SH02-COURSE-CONTRACT — How to use this course draft

Local identifier: SH02-COURSE-CONTRACT.

This course develops sheaf operations as tools for studying directions of propagation. Its writing order starts with conic objects and kernel operations; the eventual learner order follows the dependency graph in the course index. The intended audience already knows abelian and derived categories, sheaves on topological spaces, basic manifold topology and cotangent bundles. Exact open prerequisites are linked in the prerequisite lesson. Further prerequisites remain explicitly named when the available open statement does not supply the required generality or proof.

The course covers the microlocal theory of sheaves: the Fourier–Sato transformation, specialization and microlocalization, the micro-support and its functorial properties, and microlocal categories. This is the theory of M. Kashiwara and P. Schapira, Microlocal study of sheaves, Astérisque 128 (1985), Chapters 1–6; see also P. Schapira, A short review on microlocal sheaf theory (2016). The course includes the subordinate lemmas and the substantive phenomena of the classical exercises. Categorical foundations are taught in Categorical and derived tools for analytic sheaves; constructible sheaves, characteristic cycles and contact transformations in Constructible and perverse sheaves.

The advanced baseline coefficient ring is commutative with identity and finite global dimension. A theorem that specifically uses a field says so locally. There is no general assumption of constructibility, finite-rank stalks, compactness or orientability. Manifolds have finite dimension and are countable at infinity; submanifolds may be locally closed. Some operations are developed on more general locally compact spaces, and each such statement specifies its own assumptions. The finite cohomological dimension required to construct an exceptional inverse image is an assumption on proper-support direct image of abelian sheaves, not an unstated claim about arbitrary continuous maps.

We use cohomological grading: $H^i(K[n])=H^{i+n}(K)$. For a locally closed subset $A$ of $X$, $k_A$ denotes the constant sheaf on $A$ extended by zero to $X$ using its locally closed embedding. This differs from the sheaf of sections with support in $A$. The latter construction, its derived functor and its restriction to a point are typed in each lemma. A functor written $Rf_!$ uses supports proper over the target. A star or an exclamation mark cannot be changed without a theorem or a specified comparison morphism.

Orientation sheaves are retained in formulas. Choosing a local orientation can simplify a calculation, but it does not remove the transition functions from a global statement. The positive and negative pairing halfspaces, the antipodal map, the order of product orientations and the degree shifts of Fourier–Sato operations are all part of their definitions. Local calculations must return to these conventions before being used in a composition.

Each lesson separates statements proved there, exact imports, conditional proofs whose inputs remain open, and outstanding source obligations.

Examples and exercises are newly designed. Their solutions show the algebra and topology needed to check the formulas. The protected expression and organization of source examples do not enter this course. Detected source misprints or false assertions receive a visible correction and an explicit mathematical reason.

This directory is an exchange package for the existing reader. It does not provide another website. Its stable SH02-... identifiers are local course identifiers and have not been registered as official Stacks tags.