## 2019 : Introduction to Algebraic K-theory

This seminar is supervised by Prof. Jinhyun Park. Notes are in construction.

List of talks:

1. Introduction to algebraic $$K$$-theory and Grothendieck-Riemann-Roch theorem (3/18)

Abstract: Algebraic $$K$$-theory originated with the Grothendieck-Riemann-Roch theorem, a generalization of Riemann-Roch theorem to higher dimensional varieties. For this, we shall discuss the definitions of $$K_0$$-theory of a variety, its connection with intersection theory, $$\lambda$$-operation, $$\gamma$$-filtration, Chern classes and Adams operations.

2. Quillen's higher algebraic $$K$$-theory I (4/1)

Abstract: Grothendieck's $$K_0$$-group has an exact sequence $$K_0(Z)\to K_0(X)\text{}\to K_0(X\smallsetminus Z)\to 0$$ for a closed immersion $$Z\to X$$ of regular noetherian schemes, whose kernel of the leftmost map is usually nontrivial. To prolong this sequence functorially, Quillen defined higher algebraic $$K$$-groups as the higher homotopy groups of some mysterious category associated to an exact category. For this, we introduce simplicial sets, $$Q$$-construction of an exact category, higher $$K$$-groups of schemes, and their remarkable theorems, part of which extends the previously mentioned exact sequence for $$K_0$$ and relates higher $$K$$-theory with Chow groups. This is the first part of Quillen's algebraic $$K$$-theory.

3. Quillen's higher algebraic $$K$$-theory II (4/22)

Abstract: Grothendieck's $$K_0$$-group has an exact sequence $$K_0(Z)\to K_0(X)\text{}\to K_0(X\smallsetminus Z)\to 0$$ for a closed immersion $$Z\to X$$ of regular noetherian schemes, whose kernel of the leftmost map is usually nontrivial. To prolong this sequence functorially, Quillen defined higher algebraic $$K$$-groups as the higher homotopy groups of some mysterious category associated to an exact category. For this, we introduce simplicial sets, $$Q$$-construction of an exact category, higher $$K$$-groups of schemes, and their remarkable theorems, part of which extends the previously mentioned exact sequence for $$K_0$$ and relates higher $$K$$-theory with Chow groups. This is the second part of Quillen's algebraic $$K$$-theory.

4. Quillen's higher algebraic $$K$$-theory III (5/13)

Abstract: We defined Quillen's higher algebraic $$K$$-theory and examined its basic properties in previous talks. By the localization theorem and the dévissage theorem, the codimension filtration on $$\operatorname{Coh}(X)$$ for a finite dimensional noetherian scheme $$X$$ gives the Brown-Gersten-Quillen spectral sequence from page 1. If $$X$$ is a regular algebraic scheme, then the second page of this spectral sequence is given by $$E_2^{p,-q}=H^p_{Zar}(X, \underline G_{q})$$ and $$E_2^{p,-p}=CH^p(X)$$, where $$\underline G_{q}$$ denotes the Zariski sheafification of $$U\mapsto G_q(U)$$. To prove this, we employ Quillen's geometric presentation lemma. This is the third and last part of Quillen's algebraic $$K$$-theory.

5. Algebraic $$K$$-theory following Thomason-Trobaugh I : Perfect complexes (5/27)

Abstract: In the setting of Quillen’s $$K$$-theory, the $$K$$-groups are defined for the category of (algebraic) vector bundles. However, even in SGA 6, the need of replacing vector bundles by complexes quasi-isomorphic to bounded complexes of vector bundles was noticed. Such complexes are called perfect complexes and their $$K$$-theory provide better local-to-global properties, for example, Nisnevich descent or the localization sequence. In this talk, the basic notions of perfect complexes and pseudo-coherent complexes will be investigated and characterized as compact objects. We embrace as many details as possible.

6. Algebraic $$K$$-theory following Thomason-Trobaugh II : Waldhausen's $$S$$-construction (6/14)

Abstract: Quillen's $$K$$-theory is defined for categories in which a suitable notion of exactness can be spoken. In much more generality, Waldhausen introduced the $$S$$-construction for categories equipped with suitable notions of cofibrations and weak equivalences. We will discuss the definition of $$S$$-construction, its dependence on the derived category, the cofinality theorem, and its comparison to Quillen's $$K$$-theory.

