GRADED COEFFICIENTS OF EILENBERG-MACLANE SPECTRA IGOR SIKORA A. Let denote the cyclic group of order two. Using the Tate diagram we compute the ( )-graded coefficients of Eilenberg-MacLane -spectra and describe their structure as a module over the coefficients of the Eilenberg-MacLane spectrum of the Burnside Mackey functor. If the underlying Mackey functor is a Green functor, we also infer the multiplicative structure on the ( )-graded coefficients. C(3) singular spectrum Φ ; (4) Tate spectrum ; which are connected by the following commutative diagram:What makes computations by the Tate diagram feasible is that the rows are cofibre sequences and the right-hand square (known as the Tate square) is a homotopy pullback. Moreover, the coefficients of the spectra appearing in the bottom row may be computed by the homotopy orbits and homotopy fixed points spectral sequences. The foundational work on the Tate diagram is [9], where all of the details are discussed.The computational strength of the Tate diagram has been proven in various contexts -for example Greenlees use it in [8] to compute the coefficients of the Eilenberg-MacLane -spectrum Z as an ( )-graded ring, Greenlees and Meier compute the coefficients of K-theory with reality R in [10] and Hu and Kriz use it to compute the -equivariant Steenrod algebra in [13]. It was also used to compute the coefficients of Z over groups 2 with prime by Zeng in [18].( )-graded abelian group structure. The first step is describing the ( )-graded abelian group structure of ★ . We show that it is fully determined by the underlying Mackey functor . This structure is given in Theorem 6.1, which can be informally stated as follows:Theorem. The ( )-graded abelian group structure of ★ may be presented by Figure 1, where: (1) every lattice point represents a -representation, the horizontal axis describes multiplicity of the trivial -representation and the vertical axis describes multiplicity of the sign representation.(2) The empty circle at the position (0, 0) is the module ( / ).(3) The values of ★ lying on the = 0 axis are submodules of ( / ) given by the kernel of the restriction and the cokernel of the transfer. (4) The full dots in positions (−1, −1) and (1, 1) are submodules of ( / ) given by the kernel of the transfer and the cokernel of the restriction, whereas the values lying on the red/blue lines above/below them are their subquotients. (5) The values lying in blue and red areas are respectively the group cohomology and homology with coefficients in ( / ). ( 6) All other values are zero.A ★ -module structure. The category of -Mackey functors has a symmetric monoidal structure and commutative monoids with respect to this structure are called Green functors. If is a Green functor, the -spectrum is a (naive) commutative ring -spectrum and its homotopy groups form an ( )graded commutative ring. The most fundamental example of a Green functor is the Burnside Mackey functor A.Every -Mackey functor is a module over A. Therefore is a module over A and ★ is a module over the ( )-g...

A linear constraint system is specified by linear equations over the group Zd of integers modulo d. Their operator solutions play an important role in the study of quantum contextuality and non-local games. In this paper, we use the theory of simplicial sets to develop a framework for studying operator solutions of linear systems. Our approach refines the well-known group-theoretical approach based on solution groups by identifying these groups as algebraic invariants closely related to the fundamental group of a space. In this respect, our approach also makes a connection to the earlier homotopical approach based on cell complexes. Within our framework, we introduce a new class of linear systems that come from simplicial sets and show that any linear system can be reduced to one of that form. Then we specialize in linear systems that are associated with groups. We provide significant evidence for a conjecture stating that for odd d every linear system admitting a solution in a group admits a solution in Zd.