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On the canonical map to the local cohomology of a Stanley-Reisner ring. (English) Zbl 0741.55013
This paper contains two parts. In the first one, the author gives an example of a Stanley-Reisner ring \(k[\Delta]\) whose defining ideal is generated by monomials of degree at most 2 and such that the maps \[ \to F_{i+1}\to F_ i\to F_{i-1}\to\cdots\to F_ 0\to k[\Delta]\to 0, \] in a minimal free resolution of \(k[\Delta]\) as a module over a polynomial algebra \(A\), are not represented by matrices with all components of degree at most 2. (A counterexample to a question of Watanabe).
In the second part the author proves that the canonical map \(\hbox{Ext}^ i_ A(k,k[\Delta])\to H^ i_ m(k[\Delta])\) corresponds to the map induced by the inclusion map of certain subcomplexes of \(\Delta\).

55U10 Simplicial sets and complexes in algebraic topology
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
14B15 Local cohomology and algebraic geometry