Luchezar L. Avramov

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

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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 non-zero 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.
    Full-text · Article · Aug 2015
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    Luchezar L. Avramov · Winfried Bruns · Srikanth B. Iyengar

    Full-text · Article · Jun 2014 · Journal of Pure and Applied Algebra
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    Luchezar L. Avramov · Aldo Conca · Srikanth B. Iyengar
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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.
    Full-text · Article · Aug 2013 · Mathematische Annalen
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    Luchezar L. Avramov
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    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 finite-dimensional homology. It induces an equivalences of $D^{df}_{b}(A)^{op}$ and the category of perfect DG $RHom_A(K,K)$-modules, and vice-versa. Corresponding statements are proved also when $H(A)$ is simply connected and $H^{<0}(A)=0$.
    Preview · Article · May 2013
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    Luchezar L. Avramov · Srikanth B. Iyengar
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    ABSTRACT: For any non-zero 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 non-zero 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.
    Full-text · Article · Aug 2012 · Journal of Commutative Algebra
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    Luchezar L. Avramov · Srikanth B. Iyengar · Joseph Lipman
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    ABSTRACT: We prove basic facts about reflexivity in derived categories over noetherian schemes; and about related notions such as semidualizing complexes, invertible complexes, and Gorenstein-perfect maps. Also, we study a notion of rigidity with respect to semidualizing complexes, in particular, relative dualizing complexes for Gorenstein-perfect 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
    Full-text · Article · Sep 2011 · Algebra and Number Theory
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    Luchezar L. Avramov
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    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 $e-d\le 3$. For each such $R$ it is proved that there is a real number $\gamma>1$, such that $\mu^{d+i}_R\ge\gamma\mu^{d+i-1}_R$ 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.
    Preview · Article · May 2011 · Journal of Pure and Applied Algebra
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    Full-text · Article · Dec 2010 · Advances in Mathematics
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    Luchezar L. Avramov · Melvin Hochster · Srikanth B. Iyengar · Yongwei Yao
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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 non-zero finite R-module has finite flat dimension or finite injective dimension for the R-module 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 non-zero finitely generated R-module grow at the same rate, on an exponential scale, as the Betti numbers of k over R.
    Full-text · Article · Oct 2010 · Mathematische Annalen
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    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.
    Full-text · Article · Oct 2010 · Pure and applied mathematics quarterly
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    H. Ananthnarayan · Luchezar L. Avramov · W. Frank Moore
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    ABSTRACT: Given surjective homomorphisms R → T ← S of local rings, and ideals in R and S that are isomorphic to some T-module 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.
    Full-text · Article · May 2010 · Journal für die reine und angewandte Mathematik (Crelles Journal)
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    Luchezar L. Avramov · Srikanth B. Iyengar · Liana M. Sega
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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 $H_R(s)=1+es+(e-1)s^{2}$. When $H_R(s)=1+es+rs^{2}$ with $r\leq e-1$, and R is Gorenstein or $e=r+1\le 3$, it is proved that generic R-modules with $q\leq(e-1)p$ are linear. Comment: To appear in the special issue of the Journal of Commutative Algebra, dedicated to Ralf Froeberg's 65th birthday.
    Full-text · Article · May 2010 · Journal of Commutative Algebra
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    Luchezar L. Avramov · Srikanth B. Iyengar
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    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.
    Full-text · Article · Mar 2010
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    Luchezar L. Avramov
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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.
    Preview · Chapter · Mar 2010
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    Luchezar L. Avramov · Srikanth B. Iyengar
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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 d-fold 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 analog---but not a consequence---of a classical criterion of Auslander and Lichtenbaum for the freeness of finitely generated modules over regular local rings.
    Full-text · Article · Feb 2010 · Journal of Algebraic Geometry
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    Luchezar L. Avramov · Srikanth B. Iyengar · Joseph Lipman · Suresh Nayak
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    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 S-modules D, and natural reduction isomorphisms for all complexes of S-modules 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 finite-type flat map of noetherian schemes, with f!OY in place of D.
    Full-text · Article · Jan 2010 · Advances in Mathematics
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    Luchezar L. Avramov · Srikanth B. Iyengar · Joseph Lipman
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    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 C-rigid 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 C-reflexivity is a local property; it implies that a finite R-module M has finite G-dimension over R if it is locally of finite G-dimension. Comment: 31 pages. Major revisions in Sections 1 and 6. To appear in `Algebra and Number Theory'
    Full-text · Article · Jan 2010 · Algebra and Number Theory
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    Luchezar L. Avramov · Melvin Hochster

    Preview · Article · Nov 2009 · Journal of Algebra
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    Luchezar L. Avramov · Aldo Conca · Srikanth B. Iyengar
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    ABSTRACT: For R=Q/J with Q a commutative graded algebra over a field and J non-zero, 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 non-negative integer i, and also for j=2i when i>dim Q-dim R and pd_QR is finite.
    Full-text · Article · May 2009 · Mathematical Research Letters
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    Luchezar L. Avramov · Srikanth B. Iyengar · Liana M. Sega
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    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 non-Koszul modules are classified. Structure theorems are established for the graded k-algebra ExtR(k, k) and its graded module ExtR(M, k).
    Full-text · Article · Aug 2007 · Journal of the London Mathematical Society

Publication Stats

2k Citations
64.67 Total Impact Points

Institutions

  • 2002-2013
    • University of Nebraska at Lincoln
      • Department of Mathematics
      Lincoln, Nebraska, United States
  • 1994-2010
    • Purdue University
      • Department of Mathematics
      West Lafayette, Indiana, United States
  • 1992
    • IT University of Copenhagen
      København, Capital Region, Denmark
  • 1978-1992
    • Bulgarian Academy of Sciences
      • Institute of Mathematics and Informatics
      Ulpia Serdica, Sofia-Capital, Bulgaria
  • 1985-1989
    • Medical University of Sofia
      Ulpia Serdica, Sofia-Capital, Bulgaria
  • 1987
    • University of Toronto
      Toronto, Ontario, Canada