Let k be a field. Given two finite-dimensional right comodules N andM over a k–coalgebra C, the k–vector spaces ExtC(N,M) need not to be finite-dimensional. This is due to the fact that the injective right comodules appearing in the minimal injective resolution of M need not to be of finite dimension or even quasi-finite. The obstruction here is that factor comodules of quasi-finite comodules a...

Let $T=\bigl(\begin{smallmatrix}A&0\U\&B\end{smallmatrix}\bigr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings $U$ is $(B, A)$-bimodule. We prove: (1) If $U\_{A}$ ${B}U$ have finite flat dimensions, then left $T$-module $\bigl(\begin{smallmatrix}M\_1\ M\_2\end{smallmatrix}\bigr){\varphi^{M}}$ Ding projective if only $M\_1$ $M\_2/{\operatorname{im}(\varphi^{M})}$ the morphism $...

We show that every geodesic metric space admitting an injective continuous map into the plane as well planar graph has Nagata dimension at most two, hence asymptotic two. This relies on and answers a question in recent work by Fujiwara Papasoglu. conclude all three-dimensional Hadamard manifolds have three. As consequence, such are absolute Lipschitz retracts.

Let X be a smooth aane variety of dimension n > 2. Assume that the group H 1 (X; Z) is a torsion group and that (X) = 1. Let Y be a projectively smooth aane hypersurface Y C n+1 of degree d > 1, which is smooth at innnity. Then there is no injective polynomial mapping f : X ! Y: This contradicts a result of Peretz 5].

Let R be a left and right Noetherian ring and n, k any non-negative integers. R is said to satisfy the Auslander-type condition Gn(k) if the right flat dimension of the (i+1)-st term in a minimal injective resolution of RR is at most i+ k for any 0 ≤ i ≤ n− 1. In this paper, we prove that R is Gn(k) if and only if so is a lower triangular matrix ring of any degree t over R.

Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to nbp/2c for a p-th order tensor in Rnp . Previously no efficient algorithm can decompose 3rd order tensors when the rank is super-linear in the dimension. Using ideas from sum-of-squares hierarchy, we give the firs...

Let D be a division algebra over a base field k. The homological transcendence degree of D, denoted by HtrD, is defined to be the injective dimension of the algebra D⊗k D ◦. We show that Htr has several useful properties which the classical transcendence degree has. We extend some results of Resco, Rosenberg, Schofield and Stafford, and compute Htr for several classes of division algebras. The ...

Let Λ and Γ be artin algebras and ΛUΓ a faithfully balanced selforthogonal bimodule. In this paper, we first introduce the notion of k-Gorenstein modules with respect to ΛUΓ and then establish the left-right symmetry of the notion of k-Gorenstein modules, which develops a classical result of Auslander. As an application, we study the properties of dual modules relative to Gorenstein bimodules. ...

Let D be a division algebra over a base field k. The homological transcendence degree of D, denoted by HtrD, is defined to be the injective dimension of the algebra D⊗k D ◦. We show that Htr has several useful properties which the classical transcendence degree has. We extend some results of Resco, Rosenberg, Schofield and Stafford, and compute Htr for several classes of division algebras. The ...

An extension of the peripheral group and its associated structures such as the meridian and longitude to knots of arbitrary dimension and genus is studied. The analogous structures are shown to provide a complete algebraic invariant for oriented spun tori, by using the Tube map of Satoh. The algebraic invariant also provides a constraint on equivalence classes of welded knots in the preimage of...