Let $\check{C}_{\underline{x}}$ denote the Čech complex with respect to a system of elements $\underline{x} = x_1,\ldots,x_r$ of a commutative ring $R$. We construct a bounded complex $\mathcal{L}_{\underline{x}}$ of free $R$-modules and a quasi-isomorphism $\mathcal{L}_{\underline{x}} \stackrel{\sim}{\longrightarrow} \check{C}_{\underline{x}}$ and isomorphisms $\mathcal{L}_{\underline{x}} \otimes_R X \cong K^{\bullet}(\underline{x}-\underline{U}; X[\underline{U}^{-1}])$ and $\operatorname{Hom}_R(\mathcal{L}_{\underline{x}},X) \cong K_{\bullet}(\underline{x}-\underline{U};X[[\underline{U}]])$ for an $R$-complex $X$. Here $\underline{x} - \underline{U}$ denotes the sequence of elements $x_1-U_1,\ldots,x_r-U_r$ in the polynomial ring $R[\underline{U}] = R[U_1,\ldots,U_r]$ in the variables $\underline{U}= U_1,\ldots,U_r$ over $R$. Moreover $X[[\underline{U}]]$ denotes the formal power series complex of $X$ in $\underline{U}$ and $X[\underline{U}^{-1}]$ denotes the complex of inverse polynomials of $X$ in $\underline{U}$. Furthermore $K_{\bullet}(\underline{x}-\underline{U};X[[\underline{U}]])$ resp. $K^{\bullet}(\underline{x}-\underline{U}; X[\underline{U}^{-1}])$ denotes the corresponding Koszul complex resp. the corresponding Koszul co-complex. In particular, there is a bounded $R$-free resolution of $\check{C}_{\underline{x}}$ by a certain Koszul complex. This has various consequences e.g. in the case when $\underline{x}$ is a weakly pro-regular sequence. Under this additional assumption it follows that the local cohomology $H^i_{\underline{x} R}(X)$ and the left derived functors of the completion $Λ_i^{\underline{x} R}(X), i \in \mathbb{Z},$ is a certain Koszul cohomology and Koszul homology resp. This provides new approaches to the right derived functor of torsion and the left derived functor of completion with various applications.
We describe Koszul type complexes associated with a linear map from any module to a free module, and vice versa with a linear map from a free module to an arbitrary module, generalizing the classical Koszul complexes. Given a short complex of finite free modules, we assemble these complexes to what we call Koszul bicomplexes. They are used in order to investigate the homology of the Koszul complexes in projective dimension one. As in the case of the classical Koszul complexes this homology turns out to be grade sensitive. In a special setup we obtain necessary conditions for a map of free modu…
Let $R$ be a standard graded commutative algebra over a field $k$, let $K$ be its Koszul complex viewed as a differential graded $k$-algebra, and let $H$ be the homology algebra of $K$. This paper studies the interplay between homological properties of the three algebras $R$, $K$, and $H$. In particular, we introduce two definitions of Koszulness that extend the familiar property originally introduced by Priddy: one which applies to $K$ (and, more generally, to any connected differential graded $k$-algebra) and the other, called strand-Koszulness, which applies to $H$. The main theoretical res…
Let $M$ be a finite module over a noetherian ring $R$ with a free resolution of length 1. We consider the generalized Koszul complexes $\mathcal{C}_{\barλ}(t)$ associated with a map $\barλ:M\to\mathcal{H}$ into a finite free $R$-module $\mathcal{H}$ (see [IV], section 3), and investigate the homology of $\mathcal{C}_{\barλ}(t)$ in the special setup when $\grade I_M=\rank M=\dim R$. ($I_M$ is the first non-vanishing Fitting ideal of $M$.) In this case the (interesting) homology of $\mathcal{C}_{\barλ}(t)$ has finite length, and we deduce some length formulas. As an application we give a short a…
The object of the paper is the dependence of Koszul complexes and dependence of dual Koszul complexes of two systems of non-homogeneous polynomials, when one system is a part of other system, in connection with the duality in a Koszul complex established by author earlier. Whence, the dependence of Koszul complexes and dependence of dual Koszul complexes follow when one system is linearly expressed through other system. Obtaned results are used in the proof of homotopic equivalence, formulated earlier by the author, of the Koszul complex and dual Koszul complex of a system of non-homogeneous p…
The eThe existence of common zero of a family of polynomials has led to study of inertial forms whose homogeneous part of degree 0 constitutes the ideal resultant. The Koszul anc Cech cohomology groups play a fundamental role in this study. An analogueous of Hurwitz theorem is given and also we finds a MCoy theorem in a particular case of this study.
