Lectures on the Applications of Sheaves to Ring Theory

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Quillen therefore generalized Lichtenbaum's conjecture, predicting the existence of a spectral sequence like the Atiyah—Hirzebruch spectral sequence in topological K -theory. In the case studied by Lichtenbaum, the spectral sequence would degenerate, yielding Lichtenbaum's conjecture. William G. Throughout the s and early s, K -theory on singular varieties still lacked adequate foundations. While it was believed that Quillen's K -theory gave the correct groups, it was not known that these groups had all of the envisaged properties.

For this, algebraic K -theory had to be reformulated. This was done by Thomason in a lengthy monograph which he co-credited to his dead friend Thomas Trobaugh, who he said gave him a key idea in a dream. There, K 0 was described in terms of complexes of sheaves on algebraic varieties. Thomason discovered that if one worked with in derived category of sheaves, there was a simple description of when a complex of sheaves could be extended from an open subset of a variety to the whole variety. By applying Waldhausen's construction of K -theory to derived categories, Thomason was able to prove that algebraic K -theory had all the expected properties of a cohomology theory.

In , Keith Dennis discovered an entirely novel technique for computing K -theory based on Hochschild homology.


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While the Dennis trace map seemed to be successful for calculations of K -theory with finite coefficients, it was less successful for rational calculations. Goodwillie, motivated by his "calculus of functors", conjectured the existence of a theory intermediate to K -theory and Hochschild homology. He called this theory topological Hochschild homology because its ground ring should be the sphere spectrum considered as a ring whose operations are defined only up to homotopy. In the mids, Bokstedt gave a definition of topological Hochschild homology that satisfied nearly all of Goodwillie's conjectural properties, and this made possible further computations of K -groups.


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This transformation factored through the fixed points of a circle action on THH , which suggested a relationship with cyclic homology. In the course of proving an algebraic K -theory analog of the Novikov conjecture , Bokstedt, Hsiang, and Madsen introduced topological cyclic homology, which bore the same relationship to topological Hochschild homology as cyclic homology did to Hochschild homology. In , Dundas, Goodwillie, and McCarthy proved that topological cyclic homology has in a precise sense the same local structure as algebraic K -theory, so that if a calculation in K -theory or topological cyclic homology is possible, then many other "nearby" calculations follow.

The lower K-groups were discovered first, and given various ad hoc descriptions, which remain useful. Throughout, let A be a ring. The functor K 0 takes a ring A to the Grothendieck group of the set of isomorphism classes of its finitely generated projective modules , regarded as a monoid under direct sum.

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If the ring A is commutative, we can define a subgroup of K 0 A as the set. If B is a ring without an identity element , we can extend the definition of K 0 as follows. An algebro-geometric variant of this construction is applied to the category of algebraic varieties ; it associates with a given algebraic variety X the Grothendieck's K-group of the category of locally free sheaves or coherent sheaves on X.

Given a compact topological space X , the topological K-theory K top X of real vector bundles over X coincides with K 0 of the ring of continuous real-valued functions on X. The relative K-group is defined in terms of the "double" [50]. The independence from A is an analogue of the Excision theorem in homology. If A is a commutative ring, then the tensor product of projective modules is again projective, and so tensor product induces a multiplication turning K 0 into a commutative ring with the class [ A ] as identity.

Hyman Bass provided this definition, which generalizes the group of units of a ring: K 1 A is the abelianization of the infinite general linear group :. Define an elementary matrix to be one which is the sum of an identity matrix and a single off-diagonal element this is a subset of the elementary matrices used in linear algebra. The relative K-group is defined in terms of the "double" [53]. There is a natural exact sequence [54]. When A is a Euclidean domain e.

When this fails, one can ask whether K 1 is generated by the image of GL 2. For Dedekind domains with all quotients by maximal ideals finite, SK 1 is a torsion group.

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Wang's theorem states that if A has prime degree then SK 1 A is trivial, [59] and this may be extended to square-free degree. It can also be defined as the kernel of the map. For a field, K 2 is determined by Steinberg symbols : this leads to Matsumoto's theorem. One can compute that K 2 is zero for any finite field.

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For non-Archimedean local fields, the group K 2 F is the direct sum of a finite cyclic group of order m , say, and a divisible group K 2 F m. Matsumoto's theorem states that for a field k , the second K -group is given by [71] [72]. Matsumoto's original theorem is even more general: For any root system , it gives a presentation for the unstable K-theory.

This presentation is different from the one given here only for symplectic root systems. For non-symplectic root systems, the unstable second K-group with respect to the root system is exactly the stable K-group for GL A.

