ICTS

I will discuss various aspects of the derived category, and associated triangulated categories of interest in commutative algebra. 

Resources:

Some familiarity with commutative algebra, and triangulated categories. 

I will discuss classifications of thick and localising subcategories for categories of modular representations of finite groups. This requires a certain amount of machinery, for instance a theory of support for triangulated categories. In addition, I plan to talk about various dualities (local and global) which reflect a Gorenstein property of the category of finite group representations.

Resources:

D. J. Benson, S. B. Iyengar, and H. Krause, Representations of finite groups: Local cohomology and support, Oberwolfach Seminars 43, Birkhäuser Verlag, 2012, 111 pp.

H. Krause, Homological Theory of Representations, Cambridge Studies in Advanced Mathematics 195, Cambridge Univers. Press, 2021, 482 pp.

In this preparatory talk, we will discuss basics of triangulated and derived categories. We will show that the homotopy category and the derived category of an abelian category are triangulated.

Topic 1: The local cohomology theorem for group cohomology H^*(BG) by algebraic methods 
[This account follows my 1995 JPAA paper ``Commutative algebra in group cohomology'' and my 2000 JPAA paper with G.Lyubeznik]

Introduce duality (Poincare duality, and what you might do for more general graded rings). Introduce the example of the group cohomology ring.  In fact (Benson-Carlson) there is a functional equation for the Hilbert series (in the Cohen Macaulay case) and this motivates the duality statement.  Reminder of local cohomology and its basic properties. 

Forming multiperiodic resolutions in topology and algebra, directly in favourable cases or via support in general. The proof of the local cohomology theorem and the deduction of the functional equation. As a second example (from work with R.R.Bruner),  I’d like to mention the bigraded cohomology H^{**}(A) for a finite dimensional Hopf sub algebra of the Steenrod algebra, where the Adams-Margolis criterion is the way the support condition is realised. 

Topic 2: Gorenstein duality for C^*(BG)
[This gives a structural approach to the same result, following my 2006 Advances paper with Dwyer and Iyengar, with some input from my 2002 AJM paper with Dwyer]

 Differential graded algebras and spectra (description of formal properties rather than a fulll definition and construction).  The homotopical Gorenstein condition for a map R—>k. Morita theory (some of this will be covered in Iyengar’s talks). Matlis lifts. Gorenstein duality. Deducing Gorenstein duality from the Gorenstein condition in the orientable case. Gorenstein ascent. C^*(BG)—>k is Gorenstein. Deducing the local cohomology theorem. The ubiquity of other examples and perhaps the connection to Serre duality in derived algebraic geometry. 

The following are *not* prerequisites, but the material in the lectures is based on these papers.

A general reference:
J.P.C.Greenlees
``Homotopy invariant commutative algebra over fields'' Building Bridges Between Algebra and Topology, CRM IRTATCA Lectures, Birkh\"auser 2018, 103-169, arXiv:1601.024737

Other background:
J.P.C.Greenlees and J.P. May ``Derived functors of I-adic completion and local homology'' Journal of Algebra {\bf 149} (1992) 438-453. W.G.Dwyer and J.P.C.Greenlees ``Complete modules and torsion modules.''American J. Math. {\bf 124} (2002) 199-220 W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar ``Duality in algebra and topology.'' Advances in Maths. {\bf 200} (2006) 357-402, arXiv:math/0510247 J.P.C.Greenlees ``Ausoni-B\"okstedt duality'' Journal of Pure and Applied Algebra 220 (2016), pp. 1382-1402, arXiv:1406.2162 J.P.C.Greenlees and G. Stevenson ``Morita equivalences and singularity categories.'' Advances in Mathematics {\bf 365} (2020), 44pp, arXiv: 1702.07957

Topic 1: The local cohomology theorem for group cohomology H^*(BG) by algebraic methods 
[This account follows my 1995 JPAA paper ``Commutative algebra in group cohomology'' and my 2000 JPAA paper with G.Lyubeznik]

Introduce duality (Poincare duality, and what you might do for more general graded rings). Introduce the example of the group cohomology ring.  In fact (Benson-Carlson) there is a functional equation for the Hilbert series (in the Cohen Macaulay case) and this motivates the duality statement.  Reminder of local cohomology and its basic properties. 

