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Gysin maps in Balmer-Witt theory. (English) Zbl 1140.11024
The Balmer-Witt theory is a (periodic with period \(4\)) sequence of groups \( W^{i}(C)\) associated to a triangular category \(C\) with a duality. The most important case is the derived category \(C=K(X)\) of coherent sheaves of \( \mathcal{O}_{X}\)-modules on a regular scheme \(X\). Duality is usually extension of the functor \(\mathcal{H}om_{\mathcal{O}_{x}}(-,L)\) defined on locally free sheaves, where \(L\) is a line bundle. Among the arising Witt groups \( W^{i}(X,L)=W^{i}(K(X))\) there is \(W^{0}(X,\mathcal{O}_{X})\), which coincides with the usual Witt group \(W(X)\) of a scheme \(X\). For an open \( j:U\rightarrow X\) with closed complement \(u:Z\rightarrow X\) the machinery of derived categories produces Witt groups with supports \( W_{Z}^{i}(X,L)\) so there is a long exact localization sequence
\[ \cdots \rightarrow W_{Z}^{i}(X,L)\rightarrow W^{i}(X,L)\rightarrow W^{i}(U,L)\rightarrow W_{Z}^{i+1}(X,L)\rightarrow \cdots . \]
Using local algebra Stephan Gille proved that in many cases \( W_{Z}^{i+r}(X,L)\cong W^{i}(Z,u^{\ast }L)\).
In an earlier preprint {http://www.math.uiuc.edu/K-theory/0696} the author considered a partial case when \(p:E\rightarrow Z\) is a vector bundle of rank \(r\) and \(u:Z\rightarrow E\) is the zero section, producing the so-called twisted Thom isomorphism \(W_{Z}^{i+r}(E,p^{\ast }L)\cong W^{i}(Z,\det E\otimes L)\). This isomorphism is a multiplication by a Witt class of a geometric meaning. This enabled the author to compute derived Witt groups of projective spaces and split quadrics.
Now if \(N\) is the normal bundle of \(u:Z\rightarrow X\), there is a Thom isomorphism \[ W^{i}(Z,\det N\otimes u^{\ast }L)\rightarrow W_{Z}^{i+r}(N,p^{\ast }u^{\ast }L) \] and a newly defined deformation to normal cone homomorphism \(d(X,Z):W_{Z}^{i+r}(N,p^{\ast }u^{\ast }L)\rightarrow W_{Z}^{i+r}(X,L)\). The author calls a Gysin map \(W^{i}(Z,\det N\otimes u^{\ast }L)\rightarrow W^{i+r}(X,L)\) the composition of Thom isomorphism, deformation to normal cone homomorphism and the map \(W_{Z}^{i+r}(X,L) \rightarrow W^{i+r}(X,L)\) from localization exact sequence. An imposing number of good functorial properties of this Gysin map is proved. This may indicate that the notion of Gysin map is a right one.

11E81 Algebraic theory of quadratic forms; Witt groups and rings
18E30 Derived categories, triangulated categories (MSC2010)
19G12 Witt groups of rings
Full Text: DOI
[1] Arason, J., Der wittring projektiver Räume, Math. ann., 253, 205-212, (1980) · Zbl 0431.10011
[2] Balmer, P., Triangular Witt groups, part I: the 12-term localization exact sequence, \(K\)-theory, 19, 311-363, (2000) · Zbl 0953.18003
[3] Balmer, P., Triangular Witt groups, part II: from usual to derived, Math. Z., 236, 351-382, (2001) · Zbl 1004.18010
[4] Balmer, P., Witt cohomology, mayer – vietoris, homotopy invariance and the Gersten conjecture, \(K\)-theory, 23, 1, 15-30, (2001) · Zbl 0987.19002
[5] Balmer, P., Products of degenerate quadratic forms, Compos. math., 141, 1374-1404, (2005) · Zbl 1087.18008
[6] Balmer, P.; Gille, S., Koszul complexes and symmetric forms over the punctured affine space, Proc. London math. soc., 91, 3, 273-299, (2005) · Zbl 1078.11026
[7] Baum, P.; Fulton, W.; MacPherson, R., Riemann – roch and topological \(K\)-theory for singular varieties, Acta math., 143, 155-192, (1979) · Zbl 0474.14004
[8] N. Bourbaki, Algèbre homologique, AX, Hermann, Paris
[9] B. Calmès, J. Hornbostel, Witt motives, transfers and dévissage. Preprint at: http://www.math.uiuc.edu/K-theory/0786/, 2006
[10] F. Deglise, Thèse de doctorat
[11] Deglise, F., Interprétation motivique de la formule d’excès d’intersection, C. R. math. acad. sci. Paris, 338, 1, 41-46, (2004) · Zbl 1048.18004
[12] Fulton, W., Intersection theory, (1984), Springer-Verlag · Zbl 0541.14005
[13] Gille, S., A transfer morphism for Witt groups, J. reine angew. math., 564, 215-233, (2003) · Zbl 1050.11046
[14] Gille, S.; Nenashev, A., Pairings in triangular Witt theory, J. algebra, 261, 292-309, (2003) · Zbl 1016.18007
[15] A. Nenashev, On the Witt groups of projective bundles and split quadrics: Geometric reasoning. Preprint at: http://www.math.uiuc.edu/K-theory/0696/, 2004 · Zbl 1223.19005
[16] Nenashev, A., Gysin maps in oriented theories, J. algebra, 302, 200-213, (2006) · Zbl 1108.14016
[17] I. Panin (after I. Panin and A. Smirnov), Push-forwards in oriented cohomology theories of algebraic varieties: II. Preprint at: http://www.math.uiuc.edu/K-theory/0619/, 2003
[18] Panin, I., Oriented cohomology theories of algebraic varieties, \(K\)-theory, 30, 265-314, (2003), (after I. Panin and A. Smirnov) · Zbl 1047.19001
[19] A. Smirnov, Orienting structures in cohomology theories of algebraic varieties, 2004 (in Russian). (after I. Panin and A. Smirnov) Preprint in progress
[20] Voevodsky, V., Homology of schemes, Selecta math. (N.S.), 2, 1, 111-153, (1996) · Zbl 0871.14016
[21] C. Walter, Grothendieck-Witt groups of projective bundles, Preprint at: http://www.math.uiuc.edu/K-theory/0645/, 2003
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