implemented in the open source md package protomol and were tested on a water and a protein systems results were compared to those obtained using a langevin molly lm method 5 on the same systems the test results demonstrate the superiority of the new methods over lm in terms of stability accuracy and sampling efficiency this suggests that putting the mts approach in the framework of hybrid monte carlo and using the natural stochasticity offered by the generalized hybrid monte carlo lead to improving stability of mts and allow for achieving larger step sizes in the simulation of complex systems

gshmc an efficient method for molecular simulation the hybrid monte carlo hmc method is a popular and rigorous method for sampling from a canonical ensemble the hmc method is based on classical molecular dynamics simulations combined with a metropolis acceptance criterion and a momentum resampling step while the hmc method completely resamples the momentum after each monte carlo step the generalized hybrid monte carlo ghmc method can be implemented with a partial momentum refreshment step this property seems desirable for keeping some of the dynamic information throughout the sampling process similar to stochastic langevin and brownian dynamics simulations it is however ultimate to the success of the ghmc method that the rejection rate in the molecular dynamics part is kept at a minimum otherwise an undesirable zitterbewegung in the monte carlo samples is observed in this paper we describe a method to achieve very low rejection rates by using a modified energy which is preserved to high order along molecular dynamics trajectories the modified energy is based on backward error results for symplectic time stepping methods the proposed generalized shadow hybrid monte carlo gshmc method is applicable to nvt as well as npt ensemble simulations

mathbb p 1 p 1 gluing for local complete intersections we prove an analogue of the affine horrocks theorem for local complete intersection ideals of height n in r t where r is a regular domain of dimension d which is essentially of finite type over an infinite perfect field of characteristic unequal to 2 and 2n ge d 3

a riemann roch theorem in this paper we prove an analogue of this theorem for weak euler classes weak euler classes take values in weak euler class groups the definitions of the weak euler class group and weak euler classes were achieved by relaxing the definitions of the euler class group and euler classes the original definitions of the euler class group and the euler classes for smooth affine algebras over fields were given by m v nori for background literature we refer to 1 2 8 11 12 15 for noetherian commutative rings a with dima n 2 and a line bundle l on spec a the weak euler class group e0 a l has been defined in 2 as the quotient

projective modules over smooth affine varieties over real closed fields abstract let x spec a be a smooth affine variety of dimension n 2 over the field r of real numbers let p be a projective a module of rankn such that its nth chern class c n p ch 0 x is zero in this set up bhatwadekar das mandal showed amongst many other results that p a q in the case that either n is odd or the topological space x r of real points of x does not have a compact connected component in this paper we prove that similar results hold for smooth affine varieties over an arbitrary real closed field r the proof is algebraic and does not make use of tarski s principle nor of the earlier result for r

a comparison of generalized hybrid monte carlo methods with and without momentum flip the generalized hybrid monte carlo ghmc method combines metropolis corrected constant energy simulations with a partial random refreshment step in the particle momenta the standard detailed balance condition requires that momenta are negated upon rejection of a molecular dynamics proposal step the implication is a trajectory reversal upon rejection which is undesirable when interpreting ghmc as thermostated molecular dynamics we show that a modified detailed balance condition can be used to implement ghmc without momentum flips the same modification can be applied to the generalized shadow hybrid monte carlo gshmc method numerical results indicate that ghmc gshmc implementations with momentum flip display a favorable behavior in terms of sampling efficiency i e the traditional ghmc gshmc implementations with momentum flip got the advantage of a higher acceptance rate and faster decorrelation of monte carlo samples the difference is more pronounced for ghmc we also numerically investigate the behavior of the ghmc method as a langevin type thermostat we find that the ghmc method without momentum flip interferes less with the underlying stochastic molecular dynamics in terms of autocorrelation functions and it to be preferred over the ghmc method with momentum flip the same finding applies to gshmc

local coefficients and euler class groups abstract in this paper we establish an isomorphism between the euler class group e r x l for a real smooth affine variety x spec a and the 0 th homology group h 0 m c g with local coefficients in a bundle g of groups constructed from the line bundle l over m corresponding to the orientation rank 1 projective module l where m c is the compact part of the manifold m of real points in x then by steenrod s poincare duality between homology and cohomology groups with local coefficients this isomorphism is identified with the whitney class homomorphism