Name:  Irena Peeva
Employment:  Professor of Mathematics
Address:  Department of Mathematics
Cornell University
Ithaca, NY 14853, U.S.A.
E-mail:  irena@math.cornell.edu

   
 
 
 
conjectures and open problems   (pdf)
 
from my CV    (html)
 

        
               

RESEARCH INTERESTS:   My research interests are in Commutative Algebra and its connections to Algebraic Geometry, Algebraic Combinatorics, Subspace Arrangements, and Noncommutative Algebra.

Some of my research is focused on the structure of free resolutions and their applications. The study of resolutions is a beautiful and core area in commutative algebra with several challenging conjectures. The idea to associate a free resolution to a module was introduced in Hilbert's famous 1890,1893-papers. Resolutions provide a method for describing the structure of modules.




   SELECTED  PUBLICATIONS (annotated list by topic)


    On Infinite Free Resolutions:

  1. Matrix factorizations for complete intersections and minimal free resolutions, (with D. Eisenbud),
    submitted.     pdf file

      ABSTRACT:         The theory of matrix factorizations for hypersurfaces was introduced by Eisenbud thirty-five years ago. A major advance was made by Orlov, who showed that matrix factorizations could be used to study Kontsevich's homological mirror symmetry by giving a new description of singularity categories. Matrix factorizations have also proven useful for the study of cluster tilting, Cohen-Macaulay modules and singularity theory, knot theory, moduli of curves, quiver and group representations, and other topics. Starting with Kapustin and Li, who followed an idea of Kontsevich, physicists discovered amazing connections with string theory.
            Matrix factorizations of a hypersurface yield a description of the asymptotic structure of minimal free resolutions over the hypersurface, and also define a functor to the stable module category of maximal Cohen-Macaulay modules on the hypersurface. We introduce a functorial concept of matrix factorizations for complete intersections that allows us to describe the asymptotic structure of minimal free resolutions over complete intersections.
  2. Complete intersection dimension, (with L. Avramov and V. Gasharov),
    Publications Mathématiques IHES 86 (1997), 67-114.

      ABSTRACT:   In the study of minimal free resolutions over complete intersections the key idea is to relate a given infinite resolution to a finite resolution over a regular ring. The minimal free resolution of the residue field over a complete intersection was constructed by Tate and has a beautiful structure.
            We introduce a new homological dimension: CI-dimension (complete intersection dimension). The class of modules of finite CI-dimension contains all modules over a complete intersection. We obtain sharp quantitative and structural homological data for the modules of finite CI-dimension. For the study of these modules we develop some new cohomological tools and constructions over quantum symmetric algebras (in particular, we construct an extension of Manin's quantum Koszul complex). We find a place in the minimal free resolution of a module of finite CI-dimension after which asymptotically stable behavior develops; in particular, beyond this place the sequence of Betti numbers is either constant or strictly increasing, and gaps between consecutive numbers grow polynomially.
          CI-dimension localizes and stands between the projective dimension and the Gorenstein dimension (introduced by Auslander and Bridger). A fundamental homological result for modules of finite projective dimension is the Auslander-Buchsbaum Equality; we establish an analog to it for modules of finite CI-dimension. We also prove that a ring R is a complete intersection if and only if its residue field has finite CI-dimension; this result is an analog to Serre's criterion for regularity.

  3. Finite regularity and Koszul algebras, (with L. Avramov),
    American Journal of Mathematics 123 (2001), 275-281.

      ABSTRACT:         We consider positively graded commutative algebras finitely generated over a field, and finitely generated graded modules. The algebras over which all modules have finite projective dimension are characterized by the Auslander-Buchsbaum-Serre and Hilbert Syzygy Theorems, namely, the following three conditions are equivalent:
      • every module over the algebra has finite projective dimension
      • the residue field has finite projective dimension
      • the algebra is a polynomial ring in finitely many indeterminates.

            Castelnuovo-Mumford regularity is an important measure of the complexity of the differential in a graded minimal free resolution. Vanishing of the regularity of the residue field defines the class of Koszul algebras, which have received a lot of attention due to their extraordinary homological properties and to their appearance in many cases of interest in Algebraic Geometry, Algebraic Topology, Combinatorics, Commutative Algebra, and Representation Theory. We characterize those algebras over which all modules have finite Castelnuovo-Mumford regularity. We prove that the following three conditions are equivalent:
      • every module over the algebra has finite Castelnuovo-Mumford regularity
      • the residue field has finite Castelnuovo-Mumford regularity
      • the algebra is a polynomial ring in finitely many indeterminates over a Koszul algebra.

      In particular, our result shows that a conjecture by Avramov and Eisenbud (1990) holds.

  4. Boundedness versus periodicity over commutative local rings, (with V. Gasharov),
    Transactions of the American Mathematical Society 320 (1990), 569-580.
    This paper was written when I was an undergraduate student.