Unstable second K-groups in this context are defined by taking the kernel of the universal central extension of the Chevalley group of universal type for a given root system. If A is a Dedekind domain with field of fractions F then there is a long exact sequence. There is also an extension of the exact sequence for relative K 1 and K 0 : [74]. There is a pairing on K 1 with values in K 2. The above expression for K 2 of a field k led Milnor to the following definition of "higher" K -groups by. For integer m invertible in k there is a map.

This extends to. The accepted definitions of higher K -groups were given by Quillen , after a few years during which several incompatible definitions were suggested. Quillen gave two constructions, the "plus-construction" and the " Q -construction", the latter subsequently modified in different ways. Moreover, the definition is more direct in the sense that the K -groups, defined via the Q-construction are functorial by definition.

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This fact is not automatic in the plus-construction. This definition coincides with the above definition of K 0 P. More generally, for a scheme X , the higher K -groups of X are defined to be the K -groups of the exact category of locally free coherent sheaves on X.

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The resulting K -groups are usually written G n R. When R is a noetherian regular ring , then G - and K -theory coincide. Indeed, the global dimension of regular rings is finite, i. This isomorphism extends to the higher K -groups, too. A third construction of K -theory groups is the S-construction, due to Waldhausen. This is a more general concept than exact categories.

While the Quillen algebraic K -theory has provided deep insight into various aspects of algebraic geometry and topology, the K -groups have proved particularly difficult to compute except in a few isolated but interesting cases. The first and one of the most important calculations of the higher algebraic K -groups of a ring were made by Quillen himself for the case of finite fields :. Quillen proved that if A is the ring of algebraic integers in an algebraic number field F a finite extension of the rationals , then the algebraic K-groups of A are finitely generated.

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For example, for the integers Z , Borel proved that modulo torsion. In the course of proving an algebraic K -theory analog of the Novikov conjecture , Bokstedt, Hsiang, and Madsen introduced topological cyclic homology, which bore the same relationship to topological Hochschild homology as cyclic homology did to Hochschild homology. In , Dundas, Goodwillie, and McCarthy proved that topological cyclic homology has in a precise sense the same local structure as algebraic K -theory, so that if a calculation in K -theory or topological cyclic homology is possible, then many other "nearby" calculations follow.

The lower K-groups were discovered first, and given various ad hoc descriptions, which remain useful. Throughout, let A be a ring. The functor K 0 takes a ring A to the Grothendieck group of the set of isomorphism classes of its finitely generated projective modules , regarded as a monoid under direct sum. If the ring A is commutative, we can define a subgroup of K 0 A as the set. If B is a ring without an identity element , we can extend the definition of K 0 as follows. An algebro-geometric variant of this construction is applied to the category of algebraic varieties ; it associates with a given algebraic variety X the Grothendieck's K-group of the category of locally free sheaves or coherent sheaves on X.

Given a compact topological space X , the topological K-theory K top X of real vector bundles over X coincides with K 0 of the ring of continuous real-valued functions on X. The relative K-group is defined in terms of the "double" [50]. The independence from A is an analogue of the Excision theorem in homology. If A is a commutative ring, then the tensor product of projective modules is again projective, and so tensor product induces a multiplication turning K 0 into a commutative ring with the class [ A ] as identity.

Hyman Bass provided this definition, which generalizes the group of units of a ring: K 1 A is the abelianization of the infinite general linear group :. Define an elementary matrix to be one which is the sum of an identity matrix and a single off-diagonal element this is a subset of the elementary matrices used in linear algebra. The relative K-group is defined in terms of the "double" [53].

There is a natural exact sequence [54]. When A is a Euclidean domain e. When this fails, one can ask whether K 1 is generated by the image of GL 2. For Dedekind domains with all quotients by maximal ideals finite, SK 1 is a torsion group. Wang's theorem states that if A has prime degree then SK 1 A is trivial, [59] and this may be extended to square-free degree. It can also be defined as the kernel of the map.

For a field, K 2 is determined by Steinberg symbols : this leads to Matsumoto's theorem. One can compute that K 2 is zero for any finite field. For non-Archimedean local fields, the group K 2 F is the direct sum of a finite cyclic group of order m , say, and a divisible group K 2 F m. Matsumoto's theorem states that for a field k , the second K -group is given by [71] [72]. Matsumoto's original theorem is even more general: For any root system , it gives a presentation for the unstable K-theory.

This presentation is different from the one given here only for symplectic root systems. For non-symplectic root systems, the unstable second K-group with respect to the root system is exactly the stable K-group for GL A. Unstable second K-groups in this context are defined by taking the kernel of the universal central extension of the Chevalley group of universal type for a given root system. If A is a Dedekind domain with field of fractions F then there is a long exact sequence.

There is also an extension of the exact sequence for relative K 1 and K 0 : [74]. There is a pairing on K 1 with values in K 2.