Homology and Cohomology of Stacks

Throughout this lecture, we let k denote an algebraically closed field, ` a prime number which is invertible in k. In the previous, we define the `-adic cohomology H∗(X; Λ), where X is a quasi-projective k-scheme and Λ ∈ {Z`,Q`,Z/`Z}. Our first goal in this lecture is to review the corresponding theory of `-adic homology. Definition 1. Let Λ be a commutative ring, and let ModΛ denote the∞-category introduced in the previous lecture (whose objects are injective chain complexes of Λ-modules). We say that an object M ∈ ModΛ is perfect if it is dualizable with respect to the tensor product on ModΛ. That is, M is perfect if there exists another object M ∨ ∈ ModΛ together with maps e : M∨ ⊗Λ M → Λ c : Λ→M ⊗Λ M∨