Search

The monomorphism category $\mathcal S_n(\mathcal X)$ is introduced, where $\mathcal X$ is a full subcategory of the module category $A$-mod of Artin algebra $A$. The key result is a reciprocity of the monomorphism operator $\mathcal S_n$ and the left perpendicular operator $^\perp$: for a cotilting $A$-module $T$, there is a canonical construction of a cotilting $T_n(A)$-module ${\rm \bf m}(T)$, such that $\mathcal S_n(^\perp T) = \ ^\perp {\rm \bf m}(T)$. As applications, $\mathcal S_n(\mathcal X)$ is a resolving contravariantly finite subcategory in $T_n(A)$-mod with $\hat{\mathcal S_n(\mathcal X)} = T_n(A)$-mod if and only if $\mathcal X$ is a resolving contravariantly finite subcategory in $A$-mod with $\hat{\mathcal X} = A$-mod. For a Gorenstein algebra $A$, the category $T_n(A)\mbox{-}\mathcal Gproj$ of Gorenstein-projective $T_n(A)$-modules can be explicitly determined as $\mathcal S_n(^\perp A)$. Also, self-injective algebras $A$ can be characterized by the property $T_n(A)\mbox{-}\mathcal Gproj = \mathcal S_n(A)$. Using $\mathcal S_n(A)= \ ^\perp {\rm \bf m}(D(A_A))$, a characterization of $\mathcal S_n(A)$ of finite type is obtained., 20 pages

Stephenson and Vancliff recently introduced two families of quantum projective 3-spaces (quadratic and Artin–Schelter regular algebras of global dimension 4) which have the property that the associated automorphism of the scheme of point modules is finite order, and yet the algebra is not finite over its center. This is in stark contrast to theorems of Artin, Tate, and Van den Bergh in global dimension 3. We analyze the representation theory of these algebras. We classify all of the finite-dimensional simple modules and describe some zero-dimensional elements of Proj, i.e., so called fat point modules. In particular, we observe that the shift functor on zero-dimensional elements of Proj, which is closely related to the above automorphism, actually has infinite order.

The homogeneous coordinate ring of a quantum projective plane is a 3-dimensional Artin–Schelter regular algebra with the same Hilbert series as the polynomial ring in three variables; such an algebraAis a graded noncommutative analogue of the polynomial ring in three variables. WhenAis a finite module over its centerZ(A), we define the schemeS = Proj(Z(A)) and the sheaf A of OS-algebras by A(S(f)) = A[f − 1]0. The center Z of A is defined by Z(S(f)) = Z(A[f − 1]0) and following Grothendieck we may define the scheme Spec(Z). The algebrasAfall into several families, and for many of these it has been shown that Spec(Z) ≅ P2whenAis finite over its center. This paper shows that Spec(Z) ≅ P2for two more families.

Let A be a positively graded, connected affine C -algebra generated in degree one which is a finite module over a central subring R . Let A be the structure sheaf over Proj( R ) and Spec Z the central Proj. If injdim( A ) Spec Z has singularities unless the ramification locus of A is pure of codimension one. If gldim( A ) A and the singular locus of Spec Z coincide.

This paper is primarily concerned with a projective dimension of vector spaces with valuations. Let Γ be a totally ordered set with suprema. The valuated vector spaces over Γ form a pre-abelian category V. A short exact sequence 0 → A → B → C → 0 in V is proper if each element of C has a preimage with the same value. The proper projectives in V are precisely the free objects in V. An object V ϵV is free if and only if it is the coproduct of one-dimensional spaces. The main purpose of this paper is to characterize those valuated spaces having proper projective dimension n for each positive integer n. Our characterization uses the notion of separability which is defined as follows. A subspace W of V is called ℵn-separable in V if for each x ϵ V there exists a subset S of W having cardinality not exceeding ℵn such that sup{¦x + w¦: w ϵ W} = sup{¦x + s¦: s ϵ S}. Roughly speaking, we show that proj. dim.(V) ⩽ n if V has enough ℵn − 1-separable subspaces, where we have abbreviated “proper projective dimension” to “proj. dim.” More precisely, we prove that proj. dim.(V) ⩽ n if and only if V is the union of a smooth ascending chain of ℵn − 1-separable subspaces, 0 = V0 ⊆ V1 ⊆ … ⊆ Vα ⊆ … such that dim(Vα + 1Vα) ⩽ ℵn. Many other results are required before this characterization can be obtained. After it is obtained, we are able to prove the existence of valuated vector spaces having projective dimension exactly n for each positive integer n. Also, it is shown that there exist spaces having infinite projective dimension. Although our main results are homological in nature, for the most part the techniques of the paper certainly are not. One might compare what we do, for example, with Kaplansky's structural characterization of algebraically compact groups, which turn out to be those that have pure-injective dimension 1.