on the arithmetic and homology of algebras of linear type three modifications of the symmetric algebra of a module are introduced and their arithmetical and homological properties studied emphasis is placed on converting syzygetic properties of the modules into ideal theoretic properties of the algebras e g cohen macaulayness factoriality the main tools are certain fitting ideals of the module and an extension to modules of a complex of not necessarily free modules that we have used in studying blowing up rings

cohen macaulay rees algebras and degrees of polynomial relations e mail aron sunrnp vfba br 2 department of mathematics michigan state university east lansing mi 48824 usa e mail 21144bfu msu edu 3 department of mathematics rutgers university new brunswick nj 08903 usa e mail vasconce rings rutgers edu received 5 october 1993

on the depth of the fiber cone of filtrations preprint enviat per a la seva publicacio en una revista cientifica journal of algebra 15 december 1997 volume 198 issue 2 pages 428 445 https doi org 10 1006 jabr 1997 7147

nilpotence and descent in equivariant stable homotopy theory abstract let g be a finite group and let f be a family of subgroups of g we introduce a class of g equivariant spectra that we call f nilpotent this definition fits into the general theory of torsion complete and nilpotent objects in a symmetric monoidal stable category with which we begin we then develop some of the basic properties of f nilpotent g spectra which are explored further in the sequel to this paper in the rest of the paper we prove several general structure theorems for categories of module spectra over objects such as equivariant real and complex k theory and borel equivariant mu using these structure theorems and a technique with the flag variety dating back to quillen we then show that large classes of equivariant cohomology theories for which a type of complex orientability holds are nilpotent for the family of abelian subgroups in particular we prove that equivariant real and complex k theory as well as the borel equivariant versions of complex oriented theories have this property

equivariant cohomology and the maurer cartan equation let g be a compact connected lie group acting smoothly on a manifold m goresky kottwitz macpherson described a small cartan model for the equivariant cohomology of m quasi isomorphic to the standard cartan complex of equivariant differential forms in this paper we construct an explicit cochain map from the small cartan model into the standard cartan model inducing an isomorphism in cohomology the construction involves the solution of a remarkable inhomogeneous maurer cartan equation this solution has further applications to the theory of transgression in the weil algebra and to the chevalley koszul theory of the cohomology of principal bundles

homology of perfect complexes abstract it is proved that the sum of the loewy lengths of the homology modules of a finite free complex f over a local ring r is bounded below by a number depending only on r this result uncovers in the structure of modules of finite projective dimension obstructions to realizing r as a closed fiber of some flat local homomorphism other applications include as special cases uniform proofs of known results on free actions of elementary abelian groups and of tori on finite cw complexes the arguments use numerical invariants of objects in general triangulated categories introduced here and called levels they allow one to track through changes of triangulated categories homological invariants like projective dimension as well as structural invariants like loewy length an intermediate result sharpens with a new proof the new intersection theorem for commutative algebras over fields under additional hypotheses on the ring r stronger estimates are proved for loewy lengths of modules of finite projective dimension

duality in algebra and topology abstract we apply ideas from commutative algebra and morita theory to algebraic topology using ring spectra this allows us to prove new duality results in algebra and topology and to view 1 poincare duality for manifolds 2 gorenstein duality for commutative rings 3 benson carlson duality for cohomology rings of finite groups 4 poincare duality for groups and 5 gross hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon

class and rank of differential modules a differential module is a module equipped with a square zero endomorphism this structure underpins complexes of modules over rings as well as differential graded modules over graded rings we establish lower bounds on the class a substitute for the length of a free complex and on the rank of a differential module in terms of invariants of its homology these results specialize to basic theorems in commutative algebra and algebraic topology one instance is a common generalization of the equicharacteristic case of the new intersection theorem of hochster peskine p roberts and szpiro concerning complexes over commutative noetherian rings and of a theorem of g carlsson on differential graded modules over graded polynomial rings

algebra of multidimensional periodic operators with defects the procedure for calculating the spectrum of discrete periodic operators perturbed by periodic operators of smaller dimensions is obtained some properties of the corresponding algebra of operators with defects are considered these results yield explicit expressions for propagative guided and other localized spectra of different physical operators on periodic structures coupled with various defects as shown in the example of the schrodinger operator on 2d lattice with linear and point defects the explicit formula for the localized spectrum can be helpful to solve inverse problems such as a recovery of point defects

