tests of mode coupling theory in two dimensions we analyze the glassy dynamics of binary mixtures of hard disks in two dimensions predictions of the mode coupling theory mct are tested with extensive brownian dynamics simulations measuring the collective particle density correlation functions in the vicinity of the glass transition we verify four predicted mixing effects for instance for large size disparities adding a small amount of small particles at a fixed packing fraction leads to a speedup in the long time dynamics while for small size disparities it leads to a slowing down qualitative features of the nonergodicity parameters and the relaxation which both depend in a nontrivial way on the mixing ratio are found in the simulated correlators studying one system in detail we are able to determine its ideal mct glass transition point as c 0 7948 and test mct predictions quantitatively

glass transition of hard spheres in high dimensions we have investigated analytically and numerically the liquid glass transition of hard spheres for dimensions d ensuremath rightarrow ensuremath infty in the framework of mode coupling theory the numerical results for the critical collective and self nonergodicity parameters f c k d and f c s k d exhibit non gaussian k dependence even up to d 800 f c s k d and f c k d differ for k ensuremath sim d 1 2 but become identical on a scale k ensuremath sim d which is proven analytically the critical packing fraction ensuremath varphi c d ensuremath sim d 2 2 ensuremath d is above the corresponding kauzmann packing fraction ensuremath varphi k d derived by a small cage expansion its quadratic pre exponential factor is different from the linear one found earlier the numerical values for the exponent parameter and therefore the critical exponents a and b depend on d even for the largest values of d

poincare series of compressed local artinian rings with odd top socle degree abstract we define a notion of compressed local artinian ring that does not require the ring to contain a field let r m be a compressed local artinian ring with odd top socle degree s at least five and socle r m s 1 m s we prove that the poincare series of all finitely generated modules over r are rational sharing a common denominator and that there is a golod homomorphism from a complete intersection onto r

sur les groupes de transformations birationnelles des surfaces ni agissant par transformations bi rationnelles sur les surfaces complexes compactes kahl eriennes nous mon trons a que le groupe des transformations birationnelles d une surface satisfait l alternative de tits b que les actions birationnelles de groupes de kazhdan sur les surfaces sont toutes birationnellement conjugu eesa des actions homographiques sur le plan projectif et c que si f et g sont deux transformations birationnelles de surfaces qui commutent alors ou bien f pr eserve un pinceau de courbes ou bien l un des it er es g m de g m 0 concide avec un it er e f n de f n 2 z let s be any compact complex kahler surface and bir s the group of birational transformations of s we study the structure of nitely genera ted subgroups of bir s and prove three main results a bir s satises the tits alternative b if a group with kazhdan property t acts on a compact kahler surface by birational transformations then the action is conjugate to an action by linear projective transformations on the projec tive plane and c if f is an element of bir s which does not preserve any pencil of curves and if g commutes with f then a positive iterate of g coincides with an iterate of f the second statement provides a positive answer to zimmer s conjecture for birational actions on surfaces

on a conjecture of auslander and reiten abstract in studying nakayama s 1958 conjecture on rings of infinite dominant dimension auslander and reiten proposed the following generalization let be an artin algebra and m a generator such that ext i m m 0 for all i 1 then m is projective this conjecture makes sense for any ring we establish auslander and reiten s conjecture for excellent cohen macaulay normal domains containing the rational numbers and slightly more generally

p elementary subgroups of the cremona group abstract we classify up to conjugacy the subgroups of the cremona group isomorphic to z p r where p is prime and r is maximal

on the growth of the betti sequence of the canonical module we study the growth of the betti sequence of the canonical module of a cohen macaulay local ring it is an open question whether this sequence grows exponentially whenever the ring is not gorenstein we answer the question of exponential growth affirmatively for a large class of rings and prove that the growth is in general not extremal as an application of growth we give criteria for a cohen macaulay ring possessing a canonical module to be gorenstein

restricted homological dimensions and cohen macaulayness abstract the classical homological dimensions the projective flat and injective ones are usually defined in terms of resolutions and then proved to be computable in terms of vanishing of appropriate derived functors in this paper we define restricted homological dimensions in terms of vanishing of the same derived functors but over classes of test modules that are restricted to assure automatic finiteness over commutative noetherian rings of finite krull dimension when the ring is local we use a mixture of methods from classical commutative algebra and the theory of homological dimensions to show that vanishing of these functors reveals that the underlying ring is a cohen macaulay ring or at least close to being one

