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Let p be an odd prime and Λ = Z [ D 2 p ] the integral group ring of the dihedral group D 2 p of order 2p. The syzygies Ω r ( Z ) are the stable classes of the intermediate modules in a free Λ-reso...

In this note, we give a slight improvement of a result of A. K\"uronya and V. Lozovanu about higher syzygies on abelian surfaces., Comment: 6 pages

In this paper, we study commutative local rings whose residue field is a high syzygy. We are able to characterize such rings with embedding dimension at most 3 as those that are Artinian an...

In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy o...

Let I be a monomial ideal in the polynomial ring S generated by elements of degree at most d. In this paper, it is shown that, if the i-th syzygy of I has no elements of degrees j,…,j+(d−1) (where j≥i+d), then (i+1)-th syzygy of I does not have any element of degree j+d. Then we give several applications of this result, including an alternative proof for Green–Lazarsfeld index of the edge ideals of graphs as well as an alternative proof for Froberg’s theorem on classification of square-free monomial ideals generated in degree 2 with linear resolution. Among all, we deduce a partial result on subadditivity of the syzygies for monomial ideals.

Let $R = K[X_1, ..., X_n]$ be a polynomial ring over some field $K$. In this paper, we prove that the $k$-th syzygy module of the residue class field $K$ of $R$ has Stanley depth $n-1$ for $\lfloor n/2 \rfloor \leq k < n$, as it had been conjectured by Bruns et. al. in 2010. In particular, this gives the Stanley depth for a whole family of modules whose graded components have dimension greater than $1$. So far, the Stanley depth is known only for a few examples of this type. Our proof consists in a close analysis of a matching in the Boolean algebra., minor corrections, 11 pages, 4 figures

Let G be a finite group with integral group ring Λ =Z[G]. The syzygies Ωr(Z) are the stable classes of the intermediate modules in a free Λ-resolution of the trivial module. They are of significance in the cohomology theory of G via the “co-represention theorem” Hr(G, N) = Hom𝒟er(Ωr(Z), N). We describe the Ωr(Z) explicitly for the dihedral groups D4n+2, so allowing the construction of free resolutions whose differentials are diagonal matrices over Λ.

Let Syz1(𝔪) be the first syzygy of the graded maximal ideal 𝔪 of a polynomial ring K[x1,…, xn] over a field K. Using the theory of s-sequences, the dimension and depth of the symmetric algebra Sym(Syz1(𝔪)) are calculated. As a conclusion, Sym(Syz1(𝔪)) is not Cohen–Macaulay for any n ≥ 4.

Let S and R be rings and S C R a semidualizing bimodule. We investigate the relation between the G C -syzygy with the C-syzygy of a module as well as the relation between the G C -projective resolution and the projective resolution of a module. As a consequence, we get that if is an exact sequence of S-modules with all G i , G i G C -projective, such that Hom S (𝔾, T) is still exact for any module T which is isomorphic to a direct summand of direct sums of copies of S C, then Im(G 0 → G 0) is also G C -projective. We obtain a criterion for computing the G C -projective dimension of modules. When S C R is a faithfully semidualizing bimodule, we study the Foxby equivalence between the subclasses of the Auslander class and that of the Bass class with respect to C.

We describe the transpose operator for self-projective and symmetric coalgebras in terms of the syzygy and Nakayama functors.

Let R be a ring, n a fixed non-negative integer and ℱ the class of all left R-modules of FP-injective dimensions at most n. It is proved that all left R-modules over a left coherent ring R have ℱ-preenvelopes and ℱ-covers. Left (right) ℱ-resolutions and the left derived functors of Hom are used to study the FP-injective dimensions of modules and rings.

We present a constructive solution of the inverse syzygy problem over arbitrary coherent rings. By relating the existence of a kernel representation to torsionlessness instead of the more common torsionfreeness, we do not need to assume the existence of a quotient field. As a by-product, we obtain an algorithm to compute the extension groups of finitely presented modules.

In this article, we study the weak global dimension of coherent rings in terms of the left FP-injective resolutions of modules. Let R be a left coherent ring and ℱ ℐ the class of all FP-injective left R-modules. It is shown that wD(R) ≤ n (n ≥ 1) if and only if every nth ℱ ℐ-syzygy of a left R-module is FP-injective; and wD(R) ≤ n (n ≥ 2) if and only if every (n − 2)th ℱ ℐ-syzygy in a minimal ℱ ℐ-resolution of a left R-module has an FP-injective cover with the unique mapping property. Some results for the weak global dimension of commutative coherent rings are also given.

Let Λ be a monomial algebra. For any Λ-module M, we describe graphically its minimal projective resolution as a weighted graph Δ(M). If M is finitely generated we give two algorithms for computing Δ(M). We also give a short computation-like argument to reprove a famous Syzygy Theorem.

Let K be a field and sets of indetorminates, and f ∈ k [[x]]g ∈ k[[Y]] two non-zero formal power series with f ∈ (x)g ∈ (Y). Set and suppose that δg is a (y)-primary ideal (e.g., if char K = 0 and g is an isolated singularity). Write R = K[[X,Y]]/(f+g)R 1 = k[[X]]/(f) and [Rtilde]:= R/δg R. The main aim of this paper is to relate an arbitrary maximal Cohen-Macaulay (MCM for short) R-module N to the higher order syzygy , and in this way relate indecomposable MCM R-modules to higher order syzygies of certain indecomposable [Rtilde]-modules. The [Rtilde]-modules in question are deformations of MCM R-modules and are weakly liftablc. We find resolutions of the higher order syzygy modules in question which are shown to be minimal in certain situations, and express these in terms of matrix factorizations. The theory is shown to be applicable with almost complete success to singularities of KnOrrer-type and, in any case, to give detailed information about MCM R -modules. Our techniques generalise and simplify tho...

Let Λ be a left and right Noetherian ring. For a positive integer k, we give an equivalent condition that flat dimensions of the first k terms in the minimal injective resolution of Λ are less than or equal to k. In this case we show that the subcategory consisting of k-torsionfree modules is extension closed. Moreover we prove that for a Noetherian algebra every subcategory consisting of i-torsionfree modules is extension closed for any 1 ≤ i ≤ k if and only if every subcategory consisting of i-th syzygy modules is extension closed for any 1 ≤ i ≤ k. Our results generalize the main results in Auslander and Reiten [4].

The purpose of this note is to give an elementary proof of the fact discovered by Bautista and Perez that for an Artin algebra a module without selfextension is determined by its top and its first syzygy.

We prove that (a) if R is left coherent, then a finitely presented right R–module M is copure flat if and only if M is a cokernel of a flat preen-velope (b) if R is a ring and n a nonnegative integer, then r.IFD(R) ≤ n if and only if every nth yoke of each left R–module is copure flat; (c) let R be left and right coherentFP– idRR ≤ 1, n a positive integer and M a finitely presented right R–module, then M is an nth F–cosysygy of a finitely presented right R–module if and only if every finitely generated projective resolution of M is a flat resolvent of the nth syzygy; (d) if R is left coherent and n a nonnegative integer, then FP – idfRR ≤ n if and only if every projective resolution of the nth syzygy of each finitely presented left R–module is a flat resolvent.

In the theorem of Evans and Griffith on the syzygy problem. The restriction on the ring being Cohen Macaulay domain is completely eliminated.