## Algebra & Number Theory

### Adams operations on matrix factorizations

#### Abstract

We define Adams operations on matrix factorizations, and we show these operations enjoy analogues of several key properties of the Adams operations on perfect complexes with support developed by Gillet and Soulé. As an application, we give a proof of a conjecture of Dao and Kurano concerning the vanishing of Hochster’s $θ$ pairing.

#### Article information

Source
Algebra Number Theory, Volume 11, Number 9 (2017), 2165-2192.

Dates
Accepted: 9 August 2017
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513730350

Digital Object Identifier
doi:10.2140/ant.2017.11.2165

Mathematical Reviews number (MathSciNet)
MR3735465

Zentralblatt MATH identifier
06818948

#### Citation

Brown, Michael; Miller, Claudia; Thompson, Peder; Walker, Mark. Adams operations on matrix factorizations. Algebra Number Theory 11 (2017), no. 9, 2165--2192. doi:10.2140/ant.2017.11.2165. https://projecteuclid.org/euclid.ant/1513730350

#### References

• M. F. Atiyah, “Power operations in $K$-theory”, Quart. J. Math. Oxford Ser. $(2)$ 17 (1966), 165–193.
• D. J. Benson, “Lambda and psi operations on Green rings”, J. Algebra 87:2 (1984), 360–367.
• M. K. Brown, C. Miller, P. Thompson, and M. E. Walker, “Cyclic Adams operations”, J. Pure Appl. Algebra 221:7 (2017), 1589–1613.
• R.-O. Buchweitz, “Maximal Cohen–Macaulay modules and Tate-cohomology over Gorenstein rings”, preprint, Univ. Hannover, 1986, http://tinyurl.com/ragnarolaf.
• R.-O. Buchweitz and D. Van Straten, “An index theorem for modules on a hypersurface singularity”, Mosc. Math. J. 12:2 (2012), 237–259.
• H. Dao, “Decent intersection and Tor-rigidity for modules over local hypersurfaces”, Trans. Amer. Math. Soc. 365:6 (2013), 2803–2821.
• H. Dao and K. Kurano, “Hochster's theta pairing and numerical equivalence”, J. K-Theory 14:3 (2014), 495–525.
• D. Eisenbud, “Homological algebra on a complete intersection, with an application to group representations”, Trans. Amer. Math. Soc. 260:1 (1980), 35–64.
• H. Gillet and C. Soulé, “Intersection theory using Adams operations”, Invent. Math. 90:2 (1987), 243–277.
• O. Haution, Steenrod operations and quadratic forms, Ph.D. thesis, Université Paris VI, 2009, http://www.mathematik.uni-muenchen.de/~haution/these.pdf.
• M. Hochster, “The dimension of an intersection in an ambient hypersurface”, pp. 93–106 in Algebraic geometry (Chicago, 1980), edited by A. Libgober and P. Wagreich, Lecture Notes in Math. 862, Springer, Berlin, 1981.
• B. Köck, “Adams operations for projective modules over group rings”, Math. Proc. Cambridge Philos. Soc. 122:1 (1997), 55–71.
• W. F. Moore, G. Piepmeyer, S. Spiroff, and M. E. Walker, “Hochster's theta invariant and the Hodge–Riemann bilinear relations”, Adv. Math. 226:2 (2011), 1692–1714.
• D. O. Orlov, “Triangulated categories of singularities and D-branes in Landau–Ginzburg models”, Tr. Mat. Inst. Steklova 246 (2004), 240–262. In Russian; translated in Proc. Steklov Inst. Math. 246 (2004), 227–248.
• A. Polishchuk and A. Vaintrob, “Chern characters and Hirzebruch–Riemann–Roch formula for matrix factorizations”, Duke Math. J. 161:10 (2012), 1863–1926.
• J.-P. Serre, Local algebra, Springer, Berlin, 2000.
• M. E. Walker, “On the vanishing of Hochster's $\theta$ invariant”, Ann. K-Theory 2:2 (2017), 131–174.