SELECTED  PUBLICATIONS

On Infinite Resolutions:

    Complete intersection dimension    pdf file
    (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 filed over a complete intersection was constructed by Tate.
            We introduce a new homological dimension: CI-dimension (complete intersection dimension). The class of modules of finite CI-dimension contains all modules of finite projective dimension and all modules over a complete intersection. We obtain sharp quantitative and structural homological data for the modules of finite CI-dimension. Such modules provide the first class of modules of infinite projective dimension with a rich structure theory of free resolutions. 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.
    Finite regularity and Koszul algebras   pdf file
    (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: 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 considerable 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: In particular, this shows that a conjecture by Avramov and Eisenbud (1990) holds.
    Boundedness versus periodicity over commutative local rings   pdf file
    (with V. Gasharov)
    Transactions of the American Mathematical Society 320 (1990), 569-580.
    This paper was written while V. Gasharov and I were undergraduate students.
      ABSTRACT:   The Betti numbers in an infinite 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 arbitrary period of the resolution, and also modules with constant Betti numbers and no periodicity in the resolution at all.

On Hilbert Schemes:

    Connectedness of Hilbert schemes    pdf file
    (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 D. Eisenbud, G. Fløystad, and F.-O. 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 give a new proof of Hartshorne's Theorem that the classical Hilbert scheme is connected. Our construction is different than Hartshorne's: if Q is either a polynomial ring or an exterior algebra, we prove that every two strongly stable ideals in Q with the same Hilbert function are connected by a sequence of binomial gröbner deformations.
    Flips and Hilbert schemes over an exterior algebra    pdf file
    (with M. Stillman)
    Mathematische Annalen, 339 (2007), 545-557.
      ABSTRACT:   We have showed that the Hilbert scheme, that parameterizes all ideals with a fixed Hilbert function over an exterior algebra, is connected. Now, we introduce the notion of flips and show that the basic flips form a basis of the tangent space at a monomial point in the Hilbert scheme. This implies that the the tangent space has a basis consisting of directions tangent to deformations built using Gröbner basis. Such a nice structure of the tangent space is surprising since it does not hold in the polynomial case.
    Hilbert schemes and Betti numbers over a Clements-Lindström ring    pdf file
    (with S.Murai)
    submitted.
      ABSTRACT:   The structure of the Hilbert scheme is known to be very complicated, and it seems that this discouraged people from studying Hilbert schemes over quotient rings (as the proofs and constructions become significantly more intricate over quotient rings). The Clements-Lindström rings are a natural class of quotient rings to consider because Macaulay's Theorem holds over them. This paper is focused on such rings. We show that the Hilbert scheme, that parametrizes all ideals with the same Hilbert function over a Clements-Lindtröm ring W, is connected. More precisely, we prove that every graded ideal is connected by a sequence of deformations to the lex-plus-powers ideal with the same Hilbert function. Our result is an analogue of Hartshorne's theorem that Grothendieck's Hilbert scheme is connected; however our proof is different, since Hartshorne's deformations (distractions) do not work over W.
          We also prove a conjecture by Gasharov, Hibi, and Peeva that the lex ideal attains maximal Betti numbers among all graded ideals in W with a fixed Hilbert function.
    Hilbert schemes and maximal Betti numbers over Veronese rings    pdf file
    (with V. Gasharov and S. Murai)
    submitted.
      ABSTRACT:   We show that Macaulay's Theorem and Gotzmann's Persistence Theorem hold over a Veronese toric ring R. We also prove that the Hilbert scheme over R is connected; this is an analogue of Hartshorne's theorem that the Hilbert scheme over a polynomial ring is connected.
    Deformations of codimension 2 toric varieties    pdf file
    (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 generalizes a result of Arnold-Korkina-Post-Roelofs and 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:

    Generic lattice ideals   pdf file
    (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. For toric rings we define a notion of genericity which ensures nicely structured homological behavior. Our motivation comes from a strong recent result in Integer Programming. We construct the finite minimal free resolution of a generic toric ring.
    Toric Hilbert schemes   pdf file
    (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 define the structure of H and prove the universality property. The latter makes it possible to obtain the tangent space at any point on H.
          The lex ideal is a smooth point on the classical Hilbert scheme. Usually no lex ideal exists on H. However, there is another special point: we show that the toric ideal T lies on exactly one component of H and this component is reduced. In particular, 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. In sharp contrast, very little is known about the structure of the classical Hilbert scheme of a codimension two toric variety; the best result in this direction is proved by Piene and Schlessinger and says that the classical Hilbert scheme of the twisted cubic curve has two components of dimensions 12 and 15, each component is smooth but the scheme is not, the two components intersect transversally and their intersection is smooth of dimension 11.
    Rationality for generic toric rings   pdf file
    (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 o f 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. We also show that the rate of a generic toric ring is the maximum degree of a minimal generator of the corresponding toric ideal minus one; this is an analogue to a result (for monomial ideals) of Eisenbud, Reeves, and Totaro.
    How to shell a monoid?     pdf file
    (with V. Reiner and B. Sturmfels)
    Mathematische Annalen 310 (1998), 379-393.
      ABSTRACT:   We introduce two ideas how to obtain the Betti numbers of the residue field over a toric ring:
    Syzygies of codimension 2 lattice ideals     pdf file
    (with B. Sturmfels)
    Mathematische Zeitschrift 229 (1998), 163-194.
      ABSTRACT:   We introduce an approach to study the algebraic homology 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 describe combinatorially the minimal free resolution of the toric ring.
          A challenging central conjecture in Algebraic Geometry states that the Castelnuovo-Mumford regularity of a complex projective variety is bounded by the degree minus the codimension plus one. Recently Kraft showed that the conjecture implies important bounds in Invariant Theory. It is known to hold for curves (by a result of Gruson-Lazarsfeld-Peskine), Cohen-Macaulay rings, and smooth surfaces. We prove the conjecture for codimension 2 toric rings (note that we use a weaker definition of a toric variety, so such a variety is not always Cohen-Macaulay).

On Monomial Resolutions:

    The lcm-lattice    pdf file
    (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 has led to important results. 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:
    Monomial resolutions    pdf file
    (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 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". This provides a non-minimal resolution whose length is less or equal to the number of the variables (the resolution is usually much shorter/smaller than Taylor's resolution).
    Ideals containing the squares of the variables    pdf file
    (with J. Mermin and M. Stillman)
    Advances in Mathematics, 217 (2008), 2206-2230.
      ABSTRACT:   Denote by P the ideal generated by the squares of the variables in a polynomial ring. It is well known how the Hilbert function changes when we add P to a squarefree monomial ideal J; this is given by the relation between the f-vector and the h-vector. It has been an open question how the Betti numbers change. We answer this question providing a relation between the Betti numbers of J and those of J+P. Furthermore, we describe a basis of the minimal free resolution of J+P in the case when J is Borel.
            By Kruskal-Katona's Theorem, there exists a squarefree lex ideal L such that L+P has the same Hilbert function as J+P. The ideal L+P is called lex-plus-squares. It was conjectured by Herzog and Hibi that the graded Betti numbers of L+P are greater than or equal to those of J+P. Later, Graham Evans conjectured the more general lex-plus-powers conjecture that, among all graded ideals with a fixed Hilbert function and containing a homogeneous regular sequence in fixed degrees, the lex-plus-powers ideal has greatest graded Betti numbers in characteristic 0. This conjecture is very difficult and wide open. We prove it for graded ideals containing the squares of the variables.

On Subspace Arrangements:

    Cohomology of real diagonal subspace arrangements via resolutions   pdf file
    (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 completely different from the well-known method for obtaining the cohomology of such complement using a formula of Goresky and MacPherson. The new approach yields short proofs of results on r-equals arrangements, opens up the possibility to compute examples fast, and leads to results on vanishing of cohomology.

On Hilbert Functions:

    Hilbert functions and lex ideals    pdf file
    (with J. Mermin)
    Journal of Algebra 313 (2007), 642-656.
      ABSTRACT:   A well-studied and important numerical invariant of a homogeneous ideal over a graded polynomial ring S is its Hilbert function. It gives the sizes of the graded components of the ideal. In many of the recent papers and books, Hilbert functions are described using Macaulay's representation with binomials. Thus, the arguments consist of very clever computations with binomials. One of our main goals is to go back to Macaulay's original idea in 1927: there exist highly structured monomial ideals - lex ideals - that attain all possible Hilbert functions.
          Let W be a quotient of S modulo powers of the variables. We present an algebraic proof of Clements-Lindstrom's Theorem that every Hilbert function in W is attained by a lex ideal. We prove Green's Theorem over W. It is an open question whether Gotzmann's Theorem holds over W.