We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct...

This paper is devoted to the algebraic and combinatorial properties of polytopal semigroup rings defined as follows. Let P be a lattice polytope in R n , i. e. a poly-tope whose vertices have integral coordinates, and K a field. Then one considers the embedding ι : R n → R n+1 , ι(x) = (x, 1), and defines S P to be the semigroup generated by the lattice points in ι(P); the K-algebra K[S P ] is ...

X iv :m at h/ 01 10 09 7v 1 [ m at h. A G ] 9 O ct 2 00 1 LOCAL COMPLETE INTERSECTIONS IN P AND KOSZUL SYZYGIES DAVID COX AND HAL SCHENCK Abstract. We study the syzygies of a codimension two ideal I = 〈f1, f2, f3〉 ⊆ k[x, y, z]. Our main result is that the module of syzygies vanishing (schemetheoretically) at the zero locus Z = V(I) is generated by the Koszul syzygies iff Z is a local complete i...

In this paper, for noetherian commutative ring with a unit A such that every finitely generated A-module is free, and for a certain A-regular sequence f1, · · · , fp, we define Kos(A; f1, · · · , fp) the category of Koszul cubes associated to f1, · · · , fp which has a natural Waldhausen categorical structure. We also give a new algorithm of resolution process of modules by Koszul cubes. By the...

We study five different types of the homology a Lie algebra over commutative ring which are naturally isomorphic fields. show that they not rings, even Z, and connections between them. In particular, we in case is flat as module. As an auxiliary result prove Koszul complex module M principal ideal domain connects exterior symmetric powers 0→ΛnM→M⊗Λn−1M→…→Sn−1M⊗M→SnM→0 purely acyclic.

of the residue field K ∼= A/A+ as an A-module. Here ∂0 : A → K is the natural augmentation, the Fi are considered graded left free A-modules whose basis elements have degree 0, and that the resolution is linear means the boundary maps ∂n, n ≥ 1, are graded of degree 1 (unless ∂n = 0). The examples we will discuss in Section 1 are variants of the polytopal semigroup rings considered in Bruns, Gu...

We construct the super Koszul complex of a free supercommutative $A$-module $V$ rank $p|q$ and prove that its homology is concentrated in single degree it yields an exact resolution $A$. then study dual show as well isomorphic to $\Pi^{p+q} A$, with $\Pi$ parity changing functor. Finally, we that, given automorphism $V$, induced transformation on only non-trivial class by multiplication Berezin...

Let R be a noetherian ring and M a finite R-module. With a linear form χ on M one associates the Koszul complex K(χ). If M is a free module, then the homology of K(χ) is well-understood, and in particular it is grade sensitive with respect to Imχ. In this note we investigate the case of a module M of projective dimension 1 (more precisely, M has a free resolution of length 1) for which the firs...