classification of k3 surfaces with involution and maximal symplectic symmetry k3 surfaces with antisymplectic involution and compatible symplectic actions of finite groups are considered in this situation actions of large finite groups of symplectic transformations are shown to arise via double covers of del pezzo surfaces and a complete classification of k3 surfaces with maximal symplectic symmetry is obtained

huneke ulrich almost complete intersections of cohen macaulay type two abstract let r m k be a commutative noetherian local ring n 2 an integer x a 2n 1 2n 1 alternating matrix with entries from m y a 1 2n 1 matrix with entries from m i the ideal i i1 yx and a the quotient ring r i assume that the grade of i is at least 2n in this case i is a perfect ideal of grade equal to 2n and i is minimally generated by 2n 1 elements we prove that the minimal resolution of a by free r modules is a dg algebra furthermore we identify the algebra torr a k and prove that if r is regular and char k 0 or n 2 2 char k then the poincare series pma z i 0 dimk torai m k zi is a rational function for every finitely generated a module m as a consequence we deduce that if the projective dimension of m is infinite then eventually the betti numbers of m form an increasing sequence with strong exponential growth

absolute relative and tate cohomology of modules of finite gorenstein dimension we study finitely generated modules m over a ring r noetherian on both sides if m has finite gorenstein dimension mbox g dim rm in the sense of auslander and bridger then it determines two other cohomology theories besides the one given by the absolute cohomology functors rm ext n r m relative cohomology functors rm ext n mathcal g m are defined for all non negative integers n they treat the modules of gorenstein dimension 0 as projectives and vanish for n mbox g dim rm tate cohomology functors widehat rm ext n r m are defined for all integers n all groups widehat rm ext n r m n vanish if m or n has finite projective dimension comparison morphisms varepsilon mathcal g n colon rm ext n mathcal g m to rm ext n r m and varepsilon r n colon rm ext n r m to widehat rm ext n r m link these functors we give a self contained treatment of modules of finite g dimension establish basic properties of relative and tate cohomology and embed the comparison morphisms into a canonical long exact sequence 0 to rm ext 1 mathcal g m to cdots to rm ext n mathcal g m to rm ext n r m to widehat rm ext n r m to rm ext n 1 mathcal g m to cdots we show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings 2000 mathematical subject classification 16e05 13h10 18g25

nonvanishing cohomology and classes of gorenstein rings abstract we give counterexamples to the following conjecture of auslander given a finitely generated module m over an artin algebra there exists a positive integer n m such that for all finitely generated modules n if ext i m n 0 for all i 0 then ext i m n 0 for all i n m some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local gorenstein rings

on the growth of the betti sequence of the canonical module mathematische zeitschrift 2007 256 3 647 659 doi 10 1007 s00209 006 0096 x

poincare series of modules over local rings of small embedding codepth or small linking number abstract let s n k be a commutative noetherian local ring and m be a finitely generated s module then the poincare series of m is rational if any of the following holds edim s depth s 3 edim s depth s 4 and s is gorenstein s is one link from a complete intersection s is two links from a complete intersection and s is gorenstein to prove this result we assume without loss of generality that s r i with r m regular local and the key to the argument is to produce a factorization of the canonical map r s as a composition of a complete intersection r c with a golod map c s this is accomplished by invoking a theorem of avramov and backelin once one has existence of a dg algebra structure on a minimal r free resolution of s together with detailed knowledge of the structure of the induced homology algebra tor r s k h k s linkage theory provides the main technical tool for this analysis

dimensional study of the caging order parameter at the glass transition the glass problem is notoriously hard and controversial even at the mean field level little is agreed upon regarding why a fluid becomes sluggish while exhibiting but unremarkable structural changes it is clear however that the process involves self caging which provides an order parameter for the transition it is also broadly assumed that this cage should have a gaussian shape in the mean field limit here we show that this ansatz does not hold by performing simulations as a function of spatial dimension d we find the cage to keep a nontrivial form quantitative mean field descriptions of the glass transition such as mode coupling theory density functional theory and replica theory all miss this crucial element although the mean field random first order transition scenario of the glass transition is qualitatively supported here and non mean field corrections are found to remain small on decreasing d reconsideration of its implementation is needed for it to result in a coherent description of experimental observations

