Publications (83)64.67 Total impact
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ABSTRACT: Conditions on the Koszul complex of a noetherian local ring $R$ guarantee that $\mathrm{Tor}^{R}_{i}(M,N)$ is nonzero for infinitely many $i$, when $M$ and $N$ are finitely generated $R$modules of infinite projective dimension. These conditions are obtained from results concerning Tor of differential graded modules over certain trivial extensions of commutative differential graded algebras. 
Article: To HansBjørn Foxby
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ABSTRACT: Estimates are obtained for the degrees of minimal syzygies of quotient algebras of polynomial rings. For a class that includes Koszul algebra in almost all characteristics, these degrees are shown to increase by at most 2 from one syzygy module to the next one. Even slower growth is proved if, in addition, the algebra satisfies Green and Lazarsfeld's condition N_q with q > 1.  [Show abstract] [Hide abstract]
ABSTRACT: A DG algebras $A$ over a field $k$ with $H(A)$ connected and $H_{<0}(A)=0$ has a unique up to isomorphism DG module $K$ with $H(K)\cong k$. It is proved that if $H(A)$ is degreewise finite, then $RHom_A(?,K): D^{df}_{+}(A)^{op} \equiv D_{df}^{+}}(RHom_A(K,K))$ is an exact equivalence of derived categories of DG modules with degreewise finitedimensional homology. It induces an equivalences of $D^{df}_{b}(A)^{op}$ and the category of perfect DG $RHom_A(K,K)$modules, and viceversa. Corresponding statements are proved also when $H(A)$ is simply connected and $H^{<0}(A)=0$.  [Show abstract] [Hide abstract]
ABSTRACT: For any nonzero finite module M of finite projective dimension over a noetherian local ring R with maximal ideal m and residue field k, it is proved that the natural map Ext_R(k,M)>Ext_R(k,M/mM) is nonzero when R is regular and is zero otherwise. A noteworthy aspect of the proof is the use of stable cohomology. Applications include computations of Bass series over certain local rings.  [Show abstract] [Hide abstract]
ABSTRACT: We prove basic facts about reflexivity in derived categories over noetherian schemes; and about related notions such as semidualizing complexes, invertible complexes, and Gorensteinperfect maps. Also, we study a notion of rigidity with respect to semidualizing complexes, in particular, relative dualizing complexes for Gorensteinperfect maps. Our results include theorems of Yekutieli and Zhang concerning rigid dualizing complexes on schemes. This work is a continuation of part I, which dealt with commutative rings. Comment: 40 pages  [Show abstract] [Hide abstract]
ABSTRACT: The generating series of the Bass numbers $\mu^i_R=\mathrm{rank}_k \mathrm{Ext}^i_R(k,R)$ of local rings $R$ with residue field $k$ are computed in closed rational form, in case the embedding dimension $e$ of $R$ and its depth $d$ satisfy $ed\le 3$. For each such $R$ it is proved that there is a real number $\gamma>1$, such that $\mu^{d+i}_R\ge\gamma\mu^{d+i1}_R$ holds for all $i\ge 0$, except for $i=2$ in two explicitly described cases, where $\mu^{d+2}_R=\mu^{d+1}_R=2$. New restrictions are obtained on the multiplicative structures of minimal free resolutions of length 3 over regular local rings. 
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ABSTRACT: It is proved that when R is a local ring of positive characteristic, $\phi$ is its Frobenius endomorphism, and some nonzero finite Rmodule has finite flat dimension or finite injective dimension for the Rmodule structure induced through $\phi$, then R is regular. This broad generalization of Kunz's characterization of regularity in positive characteristic is deduced from a theorem concerning a local ring R with residue field of k of arbitrary characteristic: If $\phi$ is a contracting endomorphism of R, then the Betti numbers and the Bass numbers over $\phi$ of any nonzero finitely generated Rmodule grow at the same rate, on an exponential scale, as the Betti numbers of k over R.  [Show abstract] [Hide abstract]
ABSTRACT: Extending a notion defined for surjective maps by Blanco, Majadas, and Rodicio, we introduce and study a class of homomorphisms of commutative noetherian rings, which strictly contains the class of locally complete intersection homomorphisms, while sharing many of its remarkable properties.  [Show abstract] [Hide abstract]
ABSTRACT: Given surjective homomorphisms R → T ← S of local rings, and ideals in R and S that are isomorphic to some Tmodule V, the connected sum R⋕TS is defined to be the ring obtained by factoring out the diagonal image of V in the fiber product R ×TS. When T is Cohen–Macaulay of dimension d and V is a canonical module of T, it is proved that if R and S are Gorenstein of dimension d, then so is R⋕TS. This result is used to study how closely an artinian ring can be approximated by a Gorenstein ring mapping onto it. When T is regular, it is shown that R⋕TS almost never is a complete intersection ring. The proof uses a presentation of the cohomology algebra as an amalgam of the algebras and over isomorphic polynomial subalgebras generated by one element of degree 2. 
Article: Short Koszul Modules
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ABSTRACT: This article is concerned with graded modules M with linear resolutions over a standard graded algebra R. It is proved that if such an M has Hilbert series $H_M(s)$ of the form $ps^d+qs^{d+1}$, then the algebra R is Koszul; if, in addition, M has constant Betti numbers, then $H_R(s)=1+es+(e1)s^{2}$. When $H_R(s)=1+es+rs^{2}$ with $r\leq e1$, and R is Gorenstein or $e=r+1\le 3$, it is proved that generic Rmodules with $q\leq(e1)p$ are linear. Comment: To appear in the special issue of the Journal of Commutative Algebra, dedicated to Ralf Froeberg's 65th birthday.  [Show abstract] [Hide abstract]
ABSTRACT: A general method for establishing results over a commutative complete intersection local ring by passing to differential graded modules over a graded exterior algebra is described. It is used to deduce, in a uniform way, results on the growth of resolutions of complexes over such local rings. Comment: 18 pages; to appear in "Triangulated categories (Leeds, 2006)", LMS lecture notes series. 
Chapter: Infinite Free Resolutions
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ABSTRACT: This text is based on the notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matemàtica, Institut d’Estudis Catalans, July 15–26, 1996. 
Article: Detecting flatness over smooth bases
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ABSTRACT: Given an essentially finite type morphism of schemes f: X > Y and a positive integer d, let f^{d}: X^{d} > Y denote the natural map from the dfold fiber product, X^{d}, of X over Y and \pi_i: X^{d} > X the i'th canonical projection. When Y smooth over a field and F is a coherent sheaf on X, it is proved that F is flat over Y if (and only if) f^{d} maps the associated points of the tensor product sheaf \otimes_{i=1}^d \pi_i^*(F) to generic points of Y, for some d greater than or equal to dim Y. The equivalent statement in commutative algebra is an analogbut not a consequenceof a classical criterion of Auslander and Lichtenbaum for the freeness of finitely generated modules over regular local rings.  [Show abstract] [Hide abstract]
ABSTRACT: We study functors underlying derived Hochschild cohomology, also called Shukla cohomology, of a commutative algebra S essentially of finite type and of finite flat dimension over a commutative noetherian ring K. We construct a complex of Smodules D, and natural reduction isomorphisms for all complexes of Smodules N and all complexes M of finite flat dimension over K whose homology H(M) is finitely generated over S; such isomorphisms determine D up to derived isomorphism. Using Grothendieck duality theory we establish analogous isomorphisms for any essentially finitetype flat map of noetherian schemes, with f!OY in place of D.  [Show abstract] [Hide abstract]
ABSTRACT: A notion of rigidity with respect to an arbitrary semidualizing complex C over a commutative noetherian ring R is introduced and studied. One of the main result characterizes Crigid complexes. Specialized to the case when C is the relative dualizing complex of a homomorphism of rings of finite Gorenstein dimension, it leads to broad generalizations of theorems of Yekutieli and Zhang concerning rigid dualizing complexes, in the sense of Van den Bergh. Along the way, a number of new results concerning derived reflexivity with respect to C are established. Noteworthy is the statement that derived Creflexivity is a local property; it implies that a finite Rmodule M has finite Gdimension over R if it is locally of finite Gdimension. Comment: 31 pages. Major revisions in Sections 1 and 6. To appear in `Algebra and Number Theory' 
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ABSTRACT: For R=Q/J with Q a commutative graded algebra over a field and J nonzero, we relate the slopes of the minimal resolutions of R over Q and of k=R/R_{+} over R. When Q and R are Koszul and J_1=0 we prove Tor^Q_i(R,k)_j=0 for j>2i, for each nonnegative integer i, and also for j=2i when i>dim Qdim R and pd_QR is finite.  [Show abstract] [Hide abstract]
ABSTRACT: To the memory of our friend and colleague Anders Frankild Abstract. The structure of minimal free resolutions of finite modules M over commutative local rings (R, m, k) with m 3 = 0 and rankk(m 2) < rankk(m/m 2) is studied. It is proved that over generic R every M has a Koszul syzygy module. Explicit families of Koszul modules are identified. When R is Gorenstein the nonKoszul modules are classified. Structure theorems are established for the graded kalgebra ExtR(k, k) and its graded module ExtR(M, k).
Publication Stats
2k  Citations  
64.67  Total Impact Points  
Top Journals
Institutions

20022013

University of Nebraska at Lincoln
 Department of Mathematics
Lincoln, Nebraska, United States


19942010

Purdue University
 Department of Mathematics
West Lafayette, Indiana, United States


1992

IT University of Copenhagen
København, Capital Region, Denmark


19781992

Bulgarian Academy of Sciences
 Institute of Mathematics and Informatics
Ulpia Serdica, SofiaCapital, Bulgaria


19851989

Medical University of Sofia
Ulpia Serdica, SofiaCapital, Bulgaria


1987

University of Toronto
Toronto, Ontario, Canada
