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\operatorname is deprecated 

ref : github/markup#1688
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Flopsky authored Nov 3, 2023
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- [ ] Introduction
- [ ] I. Global and local cohomological invariants with respect to a closed subspace
1. [ ] The functors $\Gamma_Z$ and $\underline{\Gamma}_Z$ *(7)*
2. [ ] The functors $\operatorname{H}_Z^\bullet(X,F)$ and $\underline{\operatorname{H}}_Z^\bullet(F)$ *(6)*
2. [ ] The functors $\mathop{\text{H}}_Z^\bullet(X,F)$ and $\underline{\mathop{\text{H}}}_Z^\bullet(F)$ *(6)*
- [ ] II. Applications to quasi-coherent sheaves on preschemes *(8)*
- [ ] III. Cohomological invariants and depth
1. [ ] Reminders *(1)*
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3. [ ] Study of the case where $T$ is left exact and $T(M)$ is of finite type for all $M$ *(3)*
4. [ ] Dualising module. Dualising functor *(5)*
5. [ ] Consequences of the theory of dualising modules *(5)*
- [ ] V. Local duality and structure of the $\operatorname{H}^i(M)$
- [ ] V. Local duality and structure of the $\\mathop{\text{H}}^i(M)$
1. [ ] Complexes of homomorphisms *(3)*
2. [ ] The local duality theorem for a local regular ring *(1)*
3. [ ] Application to the structure of the $\operatorname{H}^i(M)$ *(7)*
- [ ] VI. The functors $\operatorname{Ext}(X;F,G)$ and $\underline{\operatorname{Ext}}(F,G)$
3. [ ] Application to the structure of the $\mathop{\text{H}}^i(M)$ *(7)*
- [ ] VI. The functors $\mathop{\text{Ext}}(X;F,G)$ and $\underline{\mathop{\text{Ext}}}(F,G)$
1. [ ] Generalities *(3)*
2. [ ] Application to quasi-coherent sheaves on preschemes *(2)*
- [ ] VII. Nullity criteria. Coherence conditions for the sheaves $\underline{\operatorname{Ext}}(F,G)$
- [ ] VII. Nullity criteria. Coherence conditions for the sheaves $\underline{\mathop{\text{Ext}}}(F,G)$
1. [ ] Study of $i < n$ *(5)*
2. [ ] Study of $i > n$ *(2)*
- [ ] VIII. Finiteness theorem
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2. [ ] Comparison of $\mathsf{Et}(Y)$ with $\mathsf{Et}(U)$, for varying 𝑈 *(5)*
3. [ ] Comparison of $\pi_1(X)$ with $\pi_1(U)$ *(7)*
- [ ] XI. Applications to the Picard group
1. [ ] Comparison of $\operatorname{Pic}(\widehat{X})$ with $\operatorname{Pic}(Y)$ *(1)*
2. [ ] Comparison of $\operatorname{Pic}(Y)$ with $\operatorname{Pic}(U)$, for varying $U$ *(5)*
1. [ ] Comparison of $\mathop{\text{Pic}}(\widehat{X})$ with $\mathop{\text{Pic}}(Y)$ *(1)*
2. [ ] Comparison of $\mathop{\text{Pic}}(Y)$ with $\mathop{\text{Pic}}(U)$, for varying $U$ *(5)*
3. [ ] Comparison of $\mathsf{P}(X)$ with $\mathsf{P}(U)$ *(7)*
- [ ] XII. Applications to projective algebraic schemes
1. [ ] Projective duality theorem and finiteness theorem *(7)*
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4. [ ] Algebraic structures in the category of preschemes *(15)*
5. [ ] Group cohomology *(6)*
- [ ] II. Tangent bundles. Lie algebras
1. [ ] $\underline{\operatorname{Hom}}_{Z/S}(X,Y)$ functors *(2)*
1. [ ] $\underline{\mathop{\text{Hom}}}_{Z/S}(X,Y)$ functors *(2)*
2. [ ] The preschemes $I_S(M)$ *(3)*
3. [ ] The tangent bundle, the (E) condition *(11)*
4. [ ] Tangent space of a group. Lie algebras *(15)*
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2. [ ] Pseudo-coherent complexes *(18)*
3. [ ] Link to the classical notion of coherence *(7)*
4. [ ] Perfect complexes *(12)*
5. [ ] Finite $\operatorname{Tor}$-dimension and perfection *(10)*
5. [ ] Finite $ \mathop{\text{Tor}}$-dimension and perfection *(10)*
6. [ ] Rank of a perfect complex *(9)*
7. [ ] Duality of perfect complexes *(6)*
8. [ ] Traces and cup-products *(5)*
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