Let C be a semidualizing complex over a noetherian local ring A. If there exists a local homomorphism with source A satisfying some homological properties, then C is dualizing.
We introduce the notions of Koszul $N$-complex, $\check{\mathrm{C}}$ech $N$-complex and telescope $N$-complex, explicit derived torsion and derived completion functors in the derived category $\mathbf{D}_N(R)$ of $N$-complexes using the $\check{\mathrm{C}}$ech $N$-complex and the telescope $N$-complex. Moreover, we give an equivalence between the category of cohomologically $\mathfrak{a}$-torsion $N$-complexes and the category of cohomologically $\mathfrak{a}$-adic complete $N$-complexes, and prove that over a commutative noetherian ring, via Koszul cohomology, via RHom cohomology (resp. $\oti…
Let A be a noetherian AS regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and grA the category of finitely generated graded left A-modules. Following Jorgensen, we define the Castelnuovo-Mumford regularity reg(X) of a complex X \in D^b(grA) in terms of the local cohomologies or the minimal projective resolution of X. Let A^! be the quadratic dual ring of A. For the Koszul duality functor DG : D^b(grA) -> D^b(grA^!), we have reg(X) = max {i | H^i(DG (X)) \ne 0}. Using these concepts, we study weakly Koszul modules (= componentwise linear modules) over A^…
Let R be a commutative noetherian local ring with completion R^. We apply differential graded (DG) algebra techniques to study descent of modules and complexes from R^ to R' where R' is either the henselization of R or a pointed étale neighborhood of R: We extend a given R^-complex to a DG module over a Koszul complex; we describe this DG module equationally and apply Artin approximation to descend it to R. This descent result for Koszul extensions has several applications. When R is excellent, we use it to descend the dualizing complex from R^ to a pointed étale neighborhood of R; this yields…
We extend the theory of Koszul and Buchsbaum-Eisenbud complexes to modules over commutative OI-algebras and show that they still have the familiar properties of the classical complexes. In particular, the OI-complexes are generically acyclic and often provide width-wise minimal free resolutions.
Differential modules are natural generalizations of complexes. In this paper, we study differential modules with complete intersection homology, comparing and contrasting the theory of these differential modules with that of the Koszul complex. We construct a Koszul differential module that directly generalizes the classical Koszul complex and investigate which properties of the Koszul complex can be generalized to this setting.