Forming multiperiodic resolutions in topology and algebra, directly in favourable cases or via support in general. The proof of the local cohomology theorem and the deduction of the functional equation. As a second example (from work with R.R.Bruner),  I’d like to mention the bigraded cohomology H^{**}(A) for a finite dimensional Hopf sub algebra of the Steenrod algebra, where the Adams-Margolis criterion is the way the support condition is realised. 

Topic 2: Gorenstein duality for C^*(BG)
[This gives a structural approach to the same result, following my 2006 Advances paper with Dwyer and Iyengar, with some input from my 2002 AJM paper with Dwyer]

 Differential graded algebras and spectra (description of formal properties rather than a fulll definition and construction).  The homotopical Gorenstein condition for a map R—>k. Morita theory (some of this will be covered in Iyengar’s talks). Matlis lifts. Gorenstein duality. Deducing Gorenstein duality from the Gorenstein condition in the orientable case. Gorenstein ascent. C^*(BG)—>k is Gorenstein. Deducing the local cohomology theorem. The ubiquity of other examples and perhaps the connection to Serre duality in derived algebraic geometry. 

The following are *not* prerequisites, but the material in the lectures is based on these papers.

A general reference:
J.P.C.Greenlees
``Homotopy invariant commutative algebra over fields'' Building Bridges Between Algebra and Topology, CRM IRTATCA Lectures, Birkh\"auser 2018, 103-169, arXiv:1601.024737

Other background:
J.P.C.Greenlees and J.P. May ``Derived functors of I-adic completion and local homology'' Journal of Algebra {\bf 149} (1992) 438-453. W.G.Dwyer and J.P.C.Greenlees ``Complete modules and torsion modules.''American J. Math. {\bf 124} (2002) 199-220 W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar ``Duality in algebra and topology.'' Advances in Maths. {\bf 200} (2006) 357-402, arXiv:math/0510247 J.P.C.Greenlees ``Ausoni-B\"okstedt duality'' Journal of Pure and Applied Algebra 220 (2016), pp. 1382-1402, arXiv:1406.2162 J.P.C.Greenlees and G. Stevenson ``Morita equivalences and singularity categories.'' Advances in Mathematics {\bf 365} (2020), 44pp, arXiv: 1702.07957

To begin the talk, we present some stable phenomena that serve as motivation for defining spectra, which subsequently leads to the stable homotopy category. The talk proceeds to explore the various properties and axioms satisfied by the stable homotopy category. We also examine potential candidates for this category, weighing their advantages and disadvantages. Additionally, we discuss modern categories of spectra, which are equivalent to the stable homotopy category. We conclude the talk with a discussion on the Spanier-Whitehead duality for spectra.
 

"Ambidexterity" refers to situations where we have two functors $F$ and $G$ such that $F$ is both left adjoint and right adjoint to $G$. (Some other talks in the workshop will probably involve cases where a left adjoint is some kind of twist of a right adjoint, but we will be considering cases where there is no twisting.) There are some elementary examples in the classical complex representation theory of finite groups and groupoids, and we will start by reviewing these, because that will make it easier to understand other examples by analogy. We will then discuss ambidexterity in the context of the Morava $K$-theory and $E$-theory of classifying spaces of finite groupoids, which will involve some general theory of formal groups and formal schemes, as well as the generalised character theory of Hopkins, Kuhn and Ravenel. Finally we will generalise from finite groupoids to finite $\infty$-groupoids, or equivalently $\pi$-finite spaces. This will involve some discussion of $\infty$-categories in the sense of Lurie, and the calculation of Morava $E$-theory of Eilenberg-MacLane spaces. If time permits, we will also discuss some interesting applications in chromatic homotopy theory.

