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

Research Interests:   My primary work is in Commutative Algebra, and my primary research is focused on Free Resolutions and Hilbert Functions. I have also done work on the many connections of Commutative Algebra with Algebraic Geometry, Combinatorics, Computational Algebra, Noncommutative Algebra, and Subspace Arrangements, and I remain very interested in these fields as well.

The study of free resolutions and Hilbert functions is a beautiful and core area in Commutative Algebra. It contains a number of challenging conjectures and open problems. The idea to associate a free resolution to a module was introduced by Hilbert in his famous paper Über the Theorie von Algebraischen Formen. Resolutions provide a method for describing the structure of modules.


    CV
    List of Publications
    Research Summary
    Professional Bio

   
 
Preprints:
  • Layered resolutions of Cohen-Macaulay modules   (joint with D. Eisenbud)
  • Counterexamples to the Regularity Conjecture   (joint with J. McCullough)
  •   
    Notes from my AMS Invited Address at the national Joint Mathematics Meetings in 2015
      
    Swanson's article in the Notices of the AMS (March 2017) highlighting my joint work with McCullough on the Regularity Conjecture
     
    Expository Papers:
  • Conjectures and Open Problems on Infinite Free Resolutions
  • Three Themes of Syzygies
  • Conjectures and Open Problems on Finite Free Resolutions
  •   
    Book:   Graded Syzygies, I. Peeva, Springer, 2011.

     
      
    Books (with expository papers) edited by Peeva:
      Commutative Algebra, Springer, 2013.
      Syzygies and Hilbert functions, Lect. Notes Pure Appl. Math. 254 (2007), Chapman and Hall/CRC.

           
     
     
     
                   

     
    Castelnuovo-Mumford regularity and the Eisenbud-Goto Conjecture

    Projective dimension and Castelnuovo-Mumford regularity are numerical invariants which measure the complexity of the structure of homogeneous ideals in a polynomial ring. Hilbert's Syzygy Theorem shows that projective dimension is smaller than the number of variables. A doubly exponential upper bound on the regularity is proved by Bayer-Mumford (1993) and Caviglia-Sbarra (2005). The bound is nearly sharp since the Mayr-Meyer construction leads to examples of families of ideals attaining doubly exponential regularity. In contrast, Bertram-Ein-Lazarsfeld (1991), Mumford (1993), and Chardin-Ulrich (2002) have proven that there are nice bounds on the regularity of the ideals of smooth (or nearly smooth) projective varieties. As discussed in an influential paper by Bayer and Mumford (1993), the biggest missing link between the general case and the smooth case is to obtain a decent bound on the regularity of all reduced equidimensional ideals. In particular, there has been a lot of interest in producing a bound on the regularity of all prime ideals (the ideals that define irreducible projective varieties). The longstanding Regularity Conjecture, by Eisenbud-Goto (1984), predicts the following elegant linear bound in terms of the degree:
                    reg(L)  ≤   deg(U/L) -  codim (L) + 1
    for every homogeneous non-degenerate prime ideal L in a standard graded polynomial ring U over an algebraically closed field.

    The Regularity Conjecture is proved for curves by Gruson-Lazarsfeld-Peskine (1983), completing classical work of Castelnuovo. It is also proved for smooth surfaces by Lazarsfeld (1987) and Pinkham (1986), and for most smooth 3-folds by Ran (1990). It holds if U/L is Cohen-Macaulay by a result of Eisenbud-Goto (1984). In the smooth case, Kwak (1998) gave bounds for regularity in dimensions 3 and 4 that are only slightly worse than the optimal ones. Many other special cases and related bounds have been proved as well.

    Jason McCullough and I provide counterexamples to the Regularity Conjecture . Our main theorem is much stronger and shows that the regularity of non-degenerate homogeneous prime ideals is not bounded by any polynomial function of the degree; this holds over any field k (the case k=C is particularly important). We provide a family of prime ideals Pr, depending on a parameter r, whose regularity is doubly exponential in r and whose degree is singly exponential in r. For this purpose, we introduce an approach, which starting from a homogeneous ideal I, produces a prime ideal P whose regularity, degree, projective dimension, dimension, depth, and codimension are expressed in terms of numerical invariants of I. Our construction involves two new concepts:
          (1) Rees-like algebras which, unlike the standard Rees algebras, have well-structured defining equations and minimal free resolutions;
          (2) a new homogenization technique for prime ideals which, unlike classical homogenization, preserves graded Betti numbers.

