**1. Prime ideals and the Chow group**

A brief description of the basic properties of prime ideals in Noetherian rings and the definition of the Chow group.

**2. Graded rings and Samuel multiplicity**

The theory of Hilbert polynomials of graded rings and of local rings and the definitions of multiplicities defined by the leading coefficients of Hilbert polynomials. This chapter also includes basic facts about dimension in local rings and mixed and Buchsbaum-Rim multiplicities.

**3. Complexes and derived functors**

The theory of derived functors, ordinary and double complexes, and the Koszul complex.

**4. Homological properties of rings and modules**

Cohen-Macaulay and Gorenstein rings, the definition of dimension used in Intersection Theory, and basic properties of injective modules and dualizing complexes.

**5. Intersection multplicities**

Intersection with divisors in the affine case, further properties of the Koszul complex, and Serre intersection multiplicities.

**6. The homological conjectures**

A brief outline of the homological conjectures, including Serre's conjectures (and theorems) on intersection multiplicities, the Intersection Theorem, and conjectures related to this theorem. These include the Improved New Intersection Conjecture, the Monomial Conjecture, and the Canonical Element Conjecture.

**7. The Frobenius map**

Properties of the Frobenius map for rings of positive characteristic and applications to the homological conjectures. This chapter also includes brief descriptions of Hilbert-Kunz and Dutta multiplicities and of tight closure.

**8. Projective schemes**

Basic properties of projective schemes defined by graded and multi-graded rings. Also discussed are the Hilbert polynomial for multi-graded rings, intersection with divisiors on projective schemes, and functorial properties of the Chow group for projective maps.

**9. Chern classes of locally free sheaves**

The basic properties of sheaves defined by graded modules and the theory of Chern classes and Chern characters for locally free sheaves. The push-forward of a complex of sheaves by a projective map.

**10. The Grassmannian**

The Grassmannian of rank r subsheaves of a locally free sheaf.

**11. Local Chern characters**

Definition and basic properties of local Chern characters defined by bounded complexes of locally free sheaves. Relations with Samuel multiplicity.

**12. Properties of local Chern characters**

Further properties of local Chern characters. The additivity and multiplicativity properties and the local Riemann-Roch formula.

**13. Applications and examples**

Applications to the Serre vanishing conjecture and the Intersection theorem. Examples
of negative intersection multiplicities and examples of the Todd class of a local ring.

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