K(n)-local duality for finite groups and groupoids (Strickland, https://arxiv.org/abs/math/0011109)

Ambidexterity in K(n)-Local Stable Homotopy Theory (Hopkins and Lurie, https://people.math.harvard.edu/~lurie/papers/Ambidexterity.pdf)

Ambidexterity in Chromatic Homotopy Theory (Carmeli, Schlank and Yanovski, https://arxiv.org/abs/1811.02057)

Topic 1: The local cohomology theorem for group cohomology H^*(BG) by algebraic methods 
[This account follows my 1995 JPAA paper ``Commutative algebra in group cohomology'' and my 2000 JPAA paper with G.Lyubeznik]

Introduce duality (Poincare duality, and what you might do for more general graded rings). Introduce the example of the group cohomology ring.  In fact (Benson-Carlson) there is a functional equation for the Hilbert series (in the Cohen Macaulay case) and this motivates the duality statement.  Reminder of local cohomology and its basic properties. 

Forming multiperiodic resolutions in topology and algebra, directly in favourable cases or via support in general. The proof of the local cohomology theorem and the deduction of the functional equation. As a second example (from work with R.R.Bruner),  I’d like to mention the bigraded cohomology H^{**}(A) for a finite dimensional Hopf sub algebra of the Steenrod algebra, where the Adams-Margolis criterion is the way the support condition is realised. 

Topic 2: Gorenstein duality for C^*(BG)
[This gives a structural approach to the same result, following my 2006 Advances paper with Dwyer and Iyengar, with some input from my 2002 AJM paper with Dwyer]

 Differential graded algebras and spectra (description of formal properties rather than a fulll definition and construction).  The homotopical Gorenstein condition for a map R—>k. Morita theory (some of this will be covered in Iyengar’s talks). Matlis lifts. Gorenstein duality. Deducing Gorenstein duality from the Gorenstein condition in the orientable case. Gorenstein ascent. C^*(BG)—>k is Gorenstein. Deducing the local cohomology theorem. The ubiquity of other examples and perhaps the connection to Serre duality in derived algebraic geometry. 

The following are *not* prerequisites, but the material in the lectures is based on these papers.

A general reference:
J.P.C.Greenlees
``Homotopy invariant commutative algebra over fields'' Building Bridges Between Algebra and Topology, CRM IRTATCA Lectures, Birkh\"auser 2018, 103-169, arXiv:1601.024737

Other background:
J.P.C.Greenlees and J.P. May ``Derived functors of I-adic completion and local homology'' Journal of Algebra {\bf 149} (1992) 438-453. W.G.Dwyer and J.P.C.Greenlees ``Complete modules and torsion modules.''American J. Math. {\bf 124} (2002) 199-220 W.G.Dwyer, J.P.C.Greenlees and S.B.Iyengar ``Duality in algebra and topology.'' Advances in Maths. {\bf 200} (2006) 357-402, arXiv:math/0510247 J.P.C.Greenlees ``Ausoni-B\"okstedt duality'' Journal of Pure and Applied Algebra 220 (2016), pp. 1382-1402, arXiv:1406.2162 J.P.C.Greenlees and G. Stevenson ``Morita equivalences and singularity categories.'' Advances in Mathematics {\bf 365} (2020), 44pp, arXiv: 1702.07957

In this talk I will present the idea of Landweber exactness and introduce Lubin-Tate's Universal Deformation and consequently show that Lubin-Tate spectra is Landweber exact.

Quillen's stratification theorem describes the Zariski spectrum of prime ideals of the mod p cohomology of a finite group in terms of the abelian subgroups of the group. In these talks I will explain the key points in the proof, generealizations to other cohomoloogy theories by other authors and finally a recent work (with Barthel, Heard, Naumann and Pol) with shows how Quillen stratification works for the Balmer spectrum in equivariant homotopy theory. If time permits I will analyze the dual situation of costratification.