We study the interplay between the cohomology of the Koszul complex of the partial derivatives of a homogeneous polynomial $f$ and the pole order filtration $P$ on the cohomology of the open set $U=\PP^n \setminus D$, with $D$ the hypersurface defined by $f=0$. The relation is expressed by some spectral sequences, which may be used on one hand to determine the filtration $P$ in many cases for curves and surfaces, and on the other hand to obtain information about the syzygies involving the partial derivatives of the polynomial $f$. The case of a nodal hypersurface $D$ is treated in terms of the…
The Rémond resultant attached to a multiprojective variety and a sequence of multihomogeneous polynomials is a polynomial form in the coefficients of the polynomials, which vanishes if and only if the polynomials have a common zero on the variety. We demonstrate that this resultant can be computed as a Cayley determinant of a multigraded Koszul complex, proving a key stabilization property with the aid of local Hilbert functions and the notion of filter-regular sequences. Then we prove that the Rémond resultant vanishes, under suitable hypotheses, with order at least equal to the number of com…
Let R be a commutative ring and I an ideal in R which is locally generated by a regular sequence of length d. Then, each projective R/I-module V has an R-projective resolution P. of length d. In this paper, we compute the homology of the n-th Koszul complex associated with the homomorphism P_1 --> P_0 for all n, if d = 1. This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if d = 2, we compute the homology of the complex N Sym^2 K(P.) where K and N denote the…
We prove a Cohen-Macaulay version of a result by Avramov-Golod and Frankild-Jørgensen about Gorenstein rings, showing that if a noetherian ring $A$ is Cohen-Macaulay, and $a_1,\dots,a_n$ is any sequence of elements in $A$, then the Koszul complex $K(A;a_1,\dots,a_n)$ is a Cohen-Macaulay DG-ring. We further generalize this result, showing that it also holds for commutative DG-rings. In the process of proving this, we develop a new technique to study the dimension theory of a noetherian ring $A$, by finding a Cohen-Macaulay DG-ring $B$ such that $\mathrm{H}^0(B) = A$, and using the Cohen-Macaula…
There is a well known link from the first topic in the title to the third one. In this paper we thread that link through the second topic. The central result is a criterion for the tensor nilpotence of morphisms of perfect complexes over commutative noetherian rings, in terms of a numerical invariant of the complexes known as their level. Applications to local rings include a strengthening of the Improved New Intersection Theorem, short direct proofs of several results equivalent to it, and lower bounds on the ranks of the modules in every finite free complex that admits a structure of differe…
We develop in this paper some methods for studying the implicitization problem for a rational map $ϕ: \mathbb{P}^n \to (\mathbb{P}^1)^{n+1}$ defining a hypersurface in $(\mathbb{P}^1)^{n+1}$, based on computing the determinant of a graded strand of a Koszul complex. We show that the classical study of Macaulay Resultants and Koszul complexes coincides, in this case, with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions. Under suitable hypotheses, these techniques enable us to obtain the implicit equation, up to a power, and …
A theory of cohomological support for pairs of DG modules over a Koszul complex is investigated. These specialize to the support varieties of Avramov and Buchweitz defined over a complete intersection ring, as well as support varieties over an exterior algebra. The main objects of study are certain DG modules over a polynomial ring; these determine the aforementioned cohomological supports and are shown to encode (co)homological information about pairs of DG modules over a Koszul complex. The perspective in this article leads to new proofs of well-known results for pairs of complexes over a co…
We study the ideal of maximal minors in Littlewood varieties, a class of quadratic complete intersections in spaces of matrices. We give a geometric construction for a large class of modules, including all powers of this ideal, and show that they have a linear free resolution over the complete intersection and that their Koszul dual is an infinite-dimensional irreducible representation of the orthosymplectic Lie superalgebra. We calculate the algebra of cohomology operators acting on this free resolution. We prove analogous results for powers of the ideals of maximal minors in the variety of l…
One can iteratively obtain a free resolution of any monomial ideal $I$ by considering the mapping cone of the map of complexes associated to adding one generator at a time. Herzog and Takayama have shown that this procedure yields a minimal resolution if $I$ has linear quotients, in which case the mapping cone in each step cones a Koszul complex onto the previously constructed resolution. Here we consider cellular realizations of these resolutions. Extending a construction of Mermin we describe a regular CW-complex that supports the resolutions of Herzog and Takayama in the case that $I$ has a…
Let V be a symplectic vector space of dimension 2n. Given a partition λwith at most n parts, there is an associated irreducible representation S_{[λ]}(V) of Sp(V). This representation admits a resolution by a natural complex L^λ, which we call the Littlewood complex, whose terms are restrictions of representations of GL(V). When λhas more than n parts, the representation S_{[λ]}(V) is not defined, but the Littlewood complex L^λstill makes sense. The purpose of this paper is to compute its homology. We find that either L^λis acyclic or that it has a unique non-zero homology group, which forms a…
We show that the cohomology algebra of the complement of a coordinate subspace arrangement in m-dimensional complex space is isomorphic to the cohomology algebra of Stanley-Reisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) Then we calculate the latter cohomology algebra by means of the standard Koszul resolution of polynomial ring. To prove these facts we construct an equivariant with respect to the torus action homotopy equivalence between the complement of a coordinate subspace arrangement and th…