     


     
    Free Resolutions over Complete Intersections

    Minimal free resolutions over a local complete intersection R have attracted attention ever since the elegant construction of the minimal free resolution of the residue field k by Tate in 1957. The next impressive result was Gulliksen's proof in 1974 that for every finitely generated R-module N, the Poincaré series ∑bi(N)ti (where bi(N) are the Betti numbers over R) is rational and its denominator divides (1-t2)c (where c is the codimension of R). For this purpose, he showed that ExtR(N,k) can be regarded as a finitely generated graded module over a polynomial ring k[χ 1,...,χ c], graded by deg(χ i)=2. This also implies that the even Betti numbers bi(N) are eventually given by a polynomial in i, and the odd Betti numbers are given by another polynomial. In 1989 Avramov proved that the two polynomials have the same leading coefficient and the same degree. In 1997 Avramov, Gasharov and Peeva showed that the truncated Betti sequence {bi(N)}i≥ q is either strictly increasing or constant for q≫ 0 and proved further properties of the Betti numbers. We focus on resolutions of high syzygies, since there are examples, constructed by Eisenbud, of minimal resolutions over complete intersections that have intricate structure, but exhibit stable patterns when sufficiently truncated.

    The theory of matrix factorizations was introduced by Eisenbud in 1980 to describe the asymptotic structure of minimal free resolutions over a hypersurface. A matrix factorization of a non-zero element f in a regular local ring S is a pair (d,h) of maps of finitely generated S-free modules d: A1→ A0, h: A0→ A1 such that hd = f. IdA1 and dh = f. IdA0. This concept has many other applications: for example, in the study of Cohen-Macaulay modules and Singularity Theory, Cluster Tilting, Hodge Theory, Khovanov-Rozansky Homology, Moduli of Curves, Quiver and Group Representations, and Singularity Categories. Starting with Kapustin and Li, who followed an idea of Kontsevich, physicists discovered amazing connections with String Theory. Despite all this work on applications, progress on the structure of minimal free resolutions over complete intersections was scant.

    As mentioned above, in 1980 Eisenbud described the minimal free resolutions of high syzygies over a hypersurface. In 2000, Avramov and Buchweitz analyzed the codimension 2 case. But the general case (of higher codimensions) was elusive. In contrast, non-minimal resolutions have been known for over 45 years from the work of Shamash. The condition of minimality is important. The mere existence of free resolutions suffices for foundational issues such as the definition of Ext and Tor, and there are various methods of producing resolutions uniformly (for example, the Bar resolution). But without minimality, resolutions are not unique, and the very uniformity of constructions like the Bar resolution implies that they give little insight into the structure of the modules resolved.

    Eisenbud and I have wondered, for many years, how to describe the eventual patterns in the minimal resolutions of modules over complete intersections of higher codimension (c ≥ 3). With the theory developed in
            the research monograph   Minimal Free Resolutions over Complete Intersections

            and the paper   Layered resolutions of Cohen-Macaulay modules
    we believe we have found an answer. For this purpose, we introduce a new concept of higher matrix factorization (d,h) with respect to a regular sequence; this extends the theory of matrix factorizations. We obtain the following results: Let S be a regular local ring with infinite residue field k, and let I⊂ S be an ideal generated by a regular sequence of length c. Let N be a finitely generated module over the complete intersection R := S/I, and let M be a sufficiently high syzygy of N over R. (Note that the minimal resolution of M over R is a truncation of the minimal resolution of N.) We prove that there exists a minimal higher matrix factorization (d,h), with respect to a generic choice of generators f1, ..., fc of I, such that M is its higher matrix factorization module Coker(R⊗ d). We construct the minimal free resolution of M over the complete intersection R. We also construct the minimal free resolution of M over the regular local ring S.