Tor gives the inverse to the Hilbert function of a graded algebra

The Hilbert generating function of Tor^U(A, A) gives the inverse of the Hilbert generating function of U where U is a graded A algebra.     

Tor与Ext的对偶关系

Tor与Ext是模范畴中重要的导出函子.本文首先讨论它们的某些性质,然后讨论它们之间的对偶关系.     

Characterizations of regular local rings in positive characteristics

In this note, we provide several characterizations of regular local rings in positive characteristics, in terms of the Hilbert-Kunz multiplicity and its higher Tor}$"> counterparts Tor}_i(k,{}^{f^n} R))/p^{nd}$">. We also apply the characterizations to improve a recent result by Bridgeland and Iyengar in the characteristic case. Our proof avoids using the existence of big Cohen-Macaulay modules, which is the major tool in the proof of Bridgeland and Iyengar.

     

Homology with Models and Tor

We consider the extension of the notion of a projective module to that of a projective functor relative to a model set (as in Dold, MacLane, Oberst, 1967). Then taking projective resolutions of functors, we consider the usual associated homology.We show that in some cases, including the classical simplicial homology of topological spaces, the model set can be replaced by a model set having only one element. We show that when the model set consists of a single element the homology modules can be interpreted as values of the Torsion functor. In the case of simplicial homology of topological spaces these Tors will be shown to be analogous to the Tors which occur in group homology.