      ABSTRACT:   The Betti numbers in an infinite free resolution grow polynomially or exponentially in all cases when their behavior is known. It is expected that the boundedness of the Betti numbers is a strong condition. A module with bounded Betti numbers has eventually periodic minimal free resolution if the ground ring is a group algebra, a complete intersection ring, or a Gorenstein ring of small codepth. Eisenbud conjectured (1980) that this property holds over any local noetherian ring. We provide counterexamples to Eisenbud's conjecture. We construct modules over a Gorenstein artinian ring with an arbitrary period of the resolution, and also modules with constant Betti numbers and no periodicity in the resolution at all.

    On Hilbert Schemes:

  5. Connectedness of Hilbert schemes, (with M. Stillman),
    Journal of Algebraic Geometry, 14 (2005), 193-211.

      ABSTRACT:   Hartshorne proved that the Hilbert scheme, that parameterizes subschemes of Pr with a fixed Hilbert polynomial, is connected. Recent works by Eisenbud, Fløystad, and Schreyer have inspired us to study Hilbert schemes over exterior algebras. We show that the Hilbert scheme, that parameterizes all ideals with the same Hilbert function over an exterior algebra, is connected. We also give a new proof of Hartshorne's Theorem that the classical Hilbert scheme is connected.

  6. Flips and Hilbert schemes over an exterior algebra, (with M. Stillman),
    Mathematische Annalen, 339 (2007), 545-557.
  7. Hilbert schemes and Betti numbers over a Clements-Lindström ring, (with S. Murai),
    Compositio Mathematica, to appear.
  8. Toric Hilbert schemes, (with M. Stillman),
    Duke Mathematical Journal 111 (2002), 419-449.

      ABSTRACT:   We introduce and study the toric Hilbert scheme H which parameterizes all ideals with the same multigraded Hilbert function as a given toric ideal T.
          We show that the toric ideal T is a smooth point on H. Furthermore, we consider the case when T defines a toric variety of codimension two. We prove that in this case H is two dimensional and smooth; it follows that H is the toric variety of the Gröbner fan of the toric ideal T.

  9. Deformations of codimension 2 toric varieties, (with V. Gasharov),
    Compositio Mathematica 123 (2000), 225-241.

      ABSTRACT:   The study of the ideals with the same multigraded Hilbert function as a given toric ideal T was initiated by Arnold, who showed that the structure of such ideals is encoded in continued fractions in the case when T defines a monomial curve in A3.
          Our main result solves a Conjecture of Sturmfels': we prove that if T has codimension two, then the toric Hilbert scheme has exactly one component and this component is the closure of the orbit of the toric ideal T under the torus action. For monomial curves in A3 this result was proved by Arnold, Korkina, Post, and Roelofs. Through a systematic computer search Sturmfels found an example in codimension 3 which shows that the result cannot be extended to higher codimensions.

    On Toric Varieties:

  10. Generic lattice ideals, (with B. Sturmfels),
    Journal of the American Mathematical Society, 11 (1998), 363-373.

      ABSTRACT:   Complete intersections are ideals whose generators have sufficiently general coefficients. So they might be regarded as generic among all ideals with fixed small number of generators. We introduce a different notion of genericity: toric ideals whose generators are generic with respect to their exponents -- not their coefficients. We construct the minimal free resolution of a generic toric ring; it has a beautiful structure.

  11. Rationality for generic toric rings, (with V. Gasharov and V. Welker),
    Mathematische Zeitschrift 233 (2000), 93-102.

      ABSTRACT:   We consider the infinite minimal free resolution of the residue field over a generic toric ring. Problems of rationality of Poincaré and Hilbert series were stated by several mathematicians: by Serre and Kaplansky for local Noetherian rings, by Kostrikin and Shafarevich for nilpotent algebras, by Govorov for associative graded algebras and by Serre and Moore for simply-connected complexes. The Serre-Kaplansky problem, ``Is the Poincaré series of a finitely generated commutative local Noetherian ring rational?'', was one of the central questions in Commutative Algebra for many years. Anick constructed a ring with transcendental Poincaré series which provided a counterexample to the conjecture that the Serre-Kaplansky problem has a positive answer. Our main result provides a positive answer for generic toric rings.

  12. How to shell a monoid? , (with V. Reiner and B. Sturmfels),
    Mathematische Annalen 310 (1998), 379-393.

      ABSTRACT:         We introduce two ideas on how to obtain the Betti numbers of the residue field over a toric ring:
      • We consider the toric variety as given in non-commutative variables in order to identify monomials with facets of simplicial complexes.
      • We show that a quadratic non-commutative Gröbner basis describes algebraically a uniform non-pure shelling of the toric ring.
  13. Syzygies of codimension 2 lattice ideals, (with B. Sturmfels),
    Mathematische Zeitschrift 229 (1998), 163-194.

      ABSTRACT:   We introduce an approach to study the free resolutions of toric rings via integer-points-free bodies. This leads to an upper bound 2codimension-1 on the projective dimension of an arbitrary toric ring; the existence of a bound in terms of the codimension is surprising and has no analogues for other classes of rings yet. In the codimension 2 case the integer-points-free bodies have simple structure; this makes it possible to construct the minimal free resolution of the toric ring; it has an elegant combinatorial structure.

