For a local endomorphism of a noetherian local ring we introduce a notion of entropy, along with two other asymptotic invariants. We use this notion of entropy to extend numerical conditions in Kunz' regularity criterion to every contracting endomorphism of a noetherian local ring, and to give a characteristic-free interpretation of the definition of Hilbert-Kunz multiplicity. We also show that every finite endomorphism of a complete noetherian local ring of equal characteristic can be lifted to a finite endomorphism of a complete regular local ring. The local ring of an algebraic or analytic variety at a point fixed by a finite self-morphism inherits a local endomorphism whose entropy is well-defined. This situation arises at the vertex of the affine cone over a projective variety with a polarized self-morphism, where we compare entropy with degree.

Local and category-theoretical entropies associated with an endomorphism of finite length (i.e., with zero-dimensional closed fiber) of a commutative Noetherian local ring are compared. Local entropy is shown to be less than or equal to category-theoretical entropy. The two entropies are shown to be equal when the ring is regular, and also for the Frobenius endomorphism of a complete local ring of positive characteristic. Furthermore, given a flat morphism of Cohen-Macaulay local rings endowed with compatible endomorphisms of finite length, it is shown that local entropy is "additive". Finally, over a ring that is a homomorphic image of a regular local ring, a formula for local entropy in terms of an asymptotic partial Euler characteristic is given.

We give different short proofs for a result proved by C. Mueller in [9]: Over an algebraically closed field pairs of n × n matrices whose product is symmetric form an irreducible, reduced, and complete intersection variety of dimension (3n 2 + n)/2. Our work is connected to the work of Brennan, Pinto, and Vasconcelos in [2].

Let R be a regular local ring of dimension n 5, and p a prime ideal of height 2. Let (V , O V ) be the punctured spectrum of R/p. We show that if the ring Γ (V , O V ) is Gorenstein, then R/p is complete intersection. We also show that an analog of the splitting criterion for vector bundles of small rank on projective spaces given in [N. Mohan Kumar, C. Peterson, A. Prabhakar Rao, Monads on projective spaces, Manuscripta Math. 112 (2003) 183-189; p. 185, Theorem 1] holds for vector bundles of small rank on punctured spectrum of regular local rings.