I. Group cohomology
II. Quillen stratification theorem in mod p cohomology
III. Quillen stratification theorem in equivariant homotopy theory (1)
IV. Quillen stratification theorem in equivariant homotopy theory (2) and costratification (if time permits)

D. Quillen. The spectrum of an equivariant cohomology ring I, II, Ann. of Math 94, (1971)
A. Mathew, N. Naumann, J. Noel. Derived induction and restriction theory, Geom. Topol 23, 2019.
T. Barthel, N. Castellana, D. Heard, N. Naumann, L. Pol, Quillen stratification in equivariant homotopy theory, ArXiv:2301.02212

I will discuss classifications of thick and localising subcategories for categories of modular representations of finite groups. This requires a certain amount of machinery, for instance a theory of support for triangulated categories. In addition, I plan to talk about various dualities (local and global) which reflect a Gorenstein property of the category of finite group representations.

Resources:

D. J. Benson, S. B. Iyengar, and H. Krause, Representations of finite groups: Local cohomology and support, Oberwolfach Seminars 43, Birkhäuser Verlag, 2012, 111 pp.

H. Krause, Homological Theory of Representations, Cambridge Studies in Advanced Mathematics 195, Cambridge Univers. Press, 2021, 482 pp.

"Ambidexterity" refers to situations where we have two functors $F$ and $G$ such that $F$ is both left adjoint and right adjoint to $G$. (Some other talks in the workshop will probably involve cases where a left adjoint is some kind of twist of a right adjoint, but we will be considering cases where there is no twisting.) There are some elementary examples in the classical complex representation theory of finite groups and groupoids, and we will start by reviewing these, because that will make it easier to understand other examples by analogy. We will then discuss ambidexterity in the context of the Morava $K$-theory and $E$-theory of classifying spaces of finite groupoids, which will involve some general theory of formal groups and formal schemes, as well as the generalised character theory of Hopkins, Kuhn and Ravenel. Finally we will generalise from finite groupoids to finite $\infty$-groupoids, or equivalently $\pi$-finite spaces. This will involve some discussion of $\infty$-categories in the sense of Lurie, and the calculation of Morava $E$-theory of Eilenberg-MacLane spaces. If time permits, we will also discuss some interesting applications in chromatic homotopy theory.

K(n)-local duality for finite groups and groupoids (Strickland, https://arxiv.org/abs/math/0011109)

Ambidexterity in K(n)-Local Stable Homotopy Theory (Hopkins and Lurie, https://people.math.harvard.edu/~lurie/papers/Ambidexterity.pdf)

Ambidexterity in Chromatic Homotopy Theory (Carmeli, Schlank and Yanovski, https://arxiv.org/abs/1811.02057)

"Ambidexterity" refers to situations where we have two functors $F$ and $G$ such that $F$ is both left adjoint and right adjoint to $G$. (Some other talks in the workshop will probably involve cases where a left adjoint is some kind of twist of a right adjoint, but we will be considering cases where there is no twisting.) There are some elementary examples in the classical complex representation theory of finite groups and groupoids, and we will start by reviewing these, because that will make it easier to understand other examples by analogy. We will then discuss ambidexterity in the context of the Morava $K$-theory and $E$-theory of classifying spaces of finite groupoids, which will involve some general theory of formal groups and formal schemes, as well as the generalised character theory of Hopkins, Kuhn and Ravenel. Finally we will generalise from finite groupoids to finite $\infty$-groupoids, or equivalently $\pi$-finite spaces. This will involve some discussion of $\infty$-categories in the sense of Lurie, and the calculation of Morava $E$-theory of Eilenberg-MacLane spaces. If time permits, we will also discuss some interesting applications in chromatic homotopy theory.

K(n)-local duality for finite groups and groupoids (Strickland, https://arxiv.org/abs/math/0011109)

Ambidexterity in K(n)-Local Stable Homotopy Theory (Hopkins and Lurie, https://people.math.harvard.edu/~lurie/papers/Ambidexterity.pdf)

Ambidexterity in Chromatic Homotopy Theory (Carmeli, Schlank and Yanovski, https://arxiv.org/abs/1811.02057)