7. Algebraic $$K$$-theory following Thomason-Trobaugh III : Algebraic $$K$$-theory of schemes (6/28)

Abstract: In previous talks, we investigated perfect complexes on a scheme and Waldhausen categories for which $$K$$-theory spectrum is defined. We'll define algebraic $$K$$-theory of quasi-compact quasi-separated schemes with these notions. Basic properties of algebraic $$K$$-theory will be discussed; properties include functoriality, excision property, relation with inverse limits and Poincaré duality. For this, we need various models of algebraic $$K$$-theory which give homotopy equivalent $$K$$-theory spectra most of which are proved by dependency of the derived categories of $$K$$-theory spectra.

8. Algebraic $$K$$-theory following Thomason-Trobaugh IV : The proto-localization theorem (7/26)

Let $$X$$ be a scheme and $$U$$ be its open subscheme. If $$X$$ is noetherian, then any coherent sheaf on $$U$$ always extends to $$X$$. By contrast, extension problem of algebraic vector bundles is far from being true in this naive sense; there is a counterexample even for $$(\mathbf A^3,\mathbf A^3\smallsetminus0)$$. Nevertheless, if $$X$$ is regular, then the Poincaré duality for $$K$$-theory shows that a coherent sheaf on $$X$$ extending a given algebraic vector bundle on $$U$$ is resolved by a bounded complex of algebraic vector bundles. Together with Waldhausen's resolution theorem stating that $$K$$-theory essentially depends on derived categories, this suggests that the right objects we should consider for this problem are perfect complexes. We will prove that the failure of extension of perfect complexes on $$U$$ to $$X$$ in the derived category is captured by the cokernel of $$K_0(X)\to K_0(U)$$, which is proved by Thomason-Trobaugh. As an analogue to Quillen's localization theorem for $$G$$-theory of noetherian schemes, it then directly gives the proto-localization theorem for $$K$$-theory of quasi-compact quasi-separated schemes except that the proto-localization theorem doesn't have surjectivity of $$K_0(X)\to K_0(U)$$. If possible, we will measure to what degree this map is surjective by introducing the non-connective Bass $$K$$-theory spectrum.

9. Algebraic $$K$$-theory following Thomason-Trobaugh V : Non-connective algebraic K-theory spectrum (9/17)

Algebraic $$K$$-theory of schemes has a sequence of $$K$$-groups for a closed immersion which is exact except only one place, namely, $$K_0$$ of the open immersion of the complement. Following the idea of Hyman Bass, we define a non-connective spectrum of a scheme whose non-negative part coincides with the usual algebraic $$K$$-theory defined by perfect complexes. This non-connective $$K$$-theory spectrum in particular gives a long exact sequence for a closed immersion. Our aim is to construct it, which is done by descending induction and requires the projective bundle formula that is also important regardless of our situation.

10. Algebraic $$K$$-theory following Thomason-Trobaugh VI : Descent theorems (10/8)

Recall that there is a spectral sequence strongly converging to the connective $$K$$-groups whose second page is given by the Zariski cohomology of connective $$K$$-theory sheaf. In the proof of this result by Quillen, the localization theorem is the most important ingredient. We prove an analogous statement for non-connective $$K$$-theory with both Zariski and Nisnevich cohomology for noetherian schemes of finite Krull dimension. This theorem is usually phrased as “non-connective algebraic $$K$$-theory satisfies Zariski and Nisnevich descent”. It is known that non-connective algebraic $$K$$-theory does not satisfy étale descent.

Reference:

1. Yu. I. Manin, Lectures on the K-functor in Algebraic Geometry, Uspekhi Mat. Nauk 24:5 (1969), 3-86 (Russian); English transl. in Russian Math. Surveys 24:5 (1969), 1-89.
2. D. Quillen, Higher algebraic K-theory: I, Lecture Notes in Math., vol. 341, Springer-Verlag, 1973, pp. 85–147.
3. R. W. Thomason and T. Trobaugh, Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift III, Progress in Math., vol. 88, Birkhäuser, 1990, pp. 247–435.
4. P. Berthelot, A. Grothendieck and L. Illusie, Théorie des intersections et théorème de Riemann-Roch, Lecture Notes in Math., vol. 225, Springer, 1971.
5. R. W. Thomason, Algebraic $$K$$-theory and etale cohomology, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 437–552.

Last update: Nov. 24th, 2021