Curves in Grothendieck categories. (Q1876073)

A quasischeme is, by definition, a Grothendieck category, and the objects of such a category are called modules. A subscheme \(Y\) of a scheme \(X\) is then a full subcategory, and \(Y\) is said to be closed when it is closed under subquotients and the inclusion functor \(i_*\colon X\subseteq Y\) has a left adjoint \(i^*\). If, in addition, \(i_*\) has a right adjoint \(i^!\), then the subscheme is said to be biclosed. The author considers a biclosed subscheme \(Y\) of a quasischeme \(X\) which is a regularly embedded hypersurface in the sense that there is an automorphism \(\sigma\) of \(Y\) such that the derived functor \(R^1i^!\) is naturally isomorphic to \(\sigma^{-1}\circ i^*\). A pure curve module is then defined as a module in \(X\) of Krull dimension 1 such that \(i^!C=0\). When every proper quotient of \(C\) is of finite length, it is said to be irreducible. This general framework was considered by \textit{S. P. Smith} and \textit{J. J. Zhang} [Algebr. Represent. Theory 1, No. 4, 311-351 (1998; Zbl 0947.16029)]. A pure curve module \(C\) is called well positioned (Definition 3.0.2) if (1) \(i^*C\) is of finite length, (2) all simple quotients of \(C\) are objects of \(Y\), and (3) given two elements \(p,q\) in the set \([i^*C]\) of simple subquotients of \(i^*C\) and \(n\in \mathbb{Z}\), the integers, then \(p\cong q^{\sigma^n}\) if and only if both \(n=0\) and \(p\neq q\). An \(Y\)-multistrand (Definition 3.2.4) is a pure curve module \(C\) such that the module \(C/C_3\) is a direct sum of woven modules \(B^1,\dots,B^n\) such that for \(p\in[B^j]\) and \(q\in[B^k]\) with \(j\neq k\), then \(\text{Ext}_Y^1(\sigma^m(p),\sigma^l(q))=0\) for all integers \(m,l\in\mathbb{Z}\). Here, \(C_3\) is a term of the sequence recursively defined as \(C_0=C\), and \(C_n\) as the kernel of the counit map \(C_{n-1}\to i^*C_{n-1}\). The notion of a woven module is introduced in this Definition 3.2.1: they are the finite length modules \(A\) with pairwise nonisomorphic simple subquotients such that every subquotient is either indecomposable or semisimple. Woven modules are characterized in terms of the pattern of existence of certain nonsplit short exact sequences between the simples appearing in each two consecutive levels in the Loewy series of \(A\). The precise statement (Definition 3.1.3 and Remark 3.2.2) is given in terms of a binary relation defined in \([A]\) called pointing, and its associated graph, called skeleton. The first substantial results appear in Section 5, where the smallest set \({\mathbf H}_C\) of modules in \(X\) containing a well-positioned, irreducible \(Y\)-multistrand module \(C\) and closed under submodules and nonsplit extensions by simple modules in \(Y\) is indexed by a subset \(\Lambda\) of \(\mathbb{Z}^l\) (\(l\) denotes the number of subquotients of \(i^*C\)). The set \(\Lambda\) is completely determined by the skeleton of \(C/C_3\) (Definition 5.1.1). This indexing maps each \(\alpha\in\Lambda_C\) onto an irreducible \(Y\)-multistrand module \(C(\alpha)\) in such a way that the skeleton of \(i^*C(\alpha)\) is completely determined by that of \(i^*C\) together with the automorphism \(\sigma\) and \(\alpha\) itself (Proposition 5.2.1). Some other nice properties of this indexing are also proved. The author also considers the curve category defined by \(C\), namely, the full subcategory \({\mathcal C}_C\) of \(X\) whose objects are all subquotients of direct sums of modules in \({\mathbf H}_C\). It is a closed subscheme of \(X\), and, as a Grothendieck category, \({\mathcal C}_C\) is locally Noetherian. In fact, Section 6 investigates an appropriate subset of \({\mathbf H}_C\) which turns out to be a set of Noetherian generators of \({\mathcal C}_C\) of projective dimension at most 1 (Lemmas 6.2.2 and 6.3.5, Corollary 6.3.6). Finally, the author connects the abstract theory with the research program that conceives noncommutative projective algebraic geometry as the study of (noncommutative) graded rings by replacing the projective variety by a suitable Grothendieck category. His main theorem in this setting is Theorem 7.3.4, which asserts that if the irreducible multistrand module \(C\) is such that most nontrivial quotients have \(n\) nonzero woven summands, then \({\mathcal C}_C\) is equivalent to a quotient category of the category of graded modules over an appropriate \(I\)-algebra. The localizing subcategory (in the sense of \textit{P. Gabriel} [Bull. Soc. Math. Fr. 90, 323-448 (1962; Zbl 0201.35602)]) is that of graded modules of Krull dimension less or equal than \(n-2\). All these results have a concrete start point, coming from the line modules over a Sklyanin-type algebra. From the abstract: ``\dots this extension allows examination of the category created from a line module over more general AS-regular rings than those considered by Smith and Zhang. For instance, suppose that \(C\) is a generic line module over \(R_d\), Stafford's Sklyanin-like algebra. [\dots] Then \({\mathcal C}_C\) is equivalent to the category of graded \(k[x,y]/(x^2-y^2)\) modules under the \(\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}\)-grading where \(\deg(x)=(-1,0)\) and \(\deg(y)=(-1,1)\).''