boundedness versus periodicity over commutative local rings over commutative graded local artinian rings examples are constructed of periodic modules of arbitrary minimal period and modules with bounded betti numbers which are not eventually periodic they provide counterexamples to a conjecture of d eisenbud that every module with bounded betti numbers over a commutative local ring is eventually periodic of period 2 it is proved however that the conjecture holds over rings of small length

a connection between birational automorphisms of the plane and linear systems of curves in this paper we prove that there exists a one to one correspondence between birational automorphisms of the plane and pairs of pencils of curves intersecting in a unique point as a consequence we show how to construct birational automorphisms of the plane of a certain degree d fixed in advance from some curves generating two linear systems of curves of degrees d and d where d d 2 for d 2 and d 1 otherwise in addition we also get the inverse of the birational automorphism constructed and we show that its degree is obtained from the degree of the linear system of curves as a special case we show how these results can be stated to polynomial birational automorphisms of the plane

mode coupling theory of the glass transition for confined fluids we present a detailed derivation of a microscopic theory for the glass transition of a liquid enclosed between two parallel walls relying on a mode coupling approximation this geometry lacks translational invariance perpendicular to the walls which implies that the density profile and the density density correlation function depends explicitly on the distances to the walls we discuss the residual symmetry properties in slab geometry and introduce a symmetry adapted complete set of two point correlation functions since the currents naturally split into components parallel and perpendicular to the walls the mathematical structure of the theory differs from the established mode coupling equations in bulk we prove that the equations for the nonergodicity parameters still display a covariance property similar to bulk liquids

automorphisms of finite order on rational surfaces with an appendix by i dolgachev abstract we classify minimal pairs x g for smooth rational projective surface x and finite group g of automorphisms on x we also determine the fixed locus x g and the quotient surface y x g as well as the fundamental group of the smooth part of y the realization of each pair is included mori s extremal ray theory and recent results of alexeev and also ambro on the existence of good anti canonical divisors are used

support varieties and cohomology over complete intersections quillen s geometric approach to the cohomology of finite groups 30 has revolutionized modular representation theory the ideology and techniques involved have been generalized and extended to representations of various hopf algebras culminating in the recent work of friedlander and suslin 17 on finite group schemes in this paper we develop geometric methods for the study of finite modules over a local complete intersection r if r is such a ring and m is its maximal ideal then the m adic completion r has the form q f where f is a regular sequence and q is a regular local ring that can be taken to be a ring of formal power series over a field or a discrete valuation ring the least number of equations needed to cut out r from a regular local ring equals the codimension of r where codim r r m dim r and r m denotes the minimal number of generators of an r module m in 5 a cone that is a homogeneous algebraic set vr m is attached to each finite r module m and used to study its minimal free resolution here we prove

poincare series of modules over compressed gorenstein local rings abstract given two positive integers e and s we consider gorenstein artinian local rings r whose maximal ideal m satisfies m s 0 m s 1 and rank r m m m 2 e we say that r is a compressed gorenstein local ring when it has maximal length among such rings it is known that generic gorenstein artinian algebras are compressed if s 3 we prove that the poincare series of all finitely generated modules over a compressed gorenstein local ring are rational sharing a common denominator a formula for the denominator is given when s is even this formula depends only on the integers e and s note that for s 3 examples of compressed gorenstein local rings with transcendental poincare series exist due to bogvad

algebras that satisfy auslander s condition on vanishing of cohomology auslander conjectured that every artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules the failure of this conjecture by a 2003 counterexample due to jorgensen and ega motivates the consideration of the class of rings that do satisfy auslander s condition we call them ac rings and show that an ac artin algebra that is left gorenstein is also right gorenstein furthermore the auslander reiten conjecture is proved for ac rings and auslander s g dimension is shown to be functorial for ac rings that are commutative or have a dualizing complex