on the depth of the associated graded ring of an ideal abstract let r be a local cohen macaulay ring let i be an r ideal and let g be the associated graded ring of i we give an estimate for the depth of g when g fails to be cohen macaulay we assume that i has a small reduction number and sufficiently good residual intersection properties and satisfies local conditions on the depth of some powers the main theorem unifies and generalizes several known results we also give conditions that imply the serre properties of the blow up rings

gorenstein graded rings associated to ideals let a be a noetherian local ring with the maximal ideal m and dim a d let i 6 a be an ideal in a and s hta i this article studies the question of when the associated graded ring g i i 0 i i is gorenstein to state our result we set up some notation let be an integer such that d and let j be a reduction of i generated by elements a1 a2 a we denote by rj i the reduction number of i with respect to j the analytic spread of i is i dim a m a g i then s i we always assume that the generating set a1 a2 a of j is a basic generating set for j in the sense of aberbach huneke and trung aht which means that jiaq is a reduction of iaq for all q v i with i hta q here let v i be a set of prime ideals in a containing i and ji a1 a2 ai for 0 i by aht 7 2 there always exists a basic generating set for j if the field a m is infinite let

on the gorenstein property of rees and form rings of power of ideals in this paper we determine the exponents n for which the rees ring r in and the form ring gra in are gorenstein rings where i is a strongly cohen macaulay ideal of linear type including complete and almost complete intersections or an m primary ideal in a local ring a with maximal ideal m given an ideal i in a local ring a m it is well known that the cohenmacaulayness of the rees algebra r i implies the cohen macaulayness of all rees algebras r in the same is true for the form rings gra i and gra in see 3 2 7 8 and 8 8 5 in this paper we show that in contrast to the cohenmacaulay property the gorenstein property of r in and gra in only holds for special exponents n if in particular gra i is gorenstein and r i is cohen macaulay it turns out that these special exponents are closely related to the a invariant of the form ring gra i mainly under this aspect we prove some results concerning the gorenstein property of rees and form rings of powers of i strongly cohen macaulay ideals of linear type including almost complete intersections in gorenstein rings ii m primary ideals in cohen macaulay rings and iii equimultiple prime ideals p m in a generalized cohen macaulay ring our investigations are essentially based on the explicit computation of the a invariants of form and rees rings see 2 for the above classes of idealsusing a structure theorem for the canonical module of r i in 7 we can determine in 3 the exponents n 1 for which r in and gra in are gorenstein rings see in particular theorem 3 5 a more geometrical interpretation of the results in 3 is the observation that in these situations the gorenstein property of proj r i can be deduced from the gorenstein property of a certain veronesean subring r in of r i it might be an interesting question when the gorenstein property of a blow up proj r i is inherited to an appropriate rees ring r j with proj r j proj r i over speca in 4 part i we characterize for cohen macaulay or gorenstein rings a the gorenstein property of r md i i 1 2 3 by conditions on the received by the editors june 30 1991 and in revised form april 10 1992 1991 mathematics subject classification primary 13d03 13h10 13h15 secondary 14b05 14b 1 5 1994 american mathematical society 0002 9947 94 1 00 25 per page

ideals in triangulated categories phantoms ghosts and skeleta we begin by showing that in a triangulated category specifying a projective class is equivalent to specifying an ideal i of morphisms with certain properties and that if i has these properties then so does each of its powers we show how a projective class leads to an adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence we use this to study various ideals in the stable homotopy category we examine phantom maps skeletal phantom maps superphantom maps and ghosts a ghost is a map which induces the zero map of homotopy groups we show that ghosts lead to a stable analogue of the lusternik schnirelmann category of a space and we calculate this stable analogue for low dimensional real projective spaces we also give a relation between ghosts and the hopf and kervaire invariant problems in the case ofa modules over ana ring spectrum the ghost spectral sequence is a universal coefficient spectral sequence from the phantom projective class we derive a generalized milnor sequence for filtered diagrams of finite spectra and from this it follows that the group of phantom maps fromxtoycan always be described as alim 1group the last two sections focus on algebraic examples in the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of kelly on the vanishing of composites of such maps we also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non trivially

cohen macaulay graded rings associated to ideals let g i be the associated graded ring of an ideal i in a cohen macaulay local ring a we give a sufficient condition for g i to be a cohen macaulay ring it is described in terms of the depths of a i n for finitely many n the reduction numbers of i q for certain prime ideals q and the artin nagata property of i in the sense of ulrich contemp math vol 159 pp 373 400 the main theorem unifies several results which are already known in this aspect of the theory although the statement of the main theorem is rather complicated we restate the conditions under special situations so that they are practical and simple

the depth of the associated graded ring of ideals with any reduction number abstract let r be a local cohen macaulay ring let i be an r ideal and let g be the associated graded ring of i we give an estimate for the depth of g when g is not necessarily cohen macaulay we assume that i is either equimultiple or has analytic deviation one but we do not have any restriction on the reduction number we also give a general estimate for the depth of g involving the first r l powers of i where r denotes the castelnuovo regularity of g and l denotes the analytic spread of i

homological properties of powers of the maximal ideal of a local ring abstract it is known that the powers m n of the maximal ideal of a local noetherian ring share certain homological properties for all sufficiently large integers n for example the natural homomorphisms r r m n are golod respectively small for all large n we give effective bounds on the smallest integers n for which such properties begin to hold

on a invariant formulas abstract let g i be the form ring of an ideal i of positive height in a local ring a in this work we will provide formulas for the a invariant of g i our main result will only need the assumption that a is cohen macaulay and that g i fulfills serre s condition sl where l is the analytic spread of i as a consequence of our formula we will prove upper bounds for the reduction exponent of i in the case that a is a regular local ring if g i fulfills serre s condition sl then this upper bound is l 1 and in the case that g i is even gorenstein it is l 2

approximation complexes of blowing up rings ii this paper is a sequel to 111 where we studied the relationships that hold between the arithmetical properties of the rees ring 9 i and the symmetric algebra sym l of an ideal i and some of their fibers and the depth properties of the koszul homology modules h i r on a set of generators of i the connection between these objects is realized by certain differential graded algebras the so called approximation complexes that are built out of ordinary koszul complexes these compiexes show however different sensitivities being acyclic in situations much broader than the usual context of regular sequences it has been found that certain extensions thereof d sequences and proper sequences play here that role of acyclic sequence the interplay between sequences and approximation complexes will be a basic point of view here where we give

reduction numbers and multiplicities of multigraded structures the aim of this paper is twofold first we want to relate the cohen macaulay property of r i i to the cohen macaulay propa 1 r erty of the ordinary rees rings r i j s 1 r then we determine a j the multiplicities of multi rees and form rings in particular we ask when minimal multiplicity occurs for i s s i s i the multi rees algebras r i s r i i 1 r a r a behave to some extent like the ordinary rees algebras r i in particua lar the cohen macaulay property of such multi rees algebras can be characterized in terms of conditions on the local cohomology of the form ring gr i including the condition a gr i 0 where a gr i denotes a a a w x the a invariant of gr i see hhr 1 the a invariant has in certain a w cases been related to the reduction exponent r i of the ideal i see t2 x gh moreover the cohen macaulay property of r i implies that of a w x r i for all r 1 the converse is not true in general see hhr1 but a r in certain applications the assumption r i is cohen macaulay can be a

representation dimension of exterior algebras we determine the representation dimension of exterior algebras this provides the first known examples of representation dimension 3 we deduce that the loewy length of the group algebra over f2 of a finite group is strictly bounded below by the 2 rank of the group a conjecture of benson a key tool is the use of the concept of dimension of a triangulated category

gorenstein associated graded rings of analytic deviation two ideals abstract this paper studies the question of when the associated graded ring g i n 0in in 1 of a certain ideal i in a local ring is gorenstein the main result implies for example that if a is a regular local ring 708 is a prime ideal in a with dim a p 2 and a p is a complete intersection in codimension one then the associated graded ring g p is gorenstein if and only if the reduction number of p is at most 1