Algebraic-homological constructions attached to differentials (original) (raw)
Related papers
2006
The contents of these notes originated from a talk delivered to a diversified audience at the Departamento de Matemática of my own University. The present notes expand and give full details of statements made in that occasion. Moreover, I have sought to expand as well on several algebraic notions in order to pave the way towards a more stable version even though self-sufficiency is out of question in a material of this order. If one is to single a module that stands alone in its totally device-independent nature, one'd better take the module of differentials Ω(A/k). In fact, most invariants of an algebra or scheme are attached primevally to this module or "découlent" from it by various procedures. Thus, Segre classes in algebraic geometry are ultimately defined in terms of Ω(A/k) and even the ubiquitous canonical module (dualizing sheaf) is "normally" related to Ω(A/k) by double-dualizing its top-wedge module. My objective in this Course is panorama-driven, due to the short span of lectures. However, details and examples will be worked out and, possibly, a few arguments too. 10. * Considerations about a formula of Kleiman-Plücker-Teissier for the degree (class) of the dual variety to a complete intersection.
Adelic constructions of direct images for differentials and symbols
arXiv: Algebraic Geometry, 1997
For a projective morphism of an smooth algebraic surface XXX onto a smooth algebraic curve SSS, both given over a perfect field kkk, we construct the direct image morphism in two cases: from Hi(X,Omega2X)H^i(X,\Omega^2_X)Hi(X,Omega2X) to Hi−1(S,Omega1S)H^{i-1}(S,\Omega^1_S)Hi−1(S,Omega1S) and when chark=0char k =0chark=0 from Hi(X,K2(X))H^i(X,K_2(X))Hi(X,K2(X)) to Hi−1(S,K1(S))H^{i-1}(S,K_1(S))Hi−1(S,K1(S)). (If i=2, then the last map is the Gysin map from CH2(X)CH^2(X)CH2(X) to CH1(S)CH^1(S)CH1(S).) To do this in the first case we use the known adelic resolution for sheafs Omega2X\Omega^2_XOmega2X and Omega1S\Omega^1_SOmega1S. In the second case we construct a K_2K_2K2-adelic resolution of a sheaf K2(X)K_2(X)K_2(X). And thus we reduce the direct image morphism to the construction of some residues and symbols from differentials and symbols of 2-dimensional local fields associated with pairs xinCx \in CxinC ($x$ is a closed point on an irredicuble curve CinXC \in XCinX) to 1-dimensional local fields associated with closed points on the curve SSS. We prove reciprocity laws for these maps.
A note on Chern character, loop spaces and derived algebraic geometry
Eprint Arxiv 0804 1274, 2008
In this note we present a work in progress whose main purpose is to establish a categorified version of sheaf theory. We present a notion of derived categorical sheaves, which is a categorified version of the notion of complexes of sheaves of modules on schemes, as well as its quasi-coherent and perfect versions. We also explain how ideas from derived algebraic geometry and higher category theory can be used in order to construct a Chern character for these categorical sheaves, which is a categorified version of the Chern character for perfect complexes with values in cyclic homology. Our construction uses in an essential way the derived loop space of a scheme X, which is a derived scheme whose theory of functions is closely related to cyclic homology of X. This work can be seen as an attempt to define algebraic analogs of elliptic objects and characteristic classes for them. The present text is an overview of a work in progress and details will appear elsewhere.
Contributions to Algebraic Geometry
2012
This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.
Adelic constructions for direct images of differentials and symbols
Invitation to higher local fields, 2000
Let X be a smooth algebraic surface over a perfect field k. Consider pairs x ∈ C, x is a closed point of X , C is either an irreducible curve on X which is smooth at x, or an irreducible analytic branch near x of an irreducible curve on X. As in the previous section 1 for every such pair x ∈ C we get a two-dimensional local field K x,C. If X is a projective surface, then from the adelic description of Serre duality on X there is a local decomposition for the trace map H 2 (X, Ω 2 X) → k by using a two-dimensional residue map res K x,C /k(x) : Ω 2 K x,C /k(x) → k(x) (see [P1]). From the adelic interpretation of the divisors intersection index on X there is a similar local decomposition for the global degree map from the group CH 2 (X) of algebraic cycles of codimension 2 on X modulo the rational equivalence to Z by means of explicit maps from K 2 (K x,C) to Z (see [P3]). Now we pass to the relative situation. Further assume that X is any smooth surface, but there are a smooth curve S over k and a smooth projective morphism f : X → S with connected fibres. Using two-dimensional local fields and explicit maps we describe in this section a local decomposition for the maps f * : H n (X, Ω 2 X) → H n−1 (S, Ω 1 S), f * : H n (X, K 2 (X)) → H n−1 (S, K 1 (S)) where K is the Zariski sheaf associated to the presheaf U → K(U). The last two groups have the following geometric interpretation: H n (X, K 2 (X)) = CH 2 (X, 2 − n), H n−1 (S, K 1 (S)) = CH 1 (S, 2 − n) where CH 2 (X, 2 − n) and CH 1 (S, 1 − n) are higher Chow groups on X and S (see [B]). Note also that CH 2 (X, 0) = CH 2 (X), CH 1 (S, 0) = CH 1 (S) = Pic(S), CH 1 (S, 1) = H 0 (S, O * S).
O ct 2 00 6 On N-differential graded algebras
2008
We introduce the concept of N -differential graded algebras (N-dga), and study the moduli space of deformations of the differential of an N-dga. We prove that it is controlled by what we call the (M,N)-Maurer-Cartan equation. Introduction The goal of this paper is to take the first step towards finding a generalization of Homological Mirror Symmetry (HMS) [11] to the context of N -homological algebra [5]. In [7] Fukaya introduced HMS as the equivalence of the deformation functor of the differential of a differential graded algebra associated with the holomorphic structure, with the deformation functor of an A∞-algebra associated with the symplectic structure of a Calabi-Yau variety. This idea motivated us to define deformation functors of the differential of an N -differential graded algebra. An N -dga is a graded associative algebra A, provided with an operator d : A → A of degree 1 such that d(ab) = d(a)b + (−1)ad(b) and d = 0. A nilpotent differential graded algebra (Nil-dga) wil...