    On Subspace Arrangements:

  14. Cohomology of real diagonal subspace arrangements via resolutions, (with V. Reiner and V. Welker),
    Compositio Mathematica 117 (1999), 107-123.

      ABSTRACT:   Consider the topology of the complement of a subspace arrangement. Much fewer results are known for real subspace arrangements than for complex ones. The reason for this is that Algebraic Geometry methods can be applied in the complex case, but not in the real case. We introduce an approach of expressing the cohomology of the complement of a real diagonal arrangement of subspaces by the Betti numbers of a minimal free resolution. Our approach is different from the well-known method for obtaining the cohomology of such a complement using a formula of Goresky and MacPherson.

  15. Hyperplane arrangements and linear strands in resolutions,
    Transactions of the American Mathematical Society, 355 (2003), 609-618.

    On Hilbert Functions:

  16. Hilbert schemes and maximal Betti numbers over Veronese rings, (with V. Gasharov and S. Murai),
    Mathematische Zeitschrift, 267 (2011), 155-172.
  17. Lexifying ideals, (with J. Mermin),
    Mathematical Research Letters 13 (2006), 409-422.

      ABSTRACT:   A well-studied and important numerical invariant of a homogeneous ideal (over a standard graded quotient of a polynomial ring) is its Hilbert function. It gives the sizes of the graded components of the ideal. Hilbert functions over a polynomial ring are characterized by Macaulay's Theorem. The key idea is that there exist highly structured monomial ideals - lex ideals - which attain all Hilbert functions. It is known that Macaulay's Theorem holds over an exterior algebra and over a Clements-Lindström ring. We raise the problem to describe Hilbert functions over other quotient rings, and we make the first steps to study over what other quotient rings it holds that all Hilbert functions are attained by lex ideals.

    On Monomial Resolutions:

  18. Monomial resolutions, (with D. Bayer and B. Sturmfels),
    Mathematical Research Letters 5 (1998), 36-41.

      ABSTRACT:   We introduce an idea entirely different than the well-known Stanley-Reisner correspondence: we introduce an elegant approach for resolving a monomial ideal by encoding the whole resolution (including the differential maps) into a single simplicial complex. We prove that generically such resolution is minimal and comes from the boundary of a polytope. For a non-generic ideal we introduce the technique of resolving by ``deforming to the generic case".
            Generalizing our approach, one can consider cellular resolutions. Such resolutions are built (by several authors) in many cases of interest.

  19. The lcm-lattice, (with V. Gasharov and V. Welker),
    Mathematical Research Letters 6 (1999), 521--532.

      ABSTRACT:         For many years, the Stanley-Reisner correspondence, introduced by Hochster and Reisner, was the main progress on the problem (posed by Kaplansky in the early 1960's) of finding a minimal free resolution of a monomial ideal. It has a long tradition and many applications. The Stanley-Reisner theory is based on computing the Betti numbers of a monomial ideal by simplicial complexes.
          We introduce a new approach inspired by the topological theory of subspace arrangements. We introduce the lcm-lattice of a monomial ideal. We show that it plays the same role in describing the homology of the ideal as the role of the intersection lattice in describing the cohomology of the complement of a complex subspace arrangement. Namely:
      • the lcm-lattice determines the Betti numbers (analog of the Goresky-MacPherson Formula for subspace arrangements)
      • the lcm-lattice determines the maps in the minimal free resolution up to relabeling
      • the lcm-lattice together with the additional data, which pairs of minimal monomial generators are relatively prime, determine the algebra structure of the Tor-algebra (analog of results by DeConcini-Procesi and Yuzvinsky for subspace arrangements).
  20. Ideals containing the squares of the variables, (with J. Mermin and M. Stillman),
    Advances in Mathematics, 217 (2008), 2206-2230.

      ABSTRACT:   It is well known how the Hilbert function changes when we add the squares of the variables to a monomial ideal. We describe how the minimal free resolution changes.

  21. Frames and degenerations of monomial resolutions, (with M. Velasco),
    Transactions of the American Mathematical Society, to appear.

      ABSTRACT:   We introduce the homogenization of a frame of a monomial free resolution. The frame is a complex of vector spaces. The key idea in the paper is that we prove that the problem of constructing a minimal monomial free resolution is equivalent to the problem of building its frame. Thus, the frame-concept provides the minimal free resolution of any monomial ideal, while it is known that CW-cellular resolutions cannot. Another advantage is that our approach provides a framework which allows to treat several important constructions and results on monomial resolutions by Bayer, Gasharov, Peeva, Sturmfels, Welker as particular cases.
            Using frames has the potential to extend the class of combinatorial and topological objects which can be used for constructions of monomial resolutions. Several applications of our approach are provided. For example, we introduce nearly Scarf ideals and construct their minimal free resolutions; a subsequent result by Velasco shows that nearly Scarf ideals provide the first example of a monomial ideal whose minimal free resolution is not supported by any CW-complex.

    Expository Papers:

  22. Open Problems on syzygies and Hilbert functions, (with M. Stillman),
    Journal of Commutative Algebra, 1 (2009), 159